with a focus on open problems - massachusetts institute of

14th Annual Fall Workshop

on Computational Geometry

with a Focus on Open Problems

November 19–20, 2004

MIT, Cambridge, Massachusetts, USA

Abstracts

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Topologically Sweeping the Complete Graph in Optimal Timeby Eynat Rafalin and Diane Souvaine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Dynamic Update of Half-space Depth Contoursby M. Burr, E. Rafalin, and D. L. Souvaine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Computing a Point of High Simplicial Depth in Rˆ2by Jason Burrowes-Jones and William Steiger . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Floodlight Illumination of Infinite Wedgesby Matthew Cary, Atri Rudra, Ashish Sabharwal, and Erik Vee . . . . . . . . . . . . . . . . . 7

Proximity Problems on Line Segments Spanned by Pointsby Ovidiu Daescu and Jun Luo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

INVITED TALK: Self-reconfiguring Robotsby Daniela Rus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Unfolding Smooth Prismatoidsby Nadia Benbernou, Patricia Cahn, and Joseph O’Rourke . . . . . . . . . . . . . . . . . . . 12

Folding Paper Shopping Bagsby Devin J. Balkcom, Erik D. Demaine, and Martin L. Demaine . . . . . . . . . . . . . . . . 14

Hinged Dissection of Polypolyhedraby Erik D. Demaine, Martin L. Demaine, Jeffrey F. Lindy, and Diane L. Souvaine . . . . . . 16

A 2-chain Can Interlock with a k-chainby Julie Glass, Stefan Langerman, Joseph O’Rourke, Jack Snoeyink, and Jianyuan K. Zhong 18

Grid Edge-Unfolding Orthostacks with Orthogonally Convex Slabsby Mirela Damian and Henk Meijer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

INVITED TALK: A Theorem of Tutte and 3D Mesh Parameterizationby Craig Gotsman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Drawing Equally-Spaced Curves between Two Pointsby Tetsuo Asano and Takeshi Tokuyama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A Distributed Architecture for the Visualization of Geometric Modelsby Sourav Dalal, Frank Devai, and Md Mizanur Rahman . . . . . . . . . . . . . . . . . . . . 26

Pointed Binary Encompassing Trees: Simple and Optimalby Michael Hoffmann and Csaba Toth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Hamiltonian Cycles in Sparse Vertex-Adjacency Dualsby Perouz Taslakian and Godfried Toussaint . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Trees on Tracksby Justin Cappos and Stephen Kobourov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

On the Non-Redundancy of Split Offsets in Degree Codingby Martin Isenburg and Jack Snoeyink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

ii

Fast almost-linear-sized nets for boxes in the planeby Herve Bronnimann and Jonathan Lenchner . . . . . . . . . . . . . . . . . . . . . . . . . . 36

The Metric TSP and the Sum of its Marginal Valuesby Moshe Dror, Yusin Lee, and James B. Orlin . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Optimal Area Triangulation with Angular Constraintsby J. Mark Keil and Tzvetalin S. Vassilev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Detecting Duplicates Among Similar Bit Vectors (of course, with geometric ap-plications)by Boris Aronov and John Iacono . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Approximation Algorithms for Two Optimal Location Problems in Sensor Net-worksby Alon Efrat, Sariel Har-Peled, and Joseph S. B. Mitchell . . . . . . . . . . . . . . . . . . . 43

Farthest Point from Line Segment Queriesby Ovidiu Daescu and Robert Serfling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

INVITED TALK: Computational Geometric Aspects of Musical Rhythmby Godfried Toussaint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Provably Better Moving Least Squaresby Ravikrishna Kolluri, James F. O’Brien, and Jonathan R. Shewchuk . . . . . . . . . . . . . 49

Contour Tree Simplification With Local Geometric Measuresby Hamish Carr, Jack Snoeyink, and Michiel van de Panne . . . . . . . . . . . . . . . . . . . 51

OrthoMap: Homeomorphism-guaranteeing normal-projection map between sur-facesby Frederic Chazal, Andre Lieutier, and Jarek Rossignac . . . . . . . . . . . . . . . . . . . . . 53

Parallel Guaranteed Quality Planar Delaunay Mesh Generation by ConcurrentPoint Insertionby Andrey N. Chernikov and Nikos P. Chrisochoides . . . . . . . . . . . . . . . . . . . . . . . 55

Tightening: Curvature-Limiting Morphological Simplificationby Jason Williams and Jarek Rossignac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

INVITED TALK: Compact Data Representations and their Applicationsby Moses Charikar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

NEARPT3 — Nearest Point Query in Eˆ3 with a Uniform Gridby W. Randolph Franklin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Algebraic Number Comparisons for Robust Geometric Operationsby John Keyser and Koji Ouchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

An Efficient k-center clustering Algorithm for Geometric Objectsby Guang Xu and Jinhui Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Analysis of Layered Hierarchy for Necklacesby Kathryn Bean and Sergey Bereg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

iii

Foreword

This booklet contains abstracts from talks presented at the 14th Annual Fall Workshop on Com-

putational Geometry, held on November 19–20, 2004 at MIT in Cambridge, Massachusetts, USA.In addition to 4 invited talks, we had a record number of 31 contributed talks selected among

a record number of 45 submissions.

Sponsors

• National Science Foundation

• Computer Science and Artificial Intelligence Laboratory

• Massachusetts Institute of Technology

Program and Organazing Committee

• Erik Demaine (chair, Massachusetts Institute of Technology)

• Martin Demaine (Massachusetts Institute of Technology)

• Piotr Indyk (Massachusetts Institute of Technology)

• Joseph S. B. Mitchell (Stony Brook University)

• Joseph O’Rourke (Smith College)

• Diane Souvaine (Tufts University)

• Ileana Streinu (Smith College)

History

This series of Fall Workshops on Computational Geometry was originally founded under the spon-sorship of the Mathematical Sciences Institute (MSI) at Stony Brook (with funding from the U. S.Army Research Office) and held there from 1991 through 1995. It continued during 1996–1999 un-der the sponsorship of the Center for Geometric Computing, a collaborative center of Brown, Duke,and Johns Hopkins Universities, also funded by the U.S. Army Research Office. The workshop re-turned to Stony Brook for its tenth year, and then moved to Polytechnic University, Brooklyn,NY for its eleventh. The twelfth workshop (2002) was part of the Special Focus on Computa-tional Geometry and Applications at DIMACS, while the thirteenth (2003) was part of the SpecialSemester on Computational Geometry at the Mathematical Sciences Research Institute, Berkeley.In 2004, we are proud to host the Fall Workshop on Computational Geometry at MIT, bringingthe workshop to the Boston area for the first time and returning to the original format.

iv

To p o lo g ic a lly S w e e p in g th e C o m p le te G ra p h in O p tim a l Tim e

Eynat R afalin ∗ D iane S o u v aine ∗

1 In tro d u c tio nWe present a novel, sim ple and easily im plem entable1

approach to sweep a com plete g raph of N vertices andk intersection points and report all intersections in op-tim al O(k) = O(N4) tim e and O(N2) space. O u rm ethod borrows the concept of horizon trees from thetopolog ical sweep m ethod [6] and u ses ideas from [9 ]to handle deg eneracies. The novelty of the approach isthe u se of a moving w a ll that separates the g raph intotwo reg ions at all tim es: the reg ion in front of the wallthat has k nown stru ctu re, and the reg ion behind thewall that m ay contain intersections g enerated by edg esthat the sweep process has not yet predicted. Thism ethod has applications in com pu ting the sim plicialdepth m edian of a point set in R

2 [1 ]. C ontinu ing re-search concentrates on m odifying the topolog ical sweepto work for arbitrary g raphs. For som e sparse g raphs,where the total size of the cu ts is lim ited, the proposedalg orithm m ay be particu larly eff ective.

A g raph poses challeng es for a sweep-line alg orithm ,that are not present in line arrang em ents, and thatm ost ex isting sweep techniq u es do not handle. Thestru ctu re holding the sweep-line statu s needs to be dy-nam ic, as vertices and edg es are constantly insertedand deleted. The event points now have two typesand inclu de both intersection points and g raph ver-tices. Finally, sh ort edg es not yet encou ntered by thesweep line m ay create intersections that shou ld be pro-cessed before intersections created by long edg es thathave already been detected (see Fig u re). P rocessing inthe wrong order m ay introdu ce intersections not in theg raph or ig nore others, cau sing errors.

E x isting alg orithm s for seg m ent intersection detec-tion do not work well for the com plete g raph. P la ne

sw eep [3 , 4 ] adds a log N factor to the optim al tim ecom plex ity (yielding O(N 4 log N) tim e and O(N2)space). The optim al determ inistic alg orithm for the in-tersection reporting problem [5 , 2] u sing O(n log n+k)tim e and O(n) space, is dom inated by O(k) = O(N 4)when applied to a com plete g raph and is hig hly com -plex to im plem ent. Topologica l sw eep [6, 7 , 9 ] off ers alog n im provem ent factor over the vertical line sweep onan arrang em ent of n infi nite lines, bu t does not handlefi nite line seg m ents.

∗Department o f C ompu ter S c ience, Tu fts U niversity,

M ed fo rd , M A 0 2 1 5 5 . erafalin, d ls@ c s.tu fts.ed u1O n C + + c o d e. E x perimental resu lts verify th e time

and space c omplex ities.

2 A lg o rith m O v e rv ie wL et G be a planar em bedding of a com plete g raph on Nvertices. Assu m e vertices are in g eneral position. The

com plete g raph contains ex actly n = N(N−1)2 edg es and

k, the nu m ber of seg m ent intersections, is O(N 4). Thenu m ber of edg es cu t by any sweep line is O(N 2), bu tcan be Θ (N2) with N/2 vertices on each side of theline and (N

2 )2 edg es crossing it.

We sweep G from left to rig ht to report all intersec-tion points u sing a topolog ical line (cu t): a m onotonicline in y-direction, intersecting each of the n edg es a t

most once. The seq u ence of a ctive segments, one peredg e intersected by the topolog ical line form s the cu t.A seg m ent of an edg e is delim ited by two adjacent inter-section points or by the rig htm ost/ leftm ost intersectionpoint and a vertex .

A sweep beg ins with the leftm ost cu t, which inter-sects no edg es of the g raph, and proceeds to the rig ht ina series of elem entary steps u ntil it becom es the rig ht-m ost cu t. An elem entary step com prises the sweepingof the topolog ical line past a vertex of the g raph whoseincom ing edg es are all cu rrently intersected by the cu tor past a rea d y intersection of edg es that are consec-u tive in the cu rrent cu t. There always ex ists a rea d y

intersection or a rea d y vertex , whose incom ing edg esalready lie on the cu t, u nless the cu t is righ tmost [1 0 ].

To m aintain optim al tim e com plex ity linear in thesize of the arrang em ent and space com plex ity linear inthe m ax im al size of the cu t, we bu ild u pon the con-cept of h orizon trees [6]. O u r h orizon gra p h s are oftennot trees bu t h orizon forests. The u pper (resp. lower)h orizon forest of the cu t U H F (resp. L H F) is form edby ex tending the cu t edg es to the rig ht. When twoedg es intersect, only the one of lower (resp. hig her)slope continu es to the rig ht. G iven L H F and U H F, theintersection of the rig ht delim iters of the two forestsprodu ces the cu t, which is com pu ted from the u pdatedL H F and U H F in constant tim e per active edg e. A setof seg m ents along the cu t that contain the sam e inter-section point as their rig ht endpoint, g enerates a rea d y

intersection, see [9 , 6].Inte rse ctio n e v ent po ints: The active seg m entsswitch position. To u pdate U H F (resp. L H F) after pro-cessing the intersection point of seg m ents si, . . . si+k,for each seg m ents s of the intersection apart from thefi rst (last) one, traverse the ba y form ed by the seg m entsabove (below) s, u ntil reaching the seg m ent that inter-sects the ex tension of s (see [1 0 , 6]).

1

Verte x e v e n t po in ts: The sweep processes the ver-tices of the g raph only when no intersection points areread y and in a left-to-rig ht ord er, precisely when all thevertices’ incom ing ed g es are active: In a vertex eventpoint all of the incom ing active ed g es for vertex v ared eleted from the set of active seg m ents and all its ou t-g oing ed g es are inserted . To u pd ate the horizon forestsd elete in-ed g es of v and insert its ou t-ed g es. Then u p-d ate the horizon forest, walking in cou nterclockwise or-d er arou nd the bay form ed by the previou s ed g es to fi ndthe intersection point with an active ed g e. The ed g esem anating from the new vertex to the rig ht, ad d ed tothe set of active ed g es, m ay cu t som e of the ex isting ac-tive seg m ents, and chang e the horizon forests and thecu t. To u pd ate the horizon forests, test every ex istingforest ed g e for intersection with the new seg m ent, intim e linear in the size of the cu t.A m o v in g wa ll: The set of active ed g es d oes not spanthe whole arrang em ent. C onseq u ently, the sweep canencou nter intersection points created by active ed g esbefore id entifying interm ed iary ed g es that block them .For ex am ple, Fig u re (a) contains a g raph of 7 vertices.The d otted sweep line prod u ces the bold active seg -m ents. Intersection B of 0 6 and 2 5 is read y to beprocessed , as it is the rig ht end point of the active seg -m ents associated with these ed g es. B, however, can-not be processed yet, as 0 6 and 2 5 intersect 34 beforepoint B. Fu rtherm ore, if intersection B is processed ,0 6 and 2 5 will switch position in the cu t, cau sing 34to be inserted incorrectly. Intersection points th a t a re

to th e righ t of a ny edge th a t is not yet a ctive ca nnot be

re a d y . Alternatively, consid er intersection A. G iventhe d otted sweep line, both this intersection and vertex3 are read y to be processed (in vertical sweep vertex3 wou ld be processed prior to A). H owever, the ord erof the cu t ed g es cu rrently places the ed g e 1 6 over ed g e0 5 . If vertex 3 is processed at this tim e, the u pd ate ofthe horizon forests and the cu t will be incorrect. All

intersection points th a t a re to th e left of a ll segm ents

em a na ting from vertex v m u st be processed before v is

processed.

0

1

2

3

4

6

BA

v

A

B

5

D efi ne a m oving w a ll of a position of the sweep lineas the sem i-infi nite lines correspond ing with the two ex -trem e ed g es em anating to the rig ht from the nex t sweepvertex vx. The m oving walls for all vertices can be com -pu ted in O(N log N) tim e u sing [8 ]. At any tim e, thesweep line has to be forced to the cu rrent position of

the m oving wall (by processing a read y vertex only if itis to the left of the line d efi ning the wall). Provably, aread y intersection always ex ists u nless the sweep line isalig ned with the m oving wall or has reached the rig ht-m ost cu t. To alig n the sweep line with the cu t ensu rethat no intersection that is insid e the wall is read y andthat read y intersections that aff ect the wall are sweptbefore the nex t vertex is processed .

There m ay be intersections insid e the m oving wallassociated with vx that can still be lega lly swept (e.g .intersection B in Fig u re (b)). The sweep line is forced

only to the wall form ed by the two ex trem e seg m entsem anating to the rig ht from vx and their ex tension toinfi nity instead of inclu d ing all lega l read y intersections,so intersections that are read y bu t insid e the m ovingwall are d iscard ed (Fig u re (b)). These intersections arered iscovered when vx is swept, by checking every pairof ad jacent active seg m ents in the cu t for read y inter-sections. This step is perform ed once for each vertex ,with a cost linear in the size of the active set.Ack n owle d g m e n t The au thors wish to thank Prof.Ileana S treinu , M ichael A. B u rr and R yan C olem an.

References

[1 ] G . Alou pis, S . L ang erm an, M . S oss, and G . Tou -ssaint. Alg orithm s for bivariate m ed ians and aFerm at-Torricelli problem for lines. C om p. G eom .

Th eory a nd Appl., 2 6 (1 ):6 9 – 79 , 2 0 0 3.[2 ] I. J . B alaban. An optim al alg orithm for fi nd ing

seg m ent intersections. In P roc. 1 1 th Annu . AC M

S y m pos. C om pu t. G eom ., pag es 2 1 1 – 2 1 9 , 1 9 9 5 .[3] J . L . B entley and T. A. O ttm ann. Alg orithm s for

reporting and cou nting g eom etric intersections.IE E E Tra ns. C om pu t., C -2 8 (9 ):6 4 3– 6 4 7, 1 9 79 .

[4 ] K . Q . B rown. C om m ents on “ Alg orithm s forreporting and cou nting g eom etric intersections” .IE E E Tra ns. C om pu t., C -30 :1 4 7– 1 4 8 , 1 9 8 1 .

[5 ] B . C hazelle and H . E d elsbru nner. An optim al al-g orithm for intersecting line seg m ents in the plane.J . AC M , 39 (1 ):1 – 5 4 , 1 9 9 2 .

[6 ] H . E d elsbru nner and L . J . G u ibas. Topolog icallysweeping an arrang em ent. J . C om pu t. S y st. S ci.,38 :1 6 5 – 1 9 4 , 1 9 8 9 .

[7] H . E d elsbru nner and D . L . S ou vaine. C om pu t-ing m ed ian-of-sq u ares reg ression lines and g u id edtopolog ical sweep. J . Am er. S ta tist. Assoc.,8 5 :1 1 5 – 1 1 9 , 1 9 9 0 .

[8 ] F. P. Preparata. An optim al real-tim e alg orithmfor planar convex hu lls. C om m . AC M , 2 2 (7):4 0 2 –4 0 5 , 1 9 79 .

[9 ] E . R afalin, D . S ou vaine, and I. S treinu . Topo-log ical sweep in d eg enerate cases. In P roc. 4 th

AL E N E X , volu m e 2 4 0 9 of L N C S , pag es 5 77– 5 8 8 .S pring er-V erlag , 2 0 0 2 .

[1 0 ] E . R afalin and D . L . S ou vaine. Topolog icallysweeping the com plete g raph in optim al tim e andspace. Tech. R eport 2 0 0 3-0 5 , Tu fts U niversity.

2

Dynamic Update of Half-space Depth Contours

M. Burr∗ E. Rafalin∗ D. L. Souvaine∗

Data depth is an approach to statistical analysis basedon the geometry of the data. Half-space depth1 hasbeen studied most frequently by computational geome-ters. The half-space depth of a point x relative to aset of points S = X1, ..., Xn in R

d is the minimumnumber of points of S lying in any closed half-space de-termined by a line through x [2, 11]2. Depth contours,enclosing regions with increasing depth, help to visu-alize, quantify and compare data sets. Prior work in-vestigated combinatorial properties and algorithms forcomputation of depth contours for static data sets. Wepresent a dynamic algorithm for computing the two-dimensional rank-based half-space depth contours of aset of n points in O(n log n) time per operation andin O(n2) overall space, an improvement over the staticversion of O(n2) time per operation. The same al-gorithm can compute the half-space depth of a singlepoint relative to a data set dynamically in O(log n)time and O(n) space. The algorithm does not com-pute the entire set of contours explicitly but main-tains the order (ranking) of points according to theirhalf-space depth. A constant number of contours (e.g.10%, · · · 100%) can be constructed in O(n) time fromthe sorted list of the data points, ranked by depth.Our algorithm uses generalized dynamic segment trees

to update the depth of every data point and is basedon key characterizations of the potential changes in thedepth contours upon insertions or deletions3. We onlyconsider data sets in general position.

1 PreliminariesThe statistics community produced contradictory def-initions for depth contours. The two main approacheswere termed cover and rank [9]. The cover approachdefines the contour of depth k as the boundary of theset of points in R

d with depth ≥ k (for half-space depth1 ≤ k ≤ bn

2 c). The cover-based half-space depth con-tour is provably the boundary of the intersection of allclosed half-planes containing exactly n − k + 1 datapoints whose bounding line passes through two datapoints. The rank approach defines the αth centralregion as the convex hull containing the most centralfraction of α sample points [5]. The α-central rank-

∗Department of Computer Science, Tufts University, Med-ford, MA 02155. mburr,erafalin,[email protected]. Partiallysupported by NSF grant CCF-0431027

1Also called location depth or Tukey depth.2For the reminder of the paper, every half-plane is considered

closed unless otherwise mentioned.3A detailed analysis can be found in [8] where we also present

an O(n log2 n) time and over all O(n2) space algorithm for dy-namically computing cover-based half-space depth contours.

based half-space-depth contour is constructed by sort-ing all points of the original set according to their half-space depth, yielding X[1], · · ·X[n], the ranking order

of the points and taking the convex hull of data pointsX[1], · · ·X[α]. Both approaches assign the same depthvalue to points that are members of the data set S andcreate depth contours that are nested. The main visualdifference: vertices of the rank contours are only datapoints while vertices of the cover contours can be anypoint from the data set.

Algorithms have existed for some time for construct-ing depth contours in 2 and higher dimensions under ei-ther definition. The best 2-D implementation for com-puting the ranking order of a set of points or all cover-based depth contours runs in Θ(n2) [6]. Other imple-mentations (e.g. [10, 3]) compute cover-based contours.

Much prior work exists on dynamic geometric struc-

tures (e.g. [7, 1]). To the best of our knowledge, wepresent the first dynamic algorithm for computation ofhalf-space contours, addressing prior interest [4].

2 The AlgorithmRank-based contours do not have the appealing proper-ties of the cover-based contours: e.g. a unique structurethat is relatively easy to update. Our algorithm uti-lizes the fact that a data point has equal depth valuesunder the cover and rank approaches, and computesthe rank-based contour by considering the cover-based

contours. Thus, the analysis of our algorithm for therank-based contours refers to the complexity of cover-

based contours.A key idea is, for each data point p, to consider ev-

ery directed line l passing through p and another datapoint and the associated closed half-plane Hl to itsright. Hl is represented by the unit vector vHl

associ-ated with l, see figure (a). For each point p there isa set of 2(n − 1) unit vectors, that can be thought ofas points on the unit circle, centered at p. Every pointr ∈ S\p is assigned to two antipodal vectors, rtowards

and raway (where rtowards is the vector pointing to-wards r). When a point q is inserted into or deletedfrom the data set, exactly the half-planes with associ-

ated vectors in the semi-circle counter-clockwise from

qtoward to qaway have their depth incremented or decre-mented. These half-planes are updated simultaneously,to recompute the depth of p efficiently. To do so we usethe concept of defining lines, half-planes and edges. Ifp represents a data point of depth k with respect to setS of n points, p will appear exactly once on the (cover)contour of depth k. If p is on a non-degenerate contourit has exactly two incident edges, the defining edges of

3

r

r

r

rr

r

r

2towardr

5toward

4toward

3toward

4away

3away2away

1away

5away q

r

rr r 1

p

r

2

3

4

5

r 1toward

r

1l

l 2

1v

2v

p

γ

a

q

Region 2

Region 3Region 4

The contour of depth k

Region 1

(a) (b)

p, on the contour of depth k. Every edge on any depthcontour is a sub-segment of a line created by joiningtwo data points q1, q2 of the set S. The defining lines

l1, l2 for p with respect to S are the lines containingp’s defining edges. Each defining line li, i ∈ 1, 2, bi-sects the plane into two closed half-planes containingk + 1 and n − k + 1 data points where the half-planecontaining k+1 points does not contain the k-th depthcontour. The defining half-planes Hl1 , Hl2 for p withrespect to S are the closed half-planes bounded by thedefining lines of p which contain k + 1 data points

When point q is inserted into S the cover-baseddepth of a point p ∈ S remains unchanged if q is insidep’s depth contour and can increase only if q is in theregion outside p’s depth contour. The update of everydata point p, when point q is inserted to or deleted fromS, depends on the location of q relative to the defining

lines for p4. Nine cases completely determine how p’sdepth changes and how its two defining lines are trans-

formed5, see figure (b). These updates are computedin O(log n) time for each data point p: the number ofdata points in every half-plane defined by p and eachpoint in S\p is recomputed; the defining lines of p withrespect to the new data set S ∪q or S \q are found; thenumber of data points in the defining half-plane is theupdated depth of p; knowing all half-planes containingexactly k′ + 1 data points and determined by a linethrough p and another data point makes it possible todetermine p’s new defining half-planes as well. (Notethat at least one of the defining half-planes for everydata point remains unchanged after a single insertionor deletion, see [8]).

Data Structures: For efficiency the algorithm usesnew generalized dynamic segment trees. Each tree rep-resents the half-planes passing through a data pointp ∈ S and is implemented as an augmented dynamicred-black tree.

To re-sort data points, we use a linked list of bucketsfor depths, from 0 to n/2 (the minimum to maximumpossible), to hold all data points. Bucket k holds a

4The data point to be inserted or deleted lies in one of fourregions or one of the defining lines, yielding nine cases.

5For example, if q is inserted into exactly one defining half-plane Hl2

, then p’s depth is unchanged, but Hl2is no longer a

defining half-plane. It can be shown that the vector for p’s newdefining half-plane Hm2

is the first vector found by traversingthe vectors starting from vHl2

towards vHl1whose associated

half-plane contains k + 1 data points.

linked list of data points of depth k. Upon insertion ordeletion every point q that changes its depth is movedfrom its old bucket to a new bucket. Since the depthof q changes by at most 1, the update takes O(1) time.

3 Open Questions• The lower bound for computing the half-space

depth rank of data points (and thus their rankcontours) is Ω(n log n), based on reduction to sort-ing. We are seeking a method to order data pointsaccording to their depth in o(n2).

• To the best of our knowledge, no dynamic algo-rithm for computing depth contours according toother depth measures exists. We are working on adynamic scheme to compute regression depth con-tours (envelopes of the arrangement of lines).

• Most real life experiments are high-dimensional.Since existing static algorithm for computingdepth contours for most data depth measures areexponential in dimension, dynamic approximation

algorithms for depth contours of multivariate data

are needed.

Acknowledgement: The authors wish to thank S.Venkatasubramanian, S. Krishnan and R. Liu.

References[1] Y. Chiang and R. Tamassia. Dynamic algorithms

in computational geometry. Proc. of the IEEE,80(9):1412–1434, 1992.

[2] J. Hodges. A bivariate sign test. The Annals of Math-

ematical Statistics, 26:523–527, 1955.

[3] S. Krishnan, N. H. Mustafa, and S. Venkatasubrama-nian. Hardware-assisted computation of depth con-tours. In 13th ACM-SIAM SODA, 2002.

[4] R. Liu. Private communications, May 2003. Depart-ment of Statistics, Rutgers University.

[5] R. Liu, J. Parelius, and K. Singh. Multivariate analy-sis by data depth: descriptive statistics, graphics andinference. The Annals of Statistics, 27:783–858, 1999.

[6] K. Miller, S. Ramaswami, P. Rousseeuw, T. Sellares,D. Souvaine, I. Streinu, and A. Struyf. Fast implemen-tation of depth contours using topological sweep. InProc. 12th SIAM-ACM SODA, pages 690–699, 2001.

[7] M. H. Overmars and J. van Leeuwen. Maintenance ofconfigurations in the plane. J. Comput. System Sci.,23(2):166–204, 1981.

[8] E. Rafalin, M. Burr, and D. L. Souvaine. Dynamicupdate of half-space depth contours. to appear.

[9] E. Rafalin and D. L. Souvaine. Data depth contours -a computational geometry perspective. Tech. Report2004-01, Tufts University, CS Department, May 2004.

[10] I. Ruts and P. J. Rousseeuw. Computing depth con-tours of bivariate point clouds. Comp. Stat. and Data

Analysis, 23:153–168, 1996.

[11] J. W. Tukey. Mathematics and the picturing of data.In Proc. of the Int. Cong. of Math., Vol. 2, pages 523–531. Canad. Math. Congress, Montreal, Que., 1975.

4

Computing a Point of High Simplicial Depth in R2

Jason Burrowes-Jones∗

Electrical Engineering

Howard University

William Steiger

Computer Science

Rutgers University

November 3, 2004

Abstract

Given a set S = P1, . . . , Pn of n points in R2, the simplicial depth σ(Q) of a point Q ∈ R2

is the number of open triangles ∆PiPjPk that contain Q, 1 ≤ i < j < k ≤ n. A point is Q is“deep” if σ(Q) ≥ maxi σ(Pi). We give a simple, easily implementable O(n(log n)2) deterministicalgorithm to compute a deep point and we can also guarantee that Q has depth at least cn3 fora constant c > 0.

1 Introduction and Summary

In 1974 John Tukey [11] proposed the now familiar notion of halfspace depth, generalizing from onedimension the idea of measuring depth by ranks. Since then several other mulitvariate depth measureshave been proposed, e.g., hyperplane depth [10], Oja depth [9] (and also see [5]. Here we addressthe notion of simplicial depth proposed by Regina Liu in 1990 [8]. Given a set S = P1, . . . , Pn ofn data points in general position in Rd the simplicial depth of a point Q ∈ Rd is defined to be

σ(Q) = |(i1, . . . , id+1), 1 ≤ i1 < · · · < id+1 ≤ n : Q ∈ ∆Pi1Pi2 · · ·Pid+1|, (1)

the number of open simplices whose vertices are points in S and which contain Q. This measure isaffine invariant and robust over samples from a probability distribution.

Even in R2 simplicial depth offers interesting computational and combinatorial challenges. Thesimplicial depth of a point Q can be computed in Θ(n log n): once the radial ordering of the Pi aboutQ is known, σ(Q) can be obtained in O(n) further steps [5]; the lower bound is due to Aloupis et.al.[1]. By constructing the arrangement of the lines dual to the Pi, we obtain for each Pi, the radialorder of the other points around it, and therefore can compute the simplicial depth of all points inS in time O(n2).

We call a point Q ∈ R2 “deep for S” if σ(Q) ≥ maxi σ(Pi), and by the previous observation,a deep point could be found in time O(n2). A point P ∗ ∈ R2 (not necessarily in S) of maximalsimplicial depth is called a simplicial median.

A max-depth data point (i.e., in S), though “deep” by definition, may actually have low depth(i.e., 0). However Boros and Furedi [3] showed that there is always a point Q ∈ R2 that is containedin 2/9 of the triangles determined by the points in S, but that NO point in R2 is in 1/4 + ε ofthem. This means that a simplicial median P ∗ satisfies n3/27 ≤ σ(P ∗) ≤ n3/24 + O(n2), a resultgeneralized to higher dimension by Barany [4].

It seems to be hard to find a simplicial median. The O(n4) time algorithm of Aloupis et.al. [2]is currently best. Therefore the following result may be useful and interesting.

∗Research done as a student in the DIMACS Research Experiences for Undergraduates Program at Rutgers

5

Theorem 1 Given n points in general position in the plane and ε > 0, let P ∗ denote a simplicialmedian and σ∗ its depth. Then a deep point Q having depth at least (1 − ε)σ∗ can be computed intime O(n(log n)2).

It is easy to find a point Q∗ of the claimed depth, an “approximate median”, and to do so withinthe claimed time bounds. The difficulty is to also guarantee that it is “deep”. We do this with apruning argument similar to the one used by Langerman and Steiger [7] for the case of hyperplanedepth. A main ingredient is the following

Lemma 1 Given a set S of n points in R2, in time O(n log n) a point Q′ ∈ R2 can be found, alongwith its depth δ′, and a “witness halfspace” h that contains at least cn points Pi ∈ S, c > 0, withδ(Pi) ≤ δ′.

Once we find Q′, points in S ∩ h are pruned, and the Lemma is applied again to the points inS\(S∩h), giving Q′′. We keep the deeper point. After O(log n) such steps we have Q+, a deep pointfor S. If we apply this procedure to S∪Q∗, Q∗ an approximate median, the deep point we get wouldsatisfy the claims of the Theorem.

References

[1] G. Aloupis, C. Cortes, M. Soss, and G. Toussaint. Lower Bounds for Computing Statistical Depth.Computational Statistics and Data Analysis 40, 223-229 (2002).

[2] G. Aloupis, S. Langerman, M. Soss, and G. Toussaint. Algorithms for Bivariate medians and a Fermat-Torecelli Problem for Lines. Comp. geom., theory and Application 26, 69-79 (2003).

[3] E. Boros and Z. Furedi. The Number of triangles Covering the Center of a Set. Geom. Dedicata 17,69-77 (1984).

[4] I. Barany. A Generalization of Caratheodory’s Theorem. Discrete math. 40, 141-152 (1982).

[5] J. Gill, W. Steiger, and A. Wigderson. Geometric Medians, Discrete Math. 108, 37-51. (1992).

[6] S. Jadhav and A. Mukhopadhyay. Computing a Centerpoint of a Finite Planar Set of Points in LinearTime. Discrete and Comp. Geom. 12, 291-312. (1994).

[7] S. Langerman and W. Steiger. Computing a High Depth Point in the Plane. Developments in RobustStatistics. Proceedings of ICORS 2001, R. Dutter, P. Filmoser, U. Gather, and P.J. Rousseeuw, eds.227-233 (2002).

[8] R. Liu. On a Notion of Data Depth Based on Random Simplices. Annals of Statistics 18, 405-414 (1990).

[9] H Oja. Descriptive statistics for Multivariate Distributions. Statistics and Probability Letters 1, 327-332(1983).

[10] P. Rousseeuw and M. Hubert. Depth in an Arangement of Hyperplanes. Discrete and Comp. Geom. 22167-176 (1999).

[11] John Tukey. Mathematics and the Picturing of Data. Proc. International Conf. of Mathematicians

(Vancouver, 1974) , Vol 2, 523-531 (1975).

6

Abstract, 14th Annual Fall Workshop on Computational Geometry, Boston, Nov 2004

Floodlight Illumination of Infinite Wedges

Matthew Cary† Atri Rudra† Ashish Sabharwal† Erik Vee§†Computer Science and Engr, Box 352350, University of Washington, Seattle, WA 98195-2350§IBM Almaden Research Center, Department K53/B2, 650 Harry Road, San Jose, CA 95120

cary,atri,[email protected] [email protected]

The floodlight illumination problem asks whether thereexists a one-to-one placement of n floodlights illuminat-ing infinite wedges of angles α1, . . . , αn at n locationsp1, . . . , pn in a plane such that a given infinite wedge W

of angle θ located at point q is completely illuminated bythe floodlights. We prove that this problem is NP-hard,closing an open problem from CCCG 2001 [2]. In fact, weshow that the problem is NP-complete even when αi = α

for all 1 ≤ i ≤ n (the uniform case) and θ =∑n

i=0αi

(the tight case). We discuss various approximate solu-tions and show that computing any finite approximationis NP-hard while ε-angle approximations can be obtainedefficiently. Most proofs are omitted in this abstract due tolack of space. Interested readers are referred to [1].

1 Preliminaries

A generalized wedge is a wedge with a continuous finiteregion adjacent to its apex removed. The FLOODLIGHTILLUMINATION problem on generalized wedges is as fol-lows: given n sites p1, . . . , pn, n angles α1, . . . , αn, anda generalized wedge W , determine whether there is an as-signment of angles to sites along with angle orientationsthat illuminates W . In the tight illumination problem, thesum of floodlight spans

∑

αi equals the wedge angle. Adifferent specialization is the uniform problem, where inaddition to being tight, αi = αj for all i, j.

We now look at the related problem of MONOTONEMATCHING Suppose we are given n lines in the plane,n + 1 vertical lines defining n finite width vertical slabs,and two points, one on the leftmost vertical line and oneon the rightmost. Call this an arrangement of lines andslabs, and denote it by (L, S, λ, ρ), where L is the setof lines, S ≡ s1, . . . , sn+1 is the set of vertical linesx = si forming slabs, and λ and ρ are the two specialpoints on the lines x = s1 and x = sn+1, respectively.A monotone matching in (L, S, λ, ρ) is a set of n linesegments, each a portion of a unique line and spanninga unique slab, such that the following holds: (1) the left

end point of the first segment is above λ, (2) the left end-point of each subsequent segment is above the right end-point of the segment in the previous slab, and (3) ρ isabove the right endpoint of the last segment. In the moregeneral problem of PSEUDOLINE MONOTONE MATCH-ING, one has to check whether a given arrangement ofpseuodolines1 and slabs admits a monotone matching.

The floodlight illumiination problem can be related tothe monotone matching problem through duality as de-scribed by Steiger and Streinu [3].

As a tool for our main result, we use NP-completenessof the problem of finding whether a given directed graphhas a directed disjoint cycle cover.

Theorem 1. DIRECTED DISJOINT CYCLE COVER is NP-complete, even for graphs with indegree and outdegreeeach bounded above by 3, as well as for graphs with out-degree exactly 3 and indegree at most 4.

2 Floodlight Illumination is NP-Hard

To give a flavor of the proof of our main result, we provethe following result in this abstract:

Theorem 2. PSEUDOLINE MONOTONE MATCHING isNP-complete.

The most important gadget is the forcing gadget, shownin Figure 1. This is a sequence of slabs associated withpseudolines that forces the line used previous to the gad-get to end below a chosen point, and the line used afterthe gadget to start above another chosen point.

Proof of Theorem 2. As a potential matching can easilybe verified in polynomial time, this problem is in NP. Theproof of NP-hardness is by a reduction from the boundeddegree version of DIRECTED DISJOINT CYCLE COVER(see Theorem 1).

1A pseudoline is a curve in R2 that intersects any vertical line in

exactly one point. A collection of pseudolines is a set of pseudolines notwo of which intersect more than once.

7

High sky

Low ground

Low sky

High ground

`

A B

h

s

Must start above hereMust end below here

Figure 1: The forcing gadget. The arrows show how anylines used before or after the gadget are constrained.

Suppose we are given a directed graph G = (V, E)with the outdegree of all vertices exactly 3 and the inde-gree at most 4. We will have gadgets In(v) and Out(u)for u, v ∈ V as shown in Figure 2. Let I(v) ⊂ E be inthe in-edges of v, and let O(u) ⊂ E be the out-edges ofu. By our choice of G, |I(v)| ≤ 4 and |O(u)| = 3.

O(u)I(v)

Forced end

Forced start Forced start

Forced end

Figure 2: Graph gadgets In(v) and Out(u).

Let n = |V | and m = |E|. We will use m primarypseudolines, each corresponding to an edge in E. Therewill be a number of auxiliary pseudolines used in forc-ing gadgets. The Out(·) gadgets and In(·) gadgets willbe arranged in sequence as shown in Figure 3. The pri-mary pseudoline corresponding to edge (u, v) will firstpass through Out(u), and then pass through In(v).

We claim that when arranged as in Figure 3 along withappropriate forcing gadgets for each In(·) and Out(·)gadget, exactly one e ∈ I(vi) is used in In(vi) and ex-actly one e ∈ O(vi) is not used in Out(vi), 1 ≤ i ≤ n.

A directed disjoint cycle cover of G is equivalent to apermutation π on the vertices, where π(v) is the predeces-sor of v in the cycle containing v. If such a permutationexists, then a monotone matching exists, by not selectingthe edge at Out(u) corresponding to π−1(u), and select-ing the edge corresponding to π(v) at In(v). Conversely,if a monotone matching exists, then the permutation π canbe recovered by setting π(v) equal to the edge that is usedin In(v). This completes the reduction.

The proof of NP-hardness of MONOTONE MATCH-

SKY

GROUND

In(v1)

In(v2)

In(vn)

. . .

. . .

Out(v1)

Out(v2)

Out(vn)

Figure 3: Overall view of the reduction from DIRECTEDDISJOINT CYCLE COVER.

ING with straight lines is based on the same idea but issomewhat more involved. The details can be found in [1].From the duality between monotone matching and flood-light illumination, we have

Theorem 3. FLOODLIGHT ILLUMINATION is NP-hard.The tight, restricted, and uniform versions of the problemare NP-complete.

3 Approximate Illumination

We now look at approximation algorithms to solve thefloodlight illumination problem in the tight case. Let F bean illumination of a wedge W . F is a finite-approximationif it illuminates W \ S, where S is a finite region. F is anε angle-approximation if it illuminates W \ Sε, where Sε

is a union of wedges whose total angle is at most ε. Wehave the following result:

Theorem 4. For the tight floodlight illumination problem,computing a finite-approximation is NP-hard, where asfor any ε an ε angle-approximation can be found in poly-nomial time.

References[1] M. Cary, A. Rudra, A. Sabharwal, and E. Vee. Floodlight il-

lumination of inifinite wedges. Technical Report UW-CSE-2004-10-04, University of Washington, Seattle, Oct. 2004.

[2] E. Demaine and J. O’Rourke. Open problems from CCCG2001. In Proceedings of the 13th Canadian Conference onComputational Geometry, Waterloo, Canada, Aug. 2001.

[3] W. L. Steiger and I. Streinu. Illumination by floodlights.Computational Geometry, 10:57–70, 1998.

8

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10

Self-reconfiguring Robots

Daniela Rus

Computer Science and Artificial Intelligence Lab

MIT

We wish to create versatile robots by using self-reconfiguration: hundreds

of small modules autonomously organize and reorganize as geometric struc-

tures to best fit the terrain on which the robot has to move, the shape of the

object the robot has to manipulate, or the sensing needs for the given task.

Self-reconfiguration allows large collections of small robots to actively orga-

nize as the most optimal geometric structure to perform useful coordinated

work.

A self-reconfiguring robot consists of a set of identical modules that can

dynamically and autonomously reconfigure in a variety of shapes, to best

fit the terrain, environment, and task. Self-reconfiguration leads to versatile

robots that can support multiple modalities of locomotion and manipulation.

Self-reconfiguring robots constitute large scale distributed systems. Because

the modules change their location continuously they also constitute ad-hoc

networks.

This talk will discuss the geometric challenges of creating self-reconfiguring

robots, ranging from designing hardware capable of self-reconfiguration to

developing distributed controllers and planners for such systems that are

scalable, adaptive, and support real-time behavior.

11

Unfolding S m ooth P rim sa toids

abstrac t

Nadia B enbernou ∗ Patricia C ahn† J oseph O ’R ou rke‡

Abstract

We defi ne a notion for u nfolding smooth, ru led su r-faces, and prove that every smooth prismatoid (the con-vex hu ll of two smooth cu rves lying in parallel planes),has a nonoverlapping “ volcano u nfolding .” These u n-folding s keep the base intact, u nfold the sides ou tward,splayed arou nd the base, and attach the top to the tip ofsome side rib. O u r resu lt answers a q u estion for smoothprismatoids whose analog for polyhedral prismatoids re-mains u nsolved.

In tro d u ctio n . It is a long -u nsolved problem to deter-mine whether or not every convex polyhedron can be cu talong its edg es and u nfolded fl at into the plane to a sin-g le nonoverlapping simple polyg on (see, e.g ., [O ’R 0 0 ]),the net. These u nfolding s are known as ed ge u nfo ld ings

becau se the su rface cu ts are along edg es. In this pa-per,1 we g eneralize edg e u nfolding s to certain piecewise-smooth ru led su rfaces,and show that smooth prisma-toids can always be u nfolded withou t overlap. O u r hopeis that the smooth case will inform the polyhedral case.

P y ram id s an d C o n e s. A p ry a m id is a polyhedron thatis the convex hu ll of a convex ba se polyg on B and ana pex v above the plane containing the base. The sid e

fa ces are all triang les. It is trivial to u nfold a pyramidwithou t overlap: cu t all side edg es and no base edg e.This produ ces what mig ht be called a vo lca no u nfolding .E x amples are shown in Fig . 1 (a,b) for reg u lar polyg onbases.

We g eneralize pyramids to co nes : shapes that are theconvex hu ll of a smooth convex cu rve base B lying in thex y -plane, and a point apex v above the plane. We defi nethe volcano u nfolding of a cone to be the natu ral lim-iting shape as the nu mber of vertices of base polyg onalapprox imations g oes to infi nity, and each side triang leapproaches a seg ment rib. This limiting process is illu s-trated in Fig . 1 (c). For any point b ∈ ∂B, the seg mentvb is u nfolded across the tang ent to B at b. Note that

∗Department o f M ath ematic s [email protected].†Department o f M ath ematic s [email protected].‡Department o f C ompu ter S c ience, S mith C o lleg e, N o rth amp-

to n, M A 0 1 0 6 3 , U S A . [email protected]. S u ppo rted by N S F

Disting u ish ed Teach ing S ch o lars award DU E -0 1 2 3 1 5 4 .1S ee http://arxiv.org/abs/cs/0407063 fo r th e fu ll versio n.

Fig u re 1 : U nfolding s of reg u lar pyramids (a-b) ap-proaching the u nfolding of a cone (c).

this net for a cone is no long er an u nfolding that cou ldbe produ ced by paper, becau se the area increases.

M ain R e su lt. O u r main resu lt concerns a shape knownas a p rism a to id , the convex hu ll of two convex polyg onsA and B lying in parallel planes. There is no alg orithmfor edg e-u nfolding prismatoids. O u r concentration inthis paper is on sm oo th p rism a to id s, which we defi ne asthe convex hu ll of two smooth convex cu rves A aboveand B below, lying in parallel planes. A volcano u n-folding of a smooth prismatoid u nfolds every rib seg -ment ab of the convex hu ll, a ∈ ∂A and b ∈ ∂B, acrossthe tang ent to B at b, into the x y -plane, su rrou ndingthe base B, with the top A attached to one appropri-ately chosen rib. The main resu lt of this paper is thatevery smooth prismatoid has a nonoverlapping volcanou nfolding . Fig . 2 illu strates the side u nfolding of a pris-matoid; the top A mu st be carefu lly placed tang ent tothe side u nfolding and on the convex hu ll of that u n-folding .

R e fe re n ce s

[D O 0 4 ] E rik D . D emaine and J oseph O ’R ou rke. Fo ld -

ing a nd U nfo ld ing in C o m p u ta tio na l G eo m etry .20 0 4 . M onog raph in preparation.

[O ’R 0 0 ] J oseph O ’R ou rke. Folding and u nfolding incompu tational g eometry. In D iscrete C o m p u t.

G eo m ., volu me 1 7 6 3 of Lectu re N o tes C o m p u t.

S ci., pag es 25 8 – 26 6 . S pring er-V erlag , 20 0 0 . Pa-pers from the J a pa n C o nf. D iscrete C o m p u t.

G eo m ., Tokyo, D ec. 1 9 9 8 .

12

Figure 2 : T w o v iew s of th e sid e un fold in g of a 3 D p rism atoid . T h e top A is an ellip se in a p lan e p arallel to th e b ase.

13

Folding Paper Shopping Bags

Devin. J. Balkcom∗ Erik D. Demaine†

Martin L. Demaine†

1 Introduction. In grocery stores around theworld, people fold and unfold countless paper bagsevery day. The rectangular-bottomed paper bagsthat we know today are manufactured in their 3Dshape, then folded flat for shipping and storage, andlater unfolded for use. This process was revolu-tionized by Margaret Knight (1838–1914), who de-signed a machine in 1867 for automatically gluingand folding rectangular-bottomed paper bags [8].Before then, paper bags were cut, glued, and foldedby hand. Knight’s machine effectively demolishedthe working-class profession of “paper folder”.

Our work questions whether paper bags can betruly (mathematically) folded and unfolded in theway that happens many times daily in reality. Moreprecisely, we consider foldings that use a finite num-ber of creases, between which the paper must stayrigid and flat, as if the paper were made of plastic ormetal plates connected by hinges. Such foldings aresometimes called rigid origami, being more restrictivethan general origami foldings, which allow continu-ous bending and curving of the paper and thus effec-tively uncountably infinite “creasing”. It is knownthat essentially everything can be folded by a con-tinuous origami folding [6], but that this is not thecase for rigid origami.

We prove that the rectangular-bottomed paperbag cannot be folded flat or unfolded from its flatstate using the usual set of creases that are so com-mon in reality—in fact, the bag cannot move at allfrom either its folded or unfolded state. However,we show that a different creasing of a paper bag en-ables it to fold flat from its 3D state. We also con-jecture a way to unfold a paper bag from its flatstate if it was already folded using the usual set ofcreases (by an adversary equipped with techniquesfrom origami or reality).

2 Related Work. In the mathematical literature,the closest work to rigid folding is rigidity. The fa-mous Bellows Theorem of Connelly, Sabitov, andWalz [4] says that any polyhedral piece of paperforming a closed surface preserves its volume whenfolded according to a finite number of creases. In

∗Dartmouth Computer Science Department, Pittsburgh, PA15213, [email protected]

†MIT Computer Science and Artificial Intelligence Labora-tory, 32 Vassar St., Cambridge, MA 02139, USA, edemaine,[email protected]

h

wl

Right

Back

FrontLeft

45º

Figure 1: A shopping bag with creases in the usual places.

contrast, as suggested by the existence of bellowsin the real world, it is possible to change the vol-ume using origami folding. Even more fundamen-tal are Cauchy’s rigidity theorem, Aleksandrov’s ex-tension, and Connelly’s extension [2], which all es-tablish an inability to fold a convex polyhedron us-ing a finite number of creases. (In Cauchy’s case,the creases must be precisely the edges of the poly-hedron; in Connelly’s case, any finite set of addi-tional creases can be placed; Aleksandrov’s theoremis somewhere in between.) Another result of Con-nelly1 is that a positive-curvature “corner” (the cycleof facets surrounding a vertex in a convex polyhe-dron) cannot be turned “inside-out” no matter howwe place finitely many additional creases; this resultanswers a problem of Gardner [7]. In contrast, a pa-per bag can be turned inside-out with an origamifolding (and in real life) [3].

Few papers discuss rigid origami directly. De-maine and Demaine [5] present a family of origami“bases” that can be folded rigidly. Streinu andWhiteley [9] proved that any single-vertex creasepattern can be folded rigidly—up to but not in-cluded the moment at which multiple layers of pa-per coincide. Balkcom and Mason [1] demonstratehow some classes of origami can be rigidly foldedby a robot.

3 Main Results. Figure 1 shows a shopping bagwith the usual crease pattern, and dimensions w, l,and h. For the bag shown, h > w/2, and l > w.

Our first main result states that a shopping bagcannot be folded at all with just the usual creases:

Theorem 1 A shopping bag with the usual crease pat-tern has a configuration space consisting of two isolatedpoints, corresponding to the fully-open and fully-closedconfigurations.

1Personal communication, 1998.

14

The results described in the previous section havetwo immediate consequences if we allow finitelymany additional creases. First, the Bellows Theoremimplies that, if the shopping bag had a top, no finitenumber of additional creases would allow the vol-ume of the bag to be changed. Second, because thecorners of the bag are convex, no finite number ofadditional creases would allow the shopping bag tobe turned inside-out.

Figure 2: A shortpaper bag, similarto a collapsibledepartment-storegift box.

Based on these consequences,it might seem that no finite setof additional creases would al-low a shopping bag to be foldedflat. Our second result shows theopposite. A short shopping bag,with h ≤ w/2, cannot have theusual shopping bag crease pat-tern, because the 45 creases donot intersect on the interior of theleft and right sides of the bag; seeFigure 2. In this case, we show

Theorem 2 Every short shopping bag (with h ≤ w/2)can be collapsed flat using the creases in Figure 2.

Now Theorem 2 suggests a method for folding atall shopping bag: add creases to allow the tall bagto be telescoped until it is short enough to collapseflat. Figure 3 shows an animation of our procedurefor shortening a bag by reducing h up to minw, l.Using a sequence of these operations, we show

Theorem 3 A tall shopping bag can be collapsed flatwith the addition of finitely many creases.

Figure 4: Conjec-tured creases forunfolding an al-ready folded pa-per bag.

The collapsed state of the shop-ping bag after applying the fold-ing technique described in theproof of theorem 3 is not thesame as the collapsed state ofthe shopping bag with no ad-ditional creases. This differencesuggests a more difficult ques-tion: can a collapsed shoppingbag be opened up with the ad-dition of a finite set of creases?We conjecture that it can, and pro-pose a possible crease pattern inFigure 4.

Conjecture 1 A collapsed tall shopping bag can be un-folded with the addtion of a finite number of creases.

If true, this conjecture would also offer a simplerway to flatten a tall shopping bag.

Figure 3: Procedure for shortening a rectangular tube.

References.[1] D. J. Balkcom and M. T. Mason. Introducing robotic

origami folding. In Proc. 2004 IEEE International Con-ference on Robotics and Automation, 2004.

[2] R. Connelly. The rigidity of certain cabled frame-works and the second-order rigidity of arbitrarily tri-angulated convex surfaces. Advances in Mathematics,37(3):272–299, 1980.

[3] R. Connelly. Rigidity. In Handbook of Convex Geometry,volume A, pages 223–271. North-Holland, 1993.

[4] R. Connelly, I. Sabitov, and A. Walz. The bellows con-jecture. Beitrage zur Algebra und Geometrie (Contribu-tions to Algebra and Geometry), 38(1):1–10, 1997.

[5] E. D. Demaine and M. L. Demaine. Computing ex-treme origami bases. Tech. Rep. CS-97-22, Dept. Com-puter Science, Univ. Waterloo, Canada, May 1997.

[6] E. D. Demaine, S. L. Devadoss, J. S. B. Mitchell, andJ. O’Rourke. Continuous foldability of polygonal pa-per. In Proc. 16th Canadian Conference on ComputationalGeometry, pp. 64–67, Montreal, Canada, August 2004.

[7] M. Gardner. Tetraflexagons. In The Second Scien-tific American Book of Mathematical Puzzles & Diversions,ch. 2, pp. 24–31. University of Chicago Press, 1987.

[8] M. E. Knight. Paper bag machine. U.S. Patent 116,842,July 11 1871.

[9] I. Streinu and W. Whiteley. The spherical carpen-ter’s rule problem and conical origami folds. In Proc.11th Annual Fall Workshop on Computational Geometry,Brooklyn, New York, November 2001.

15

Hinged Dissection of PolypolyhedraErik D. Demaine∗ Martin L. Demaine∗

Jeffrey F. Lindy† Diane L. Souvaine‡

1 Introduction. A dissection of two figures (solid2D or 3D shapes, e.g., polygons or polyhedra) is a wayto cut the first figure into finitely many (compact) piecesand to rigidly move those pieces to form the second fig-ure. It is well-known that any two polygons of the samearea have a dissection, but not every two polyhedra of thesame volume have a dissection [3, 5].

Figure 1: Hinged dissec-tion of square and equilat-eral triangle, described by Du-deney [5].

A hinged dissection oftwo figures is a dissection inwhich the pieces are hingedtogether at points (in 2Dor 3D) or along edges (in3D), and there is a mo-tion between the two figuresthat adheres to the hinging,keeping the hinge connec-tions between pieces intact.While a few hinged dissec-tions such as the one in Fig-ure 1 are quite old (1902), hinged dissections have re-ceived most of their study in the last few years; see [6, 4].It remains open whether every two polygons of the samearea have a hinged dissection, or whether every two poly-hedra that have a dissection also have a hinged dissection.

Figure 2: Two polycubes oforder 8, which have a 24-piece edge-hinged dissectionby our results.

2 Results. In this pa-per we develop a broad fam-ily of 3D hinged dissec-tions for a class of poly-hedra called polypolyhedra.For a polyhedron P withlabeled faces, a polypoly-hedron of type P is aninterior-connected non-self-intersecting solid formed byjoining several rigid copiesof P wholly along identically labeled faces. (Such join-ings are possible only for reflectionally symmetric faces.)Figure 2 shows two polycubes (where P is a cube).

For every polyhedron P and positive integer n, we de-velop one hinged dissection that folds into all (exponen-tially many) n-polyhedra of type P . The number of pieces

∗Computer Science and Artificial Intelligence Laboratory, Mas-sachusetts Institute of Technology, Cambridge, MA, USA, email:edemaine, [email protected]. First author supported in part by NSFCAREER award CCF-0347776.

†Courant Institute of Mathematical Sciences, New York University,New York, NY, USA, email: [email protected]. Work begun when au-thor was at Tufts University. Supported by NSF grant EIA-99-96237.

‡Department of Computer Science, Tufts University, Medford, MA,USA, email: [email protected]. Supported by NSF grant EIA-99-96237.

in the hinged dissection is linear in n and the combinato-rial complexity of P . For polyplatonics, we give partic-ularly efficient hinged dissections, tuning the number ofpieces to the minimum possible among a natural class of“regular” hinged dissections of polypolyhedra. For poly-parallelepipeds (where P is any fixed parallelepiped), wegive hinged dissections in which every piece is a scaledcopy of P . All of our hinged dissections are hinged alongedges and form a cyclic chain of pieces, which can be bro-ken into a linear chain of pieces.

Our results generalize analogous results about hingeddissections of “polyforms” in 2D [4].

Like most previous theoretical work in hinged dissec-tions, we do not know whether our hinged dissections canbe folded from one configuration to another without self-intersection. However, we prove the existence of such mo-tions for the most complicated gadget, the twister.

3 Proof Overview. Our construction of a hinged dis-section of all n-polyhedra of type P divides into two parts.First, we find a suitable hinged dissection of the base poly-hedron P . The exact constraints on this dissection vary,but two necessary properties are that the hinged dissec-tion must be (1) cyclic, forming a closed chain of pieces,and (2) exposed in the sense that, for every face of P , thereis a hinge in H that lies on the face (either interior to theface or on its boundary). For platonic solids, these hingeswill be edges of the polyhedron; in the general case, weplace these hinges along faces’ lines of reflectional sym-metry. Second, we repeat n copies of this hinged dissec-tion of P , spliced together into one long closed chain. Fi-nally, we prove that this new hinged dissection can foldinto all n-polyhedra of type P , by induction on n.

3.1 Platonic Solids. Figure 3 shows an exposedcyclic hinged dissection of each of the platonic solids.Basically, each piece comes from carving the k-sided pla-tonic solid into k face-based pyramids with the platonicsolid’s centroid as the apex. As drawn, these hinged dis-sections consist of k pieces, but by merging consecutivepairs of pieces along their common face, the number ofpieces can be reduced to k/2 pieces while maintaining ex-posed hinges. These exposed hinged dissections have thefewest possible pieces, subject to the exposure constraint,because a hinge can simultaneously satisfy at most twofaces of the original polyhedron.

3.2 General Case. In the general case, we use a 3Dgeneralization of the straight skeleton [1] to decomposea given polyhedron into a collection of cells, exactly onecell per facet, such that exactly one cell is incident to eachfacet. These cells form the pieces in an exposed hingeddissection. For these pieces can be connected togetherinto a cyclic hinged dissection, we need to first arrangefor the polyhedron P to have a Hamiltonian dual graph.

In fact, we make two main modifications to P ’s sur-

16

edemaine

mdemaine

@mit.edu

[email protected]

[email protected]

cube

icosahedron

tetrahedron

octahedron

dodecahedron

Figure 3: Unfolded exposed cyclic hinged dissections of theplatonic solids. The bold lines indicate a pair of edges that arejoined by a hinge but have been separated in this figure to permitunfolding. The dashed lines denote all other hinges betweenpieces. In the unfolding, the bases of all of the pyramid pieceslie on a plane, and the apexes lie above that plane (closer to theviewer).

face. First, we divide each reflectionally symmetric faceof P along one of its lines of symmetry, producing a poly-hedron P ′. Second, we divide each face of P ′ so that

Figure 4: Hamiltonian refine-ment of five faces in a hypo-thetical polyhedron.

any spanning tree of thefaces in P ′ translates intoa Hamiltonian cycle in theresulting polyhedron P ′′.This reduction is similar tothe Hamiltonian triangula-tion result of [2] as well asa refinement for hinged dis-section of 2D polyforms [4,Section 6]. We conceptu-ally triangulate each face fof P ′ using chords (thoughwe do not cut along the edges of that triangulation). Then,for each triangle, we cut from an arbitrarily chosen inte-rior point to the midpoints of the three edges. Figure 4shows an example. For any spanning tree of the facesof P ′, we can walk around the tree (follow an Euleriantour) and produce a Hamiltonian cycle on the faces of P ′′.

In particular, we can start from the matching on thefaces of P ′ from the reflectionally symmetric pairing, andchoose a spanning tree on the faces of P ′ that containsthis matching. Then the resulting Hamiltonian cycle inP ′′ crosses a subdivided edge of every line of symme-try. (In fact, the Hamiltonian cycle crosses every subdi-vided edge of every line of symmetry.) Thus, in the ex-posed cyclic hinged dissection of the Hamiltonian poly-hedron P ′′, there is an exposed hinge along every line ofsymmetry. Therefore all joinings between copies of P ′′

can use these hinges.3.3 Putting Pieces Together. We use induction to

prove that the nth repetition of the exposed cyclic hingeddissection of P described above can fold into any n-polyhedron of type P . The base case of n = 1 is trivial.

Given an n-polyhedron Q of type P , one copy P1 of Pcan be removed to produce an (n−1)-polyhedron Q′. Byinduction, the (n − 1)st repetition of the exposed hingeddissection can fold into Q′. Also, P1 itself can be decom-posed into an instance of the exposed hinged dissection.Our goal is to merge these two hinged dissections. Essen-tially, we show that the hinged dissections can be placedagainst the shared face between P1 and Q′ in such a waythat (1) a hinge of the exposed hinged dissection of P1

coincides with a hinge of the hinged dissection of Q′, and(2) the four pieces involved in these two hinges can be re-hinged so that all pieces are connected in a single cycle,and that cycle is exactly the nth repetition of the exposedhinged dissection of P .

3.4 Mutually Rotated Base Polyhedra: Twisters.If a face is k-fold symmetric for k ≥ 3, then there areseveral ways to glue two copies of P along this face.These different gluings produce different polypolyhedraif P itself is not k-fold symmetric. However, only oneof the gluings can be produced by the inductive argumentdescribed above, because only one relative rotation willalign the hinges that lie along the one chosen line of sym-metry.

5

67810

11

1213

1415 16 1 2

3

4

9

Figure 5: This 32-piecetwister gadget allows turns ofone-quarter of a twist. Al-though the pieces look twodimensional, they have thick-ness (they are prisms). Thegaps between pieces 8 and 9in subfigure (a) and betweenthe top and bottom layers arefor visual clarity only; in fact,the two layers are flush. Solidsegments denote lengthwisehinges on the “inside” layer;dashed segments denote tinyhinges on the perimeter.

To enable these kindsof joinings, we embed thetwister gadget shown inFigure 5 beneath each faceof P ′′ that has k-fold sym-metry for k ≥ 3. Thisgadget consists of 8k cycli-cally hinged pieces that al-low any integer multiple of1/k rotation of one set ofpieces with respect to theother pieces.

References.[1] O. Aichholzer, F. Auren-

hammer, D. Alberts, andB. Gartner. A noveltype of skeleton for poly-gons. J. Univ. Comput.Sci., 1(12):752–761, 1995.

[2] E. M. Arkin, M. Held,J. S. B. Mitchell, and S. S.Skiena. Hamiltonian triangulations for fast rendering. TheVisual Computer, 12(9):429–444, 1996.

[3] V. G. Boltianskii. Hilbert’s Third Problem. V. H. Winston& Sons, 1978.

[4] E. D. Demaine, M. L. Demaine, D. Eppstein, G. N. Freder-ickson, and E. Friedman. Hinged dissection of polyominoesand polyforms. Computational Geometry: Theory and Ap-plications. To appear. http://arXiv.org/abs/cs.CG/9907018.

[5] G. N. Frederickson. Dissections: Plane and Fancy. Cam-bridge University Press, November 1997.

[6] G. N. Frederickson. Hinged Dissections: Swinging & Twist-ing. Cambridge University Press, August 2002.

17

http://arXiv.org/abs/cs.CG/9907018

A 2 -c h ain C an In te rlo c k w ith a k-c h ain

abstrac t

Julie G lass∗ S tefan L ang erm an† Joseph O ’R ourke‡ Jack S noeyink§ Jianyuan K . Z hong ¶

Abstract

O ne of the open problem s posed in [3] is: what is them inim al num ber k such that an open, fl ex ible k-chaincan interlock with a fl ex ible 2 -chain? In this paper,we establish the assum ption behind this problem , thatthere is ind eed som e k that achieves interlocking . Weprove that a fl ex ible 2 -chain can interlock with a fl ex ible,open 16-chain.

1 In tro d u ctio n

A po ly go n a l ch a in (or just ch a in ) is a linkag e of rig idbars (line seg m ents, ed g es) connected at their end points(joints, vertices), which form s a sim ple path (an o pen

ch a in ) or a sim ple cycle (a clo sed ch a in ). A fo ldin g ofa chain is any reconfi g uration obtained by m oving thevertices so that the leng ths of ed g es are preserved andthe ed g es d o not intersect or pass throug h one another.T he vertices act as universal joints, so these are fl ex ible

ch a in s. If a collection of chains cannot be separated byfold ing s, the chains are said to be in terlocked.

Interlocking of polyg onal chains was stud ied in [4 , 3],establishing a num ber of results reg ard ing which col-lection of chains can and cannot interlock. O ne of theopen problem s posed in [3] asked for the m inim al k suchthat a fl ex ible open k-chain can interlock with a fl ex ible2 -chain. An unm entioned assum ption behind this openproblem is that there is som e k that achieves interlock-ing . It is this q uestion we ad d ress here, showing thatk = 16 suffi ces.

It was conjectured in [3] that the m inim al k satisfi es6 ≤ k ≤ 11. T his conjecture was based on a construc-tion of an 11-chain that likely d oes interlock with a 2 -chain. We em ploy som e id eas from this construction inthe ex am ple d escribed here, but for a 16-chain. O urm ain contribution is a proof that k = 16 suffi ces. It

∗Dept. o f M ath . & C o m pu t. S c i., C alif. S tate U n iv . H ayward ,

H ayward , C A 9 4 5 4 2 . [email protected]†U n iv . L ibre d e Bru x elles, Departem en t d ’in fo rm atiq u e, U L B

C P2 1 2 , Bru x elles, Belg iu m . [email protected]‡Dept. C o m pu t. S c i., S m ith C o lleg e, N o rth am pto n , M A

0 1 0 6 3 . [email protected] Partially su ppo rted by N S F DT S

award DU E -0 1 2 3 1 5 4 .§Dept. C o m pu t. S c i., U n iv . N o rth C aro lin a, C h apel H ill, N C

2 7 5 9 9 . [email protected]¶Dept. o f M ath . & S tatistic s C alif. S tate U n iv ., S acram en to

6 0 0 0 J S t., S acram en to , C A 9 5 8 1 9 . [email protected]

appears that using m ore bars m akes it easier to obtaina form al proof of interlocked ness.1

R esults from [3] includ e:

1. T wo open 3-chains cannot interlock.

2 . N o collection of 2 -chains can interlock.

3. A fl ex ible open 3-chain can interlock with a fl ex ibleopen 4 -chain.

T his third result is crucial to the construction wepresent, which establishes our m ain theorem , that a 2 -chain can interlock a 16-chain (T heorem 1 below.)

2 Id e a o f P ro o f

We fi rst sketch the m ain id ea of the proof. If we couldbuild a rig id trapezoid with sm all ring s at its four ver-tices (T1, T2, T3, T4), this could interlock with a 2 -chain,as illustrated in Fig ure 1(a). For then pulling vertex v

of the 2 -chain away from the trapezoid would necessar-ily d im inish the half apex ang le α, and pushing v d owntoward the trapezoid would increase α. B ut the onlyslack provid ed for α is that d eterm ined by the d iam eterof the ring s. We m ake as our subg oal, then, build ingsuch a trapezoid .

α

(c)(a)

v

b

a

b'

a'

(b)

51

41

T1

T3T4

T2

wu

Fig ure 1: (a) A rig id trapezoid with ring s would inter-lock with a 2 -chain; (b) An open chain that sim ulates arig id trapezoid ; (b) Fix ing a crossing of a a ′ with b b ′.

We can construct a trapezoid with four links, andrig id ify it with two crossing d iag onal links. In fact, only

1S ee http://arxiv.org/abs/cs.CG/0410052 fo r th e fu ll paper.

18

one d iag onal is necessary to rig id ify a trapezoid in theplane, bu t clearly a sing le d iag onal leaves the freed omto fold along that d iag onal in 3D . This freed om will berem oved by the interlocked 2 -chain, however, so a sing led iag onal su ffi ces. To create this rig id ifi ed trapezoid witha sing le open chain, we need to em ploy 5 links, as shownin F ig u re 1(b). B u t this will only be rig id if the linksthat m eet at the two vertices incid ent to the d iag onalare tru ly “ pinned ” there. In g eneral we want to take onesu bchain aa′ and pin its crossing with another su bchainb b ′ to som e sm all reg ion of space. S ee F ig u re 1(c) forthe id ea.

This pinning can be achieved by the “ 3/4-tang le” in-terlocking from [3], resu lt (3) above; see F ig u re 2 .

Cy

xB D

Ew

A

z

F ig u re 2 : F ig . 6 from [3].

S o the id ea is replace the two critical crossing s witha sm all copy of this confi g u ration. This can be accom -plished with 7 links per 3/4-tang le, bu t sharing with theincid ent incom ing and ou tg oing trapezoid links poten-tially red u ces the nu m ber of links need ed per tang le. Wehave achieved 5 links at one tang le and 4 at the other.The other two vertices of the trapezoid need to sim u latethe ring s in F ig u re 1(a), and this can be accom plishedwith one ex tra link per vertex . Tog ether with the 5links for the m ain trapezoid skeleton, we em ploy a totalof 5 + (5 + 4 + 1 + 1) = 16 links.

The fi nal constru ction, shown in F ig u re 3, establishesou r m ain resu lt:

Theorem 1 The 2 -lin k cha in is in terlocked w ith the 1 6 -

lin k tra pezo id cha in .

3 D isc u ssio n

We d o not believe that k = 16 is m inim al. We haved esig ned two d iff erent 11-chains both of which appearto interlock with a 2 -chain. H owever, both are basedon a triang u lar skeleton rather than on a trapezoid alskeleton, and place the apex v of the 2 -chain close tothe 11-chain. It seem s it will req u ire a d iff erent prooftechniq u e to establish interlocking , for the sim plicity ofthe proof presented here relies on the vertices of the2 -chain rem aining far from the entang ling chain.

A nother d irection to ex plore is closed chains, forwhich it is reasonable to ex pect fewer links. R eplac-

T1

T3

T4

T2

F ig u re 3: A n open 16-chain form ing a nearly rig id trape-zoid , interlocked with a 2 -chain (red ).

ing the 3/4-tang les with “ knitting need les” confi g u ra-tions [2 ][1] prod u ces a closed chain that appears inter-locked , bu t we have not d eterm ined the m inim u m nu m -ber of links that can achieve this.

A c k n o w le d g e m e n ts

We th a n k E rik D em a in e fo r d isc u ssio n s th ro u g h o u t th iswo rk , th e p a rtic ip a n ts o f th e D IM AC S R ec o n n ec t Wo rk -sh o p h eld a t S t. M a ry ’s C o lleg e in J u ly 2 0 0 4 fo r h elp fu ld isc u ssio n s, a n d G illia n B ru n et a n d M eg h a n Irv in g fo r p h y s-ic a l m o d el c o n stru c tio n : http://cs.smith.edu/~orourke/

Interlocked/Linkage.Model.html.

R e fe re n c e s

[1 ] T. B ied l, E . D em a in e, M . D em a in e, S . L a z a rd , A. L u -b iw, J . O ’R o u rk e, M . O v erm a rs, S . R o b b in s, I. S trein u ,G . To u ssa in t, a n d S . Wh itesid es. Locked a n d u n locked

poly gon a l ch a in s in 3 D . D isc rete C o m p u t. G eo m .,2 6 (3):2 6 9 – 2 8 2 , 2 0 0 1 .

[2 ] J . C a n ta rella a n d H . J o h n sto n , N on trivia l em beddin gs

of poly gon a l in terva ls a n d u n kn ots in 3-spa ce, J o u rn a lo f K n o t th eo ry a n d Its R a m ifi c a tio n s, 7 (8 ): 1 0 2 7 – 1 0 39 ,1 9 9 8 .

[3] E . D . D em a in e, S . L a n g erm a n , J . O ’R o u rk e, a n dJ . S n o ey in k , In terlocked O pen Lin ka ges with Few

J oin ts, P ro c . 1 8 th AC M S y m p o s. C o m p u t. G eo m ., 1 8 9 –1 9 8 , 2 0 0 2 .

[4 ] E . D . D em a in e, S . L a n g erm a n , J . O ’R o u rk e, a n dJ . S n o ey in k , In terlocked open a n d closed lin ka ges with

few join ts, C o m p . G eo m . Th eo ry Ap p l., 2 6 (1 ): 37 – 4 5 ,2 0 0 3.

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21

A Theorem of Tutte and 3D Mesh Parameterization

Craig Gotsman

Technion – Israel Institute of Technology and

Harvard University

Parameterization of 3D manifold mesh data involves embedding the mesh in some natural parametric domain, such as the plane or the sphere. Parameterization is important for many applications in geometry processing, including texture mapping, remeshing and morphing. The main objective is to generate a bijective mapping between the mesh surface and the parametric domain, which minimizes the distortion incurred in the transition in some meaningful sense. Examples of possible distortion are metric (edge length) distortion, conformal (angular) distortion and authalic (area) distortion.

A classical theorem of Tutte [7], originally designed to draw planar graphs, shows how to embed a manifold graph with the topology of a disk in the plane. This is achieved by fixing its boundary to a convex shape, and then solving a set of linear equations for the positions of the interior vertices. These equations express the fact that every interior vertex is positioned at the centroid of its neighbors. This basic method was later generalized by Floater [2] to arbitrary convex combinations, and Tutte’s method could then be used to embed a 3D mesh in the plane, controlling the distortion by using convex weights derived from the geometry of the mesh. This class of embeddings are harmonic solutions of a discrete Laplace equation, namely requiring the weighted graph Laplacian operator to vanish at all interior vertices, with convex boundary conditions.

While Tutte’s basic method remains a popular parameterization method, the constraint of a convex boundary is very severe, in most cases introducing unnecessary distortion into the result. Beyond that, it does not provide a satisfactory method to parameterize closed genus-0 meshes and meshes with higher genus. In this talk I will briefly survey some recent work of mine with colleagues on various generalizations of Tutte’s method which overcome these problems.

Gortler, Gotsman and Thurston [3] provided conditions under which Tutte’s method produces bijective embeddings even when the boundary is non-convex. This was used by Karni, Gotsman and Gortler [5] to generate free-boundary planar embeddings with constraints.

Gotsman, Gu and Sheffer [4] showed how to generalize the theory of Tutte to embed a closed genus-0 mesh on the sphere. This relies on recent algebraic characterizations of convex embeddings due to Colin de Verdiere [1], and related eigenvector constructions due to Lovasz and Schrijver [6]. In practice it involves solving a set of quadratic equations.

Inspired by recent work on discrete vector calculus, Gortler, Gotsman and Thurston [3] showed how the concept of a one-form from differential geometry can be defined on a discrete mesh. When such a one-form is harmonic, is may be used to generate bijective embeddings in the plane, in analogy to the Tutte method. The theory culminates in a discrete version of the Hopf-Poincare Index theorem, which may be used to provide a simple proof of the Tutte theorem, Moreover, it shows very simply (using a mere counting argument) how to generate a doubly-periodic embedding of the torus in the plane.

22

References

[1] Y. Colin de Verdiere. Sur un nouvel invariant des graphes et un critere de planarite.Journal of Combinatorial Theory B 50:11-21, 1990. [English translation: On a new graph invariant and a criterion for planarity, in: Graph Structure Theory (N. Robertson, P. Seymour, Eds.) Contemporary Mathematics, AMS, pp. 137-147, 1993.]

[2] M.S. Floater. Parameterization and smooth approximation of surface triangulations.Computer Aided Geometric Design, 14:231–250, 1997.

[3] S. J. Gortler, C. Gotsman and D. Thurston. One-forms on meshes and applications to 3D mesh parameterization. Harvard University CS TR-12-04, 2004.

[4] C. Gotsman, X. Gu and A. Sheffer. Fundamentals of spherical parametrization for 3D meshes. ACM Transactions on Graphics, 22(3):358-363 (Proc. SIGGRAPH), 2003.

[5] Z. Karni, C. Gotsman and S. Gortler. Free-boundary parametrization in the presence of constraints. Preprint, 2004.

[6] L. Lovasz and A. Schrijver. On the nullspace of a Colin de Verdiere matrix. Annalesde l'Institute Fourier 49:1017-1026, 1999.

[7] W.T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13(3):743-768, 1963.

23

Drawing Equally-Spaced Curves between Two Points

Tetsuo Asano 1 and Takeshi Tokuyama 2 1 School of Information Science, JAIST, Japan

2 Graduate School of Information Sciences, Tohoku University, Japan.

Given two points a and b in the plane, draw n equally-spaced curves, C1 C2 Cn between them. More for-mally, those curves must satisfy the equi-distant prop-erty: for any point p on any such curve C i 1 i nthe two adjacent curves Ci 1 and Ci ! 1 are equally dis-tant, that is, d " p Ci 1 #%$ d " p Ci ! 1 # , where d " p C # isthe distance from p to the point on a curve C that isclosest to p. We define C0 $ a and Cn ! 1 $ b as de-generate curves. So, we have d " p C0 #%$ d " p a # andd " p Cn ! 1 #$ d " p b # .

This problem is related to wire routing in printed cir-cuit boards. In a new production technology more thanone wires can be put between two pins. Then, it is de-sired to draw those wires so that they are equally spaceddue to electrical constraints.

It is easy to draw one curve (line) between two points.It is simply a perpendicular bisector of the two points. Itis also rather easy to draw three equally-spaced curves.The middle one is the perpendicular bisector of the twopoints and the remaining two curves are parabolas, eachof which is a curve equidistant from a point and a line.

The case n $ 2 is not easy. In this short paper weprove that we can draw two such equally spaced curvesalthough we have no analytic equations for the curves.

& '(*)+ ,-./01 2)3*4516*-798:2;)

For simplicity but without loss of generality we fix thetwo points a and b: a $ " 3 0# and b $ "< 3 0 # . Wedenote the two curves by Ca and Cb, which are charac-terized as follows:(1) For any point p on Ca, d " p a#=$ d " p Cb # , and(2) for any point p on Cb, d " p b #>$ d " p Ca # .

The curves Ca and Cb must pass through thepoints " 1 0 # and "?< 1 0 # , respectively, and the distanced " p Ca # , for example, is given as the length of a linesegment directed from p perpendicularly to the curveCa.

Now, take any point p on Ca. Then, we have d " p a #$d " p Cb # It implies that the circle centered at p with thepoint a on it must be tangent to the curve Cb at a uniquepoint, which is denoted by q p. The point qp on Cb iscalled an image of the point p on Ca.

Fig. 1 shows this tangential property.

ab

p

qp

CaCb

Figure 1: Tangential property.

By this tangential property we can guess a roughshapes of Ca and Cb, that is, they are smooth and convextoward the origin since they are envelopes of circles. Asa point p goes to infinity along the curve Ca, the radiusof its associated circle approaches to infinity, and finallyit converges to a line.

@ A(;BCD*EF798:2;)

The properties described above suggest equations spec-ifying the curves Ca and Cb, but it seems to be hard toobtain analytic representations of the curves. Our strat-egy here is to compute rough shapes of the curves. Forthat purpose, we partition the plane into small squaresof side length ε and remove all squares that cannot in-tersect the curves (see Fig. 2 for illustration). Due to thesymmetry of the curves Ca and Cb, we only consider Ca.

Given a value of ε G 0, the plane is partitionedinto squares. Each such square is specified by two-dimensional indices, as

si H j $JI iε " i K 1# ε LM I jε " j K 1 # ε L (1)

We start with finding a square s0 H j which contains thepoint " 1 0 # , the intersection of the curve Ca with the x-axis, and then remove all other squares with i $ 0. Ati $ 1, we take squares near the square s0 H j containing" 1 0 # and check their feasibility.

Feasible squares are defined as follow:(1) The square containing the point " 1 0 # is feasible.(2) A square si H j is feasible if there exists a feasiblesquare si NDH j N such that

(i) i O;P i,

24

a

dmax(si,j,si′,j′)

dmin (si,j,si′,j′)

si′,j′

si,j

dmin (si,j,a)

Figure 2: Feasible squares with related distances.

Figure 3: Two curves drawn by our approximation algo-rithm.

(ii) dmin si j a dmax si j a dmin si j si j dmax si j si j /0, and

(iii) there exists no square si” j” such thatdmax si j si” j” dmin si j a ,

where note that if a square si j on the curve Ca is feasiblethen the square si j is a feasible square on the curve Cb.dmin si j a and dmax si j a are the minimum and max-imum distances between the square si j and the point a,and dmin s s and dmax s s are the minimum and max-imum distances between two squares s and s .

Once we have feasible squares si j si j 1 si k for i,a set of feasible squares for i 1 should start somewherearound j and end somewhere near k on i 1. Fig. 3shows our implementation result.

"!$#&%(')"'(*,+!)- .,'(+- 0/(1There are some curves and lines that characterize theequally-spaced curves Ca and Cb. Because of sym-

metricity of the curves we only consider the upperhalves of Ca and Cb.

What is an optimal pair of half lines to approximatethem to minimize the error? Because of their symmtric-ity, we can assume that a pair y cx d x 2 0 andy 43 cx d x 5 0 is optimal, where c d 6 0. An er-ror for a point p x y on y cx d is given by thedifference between the distance from p to a and thelength of a perpendicular line segment from p to the liney 73 cx d. If we denote the other endpoint of the seg-ment by x y , then the difference d is given by

d 8 x 3 1 2 y2 398 x 3 x 2 y 3 y 2 x 3 1 2 y2 3 x 3 x: 2 3 y 3 y; 2< x 3 1 2 y2 < x 3 x 2 y 3 y 2

For this difference to converge as x goes to infinity, thecoefficient of x2 in the numerator must be 0, that is, c 1.

Now, the difference is simplified to

d 2 d 3 1 x 1 d2< x 3 1 2 x d 2 = 2x2

It is minimized when d 1. In fact, when c d 1,the difference converges to 0 as x goes to infinity.

We have shown that an optimal pair of half lines toapproximate the curves is given by y x 1 x 2 0 andy >3 x 1 x 5 0. However, it does not imply that theyare asymptotes of the curves. For any point p on Ca itscorresponding point q p on Cb that is closest to p on Cbis defined as the image of p. If we move a point p alongCa toward infinity, its image also moves on Cb in thedirection away from the origin. Then, does it also go toinfinity? The answer is no. Images cannot go beyondsome point q∞ on Cb.

This also means that if a point p on Ca is sufficientlyfar away from the origin then it must be close to thebisector of the two points a and q∞. So, the bisectorscan be considered as the asymptotes of our curves.

? @(A(+BAC"!EDF-"!G*,+B-0/We are now working on a Voronoi diagram based on thecurves defined here. We call it a Voronoi diagram withneutral zones. We have obtained preliminary results.

H&*(IJ/ )KML!GNCO0P!)/(+The problem was originally given by Dr. Hiroshi Mu-rata, a visiting professor, The University of Kitakyushu,Japan. The authors would like to thank Benjamin Doerr,Alexander Wolff, and Sergei Bereg for their discussions.This work is partially supported by Ministry of Educa-tion, Science, Sports and Culture, Grant-in-Aid for Sci-entific Research on Priority Areas.

25

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27

Pointed Binary Encompassing Trees: Simple and Optimal

Michael Hoffmann

Institute for Theoretical Comp. Sci.

ETH Zurich, CH-8092, Switzerland

[email protected]

Csaba D. Toth

Department of Mathematics

MIT, Cambridge, MA 02139

[email protected]

Abstract. For n disjoint line segments in the planewe can construct a binary encompassing tree suchthat every vertex is pointed, what’s more, at everysegment endpoint all incident edges lie in a halfplanedefined by the incident input segment. Our algorithmruns in O(n log n) time which is known to be optimalin the algebraic computation tree model.

Introduction. Interconnection graphs of disjointline segments in the plane are fundamental structuresin computational geometry, and often more complexobjects are modelled by their boundary segments orpolygons. One particularly well-studied example isa crossing-free spanning graph: the encompassing

graph for disjoint line segments in the plane is a con-nected planar straight line graph (Pslg) whose ver-tices are the segment endpoints and that containsevery input segment as an edge.

A simple construction shows that not every set of n

disjoint segments in the plane admits an encompass-

ing path. But there is always a path that encompassesΘ(log n) segments and does not cross any other inputsegment [6]. The question, whether encompassingtrees of bounded degree exist, was answered in theaffirmative by Bose and Toussaint [4]. Later Bose,Houle and Toussaint [3] constructed an encompass-ing tree of maximal degree three in O(n log n) timeand proved that the runtime is optimal.

Recently, Hoffmann, Speckmann, and Toth [5]have shown that for every set of disjoint segments apointed binary encompassing tree can be constructedin O(n4/3 log n) time. A Pslg is pointed iff at everyvertex p all incident edges lie on one side of a linethrough p.

Pointed Pslgs are tightly connected to mini-mum pseudo-triangulations, which have numerousapplications in motion planning [10], kinetic datastructures [8], collision detection [1], and guard-ing [9]. Streinu [10] showed that a minimum pseudo-triangulation of V is a pointed Pslg on the vertexset V with a maximal number of edges. As opposedto triangulations, there is always a bounded degreepseudo-triangulation of a set of points in the plane [7].A bounded degree pointed encompassing tree for dis-joint segments leads to a bounded degree pointed en-compassing pseudo-triangulation, due to a result ofAichholzer et al. [2].

In this paper, we improve all previous results onencompassing trees of n segments and give a simplealgorithm to construct a pointed binary encompass-ing tree in optimal O(n log n) time. Moreover, forevery vertex of the tree all incident edges lie on oneside of the line through the incident input segment.

Theorem 1 For a set S of n disjoint line segments

in the plane, we can build in O(n log n) time a binary

encompassing tree such that for every segment end-

point p of every input segment pq the edges incident

to p lie in a halfplane bounded by the line through pq.

Tunnel Graphs. The free space around the seg-ments can be partitioned into n + 1 convex cells1 bythe following well known partitioning algorithm: Forevery segment endpoint p of every input segment pq,extend pq beyond p until it hits another input seg-ment, a previously drawn extension, or to infinity.

1For simplicity, we assume that no three segment endpointsare collinear.

28

(a) Partition. (b) Tunnel Graph. (c) Encompassing Tree.

p

q

a

b

c

(d) Disconnected.

Fig. 1: An example for a partition with an assignment (a), the corresponding tunnel graph (b), and theresulting tree (c). A partition for which no assignment gives a connected tunnel graph (d).

Consider a set of segments S and a convex partitionP (S) obtained by the above algorithm. Let us assignevery p to an incident cell τ(p) of the partition. A keytool in our proof is the tunnel graph T (S, P (S), τ) ofP (S) and the assignment τ , defined as follows: Thenodes of T correspond to the convex cells of P (S),two nodes a and b are connected by an edge iff thereis a segment pq ∈ S such that τ(p) = a and τ(q) = b.It is easy to see that the tunnel graph is planar. AsT has n + 1 nodes and n edges, it is connected iff itis a tree.

Theorem 2 For any set S of n disjoint line seg-

ments, we can construct in O(n log n) time a convex

partition P (S) and an assignment τ such that the

tunnel graph T (S, P (S), τ) is a tree.

We note that Theorem 2 does not hold for every par-tition: Fig. 1(d) shows 7 disjoint line segments anda convex partition such that there is no assignmentfor which the tunnel graph is connected. The proofof Theorem 2 can be found in the full version of thispaper. Our main result follows from Theorem 2.

Proof of Theorem 1. Consider a partition P (S)and an assignment τ provided by Theorem 2. In eachcell connect the segment endpoints assigned to it bya simple path.

The resulting graph is clearly a Pslg that en-compasses the input segments. The maximal degreeis three because we add at most two new edges at

every segment endpoint. It remains to prove con-nectivity. Let p and r be two segment endpoints.We know that the tunnel graph is connected, sothere is an alternating sequence of cells and segments(a1 = τ(p), p1q1, a2, . . . , pk−1qk−1, ak = τ(r)) suchthat τ(pi) = ai and τ(qi) = ai+1, for every i. Asall segment endpoints assigned to the same cell areconnected, this path corresponds to a path in the en-compassing graph. 2

References

[1] P. K. Agarwal, J. Basch, L. J. Guibas, J. Hershberger, andL. Zhang, Deformable free space tilings for kinetic collisiondetection, in Proc. 4th WAFR, 2001, 83–96.

[2] O. Aichholzer, M. Hoffmann, B. Speckmann, and Cs. D. Toth,Degree bounds for constrained pseudo-triangulations, in:Proc. 15th CCCG, 2003, pp. 155–158.

[3] P. Bose, M. E. Houle, and G.T. Toussaint, Every set of dis-joint line segments admits a binary tree, Discrete Comput

Geom. 26 (2001), 387–410.

[4] P. Bose and G. T. Toussaint, Growing a tree from itsbranches, J. Algorithms 19 (1995), 86–103.

[5] M. Hoffmann, B. Speckmann, and Cs. D. Toth, Pointed bi-nary encompassing trees, in Proc. 9th SWAT, 2004.

[6] M. Hoffmann and Cs. D. Toth, Alternating paths throughdisjoint line segments, Inf. Proc. Letts. 87 (2003), 287–294.

[7] L. Kettner, D. Kirkpatrick, A. Mantler, J. Snoeyink,B. Speckmann, and F. Takeuchi, Tight degree bounds forpseudo-triangulations of points, Comput. Geom. TheoryAppl. 25 (2003), 1–12.

[8] D. Kirkpatrick and B. Speckmann, Kinetic maintenance ofcontext-sensitive hierarchical representations for disjoint sim-ple polygons, in Proc. 18th SoCG, 2002, pp. 179–188.

[9] B. Speckmann and Cs. D. Toth, Allocating vertex π-guardsin simple polygons, 14th SODA, 2003, pp. 109–118.

[10] I. Streinu, A combinatorial approach to planar non-collidingrobot arm motion planning, 41st FOCS, 2000, pp. 443–453.

29

Hamiltonian Cycles in Sparse Vertex-Adjacency Duals

Perouz Taslakian and Godfried ToussaintSchool of Computer Science

McGill UniversityMontreal, Quebec, Canada

perouz, [email protected]

1 Introduction

In computer graphics, a common way of represent-ing objects is with their triangulations. The perfor-mance of operations executed on these objects dependshighly on how their surfaces are triangulated and howthese triangles are transmitted to the processing engine.Thus, to speed up many operations, such as renderingor compression [9, 10], it is desirable that triangles bearranged such that their adjacency information is pre-served.

In this paper we present a sparse vertex-adjacencydual of a polygon triangulation, which is a graph thatpreserves the vertex-adjacency information of the tri-angles and contains a Hamiltonian cycle. The size ofthis graph is linear in the number of polygonal vertices.

Effort has been made to design algorithms that pro-duce Hamiltonian triangulations, where the dual graphof the triangulation is a path. In [1], Arkin et al. showthat any set of n points has a Hamiltonian triangula-tion and describe two algorithms which construct suchtriangulations. They also show that the problem of de-termining whether a polygon (with holes) has a Hamil-tonian triangulation is NP-complete. In the same paper,a sequential triangulation of a set of points is defined tobe a Hamiltonian triangulation whose dual graph con-tains a Hamiltonian path, and it is proved that suchtriangulations do not always exist for any given set ofpoints.

Hamiltonian properties of general triangulationshave been studied extensively. Various results that con-struct a Hamiltonian cycle in a given triangulation canbe classified based on the model considered. In onemodel, the given triangulation is allowed to be modi-fied by adding new vertices or Steiner points. In [7],Gopi and Eppstein present an algorithm for construct-ing a Hamiltonian cycle in a given triangulation by in-serting new vertices within existing triangles.

In the second model, the input triangulation cannot

be modified. In this case, the problem is that of arrang-ing adjacent triangles in some order such that the result-ing graph contains a Hamiltonian cycle. An importantproperty here is how adjacency is defined. In the dualgraph of a triangulation, adjacency is defined as edge-adjacency where two triangles are adjacent when theyshare an edge. Unfortunately, it is not always possibleto find Hamiltonian cycles in the dual graph.

Hamiltonian cycles in triangulations are studiedwhen adjacency is defined as vertex-adjacency, wheretwo triangles are considered to be adjacent if they shareat least one vertex. In [5], a triangulation is repre-sented with a vertex-facet incidence graph which hasa vertex f for each facet (triangle), a vertex v foreach triangle-vertex and an edge (v, f) whenever v isa vertex of triangle f . A facet cycle is defined bya walk (v0, f1, v1, f2, v2, . . . , fk, vk, f0, v0) where noarc is repeated and that includes each facet vertex ex-actly once, but may repeat triangle-vertices. The au-thors prove that any triangulation has a facet cycle ifit is not a checkered polygonal triangulation - that is ifit does not have a 2-coloring of the triangles such thatevery white triangle is adjacent to three black ones.

A similar result under the same facet cycle modelis found in [2]. Here, Bartholdi III and Goldsman re-fer to general triangulation as Triangulated IrregularNetworks (TINs). The authors describe an algorithm toconstruct a cycle in a 2-adjacent TIN (a triangulationin which each triangle shares an edge with at least twoother triangles). Their algorithm runs in O(n2) time inthe worst case.

In [4], Chen, Grigni and Papadimitriou define themap graph of a planar subdivision P (or a map) to be agraph G where the vertices of G correspond to the facesof P and two vertices u and v are adjacent if their corre-sponding faces in P share any point on their boundary.This characterization is equivalent to the dual graph ofa triangulation in which two vertices u and v of thedual are connected by an edge whenever the triangles

30

corresponding to u and v share a triangular edge or atriangular vertex. Chen et al. study sparsity and color-ing of map graphs.

Bartholdi III and Goldsman [3] introduce the sameconcept of a map graph that they call the vertex-adjacency dual of a general triangulation. The authorsshow that the vertex-adjacency dual contains a Hamil-tonian cycle, and they describe a linear time algorithmto construct such a cycle. Here we note that the modeldescribed in [3] is a variation of the facet-cycle modeldescribed in [5]: In the facet cycle model a continuouswalk enters every triangle from a vertex v and leavesfrom a different vertex u. In the vertex-adjacency dualmodel, u and v are allowed to be the same vertex. Thevertex-adjacency dual described in [3] can have O(n2)edges in the worst case. Here we consider linear sizesubgraphs of the vertex-adjacency dual that still containHamiltonian cycles, and that may be computed in lin-ear time. We call such graphs sparse vertex-adjacencyduals.

2 Constructing a Sparse Vertex-Adjacency Dual

Here we illustrate an approach with the simple case ofsequential triangulations [6].

Let P be a simple polygon with n vertices and let TP

be a sequential triangulation of P . We will refer to thevertices of P as polygonal vertices and to the verticesof the dual graph D of TP as dual vertices.

To construct the sparse vertex-adjacency dual ofTP , first construct its dual graph D, which in thiscase is a path. Then, for every polygonal vertexvP , if vP is shared by k > 2 consecutive trianglest1, t2, . . . , tk, insert an edge between the first andlast triangles t1 and tk. The resulting graph G is asparse vertex-adjacency dual. To show that it containsa Hamiltonian cycle, consider the following. In thedual graph D of TP every consecutive vertices u, v

and w are vertex-adjacent. Thus, connecting every u tow in the sparse vertex-adjacency dual is equivalent toconnecting the vertices which are at distance 2 apart.The resulting graph is known as the square of D. In[8], it is shown that if removing the leaves of a tree T

produces a path, then the square of T is Hamiltonian.In our case, our tree is the dual graph D, which willstill be a path if we remove its leaves. Thus, from theresults in [8] we can conclude that our construction fora sequential triangulation yields a Hamiltonian cycle(figure 1).

Figure 1: The sparse vertex-dual of a sequential tri-angulation is equivalent to the square D2 of the dualgraph D.

We can show that for any serpentine triangulation,the above construction will produce a graph that con-tains a Hamiltonian cycle. For general polygonal tri-angulations however, this construction needs a slightmodification in order to contain such a cycle while pre-serving adjacency information of the triangles.

References

[1] E. M. Arkin, M. Held, J. S. B. Mitchell, and S. S.Skiena. Hamiltonian triangulations for fast rendering.Visual Computing, 12(9):429–444, 1996.

[2] J. J. Bartholdi III and P. Goldsman. Multiresolu-tion indexing of triangulated irregular networks. IEEETransactions on Visualization and Computer Graphics,10(3):1–12, 2004.

[3] J. J. Bartholdi III and P. Goldsman. The vertex-adjacency dual of a triangulated irregular network hasa hamiltonian cycle. Operations Research Letters,32:304–308, 2004.

[4] Z. Chen, M. Grigni, and C. H. Papadimitriou. Mapgraphs. Journal of the ACM (JACM), 49(2):127–138,2002.

[5] E. D. Demaine, D. Eppstein, J. Erickson, G. W. Hart,and J. O’Rourke. Vertex-unfoldings of simplicial man-ifolds. In Proceedings of the eighteenth annual sym-posium on Computational geometry, pages 237–243,Barcelona, Spain, 2002.

[6] R. Flatland. On sequential triangulations of simplepolygons. In Proceedings of the 16

th Canadian Con-ference on Computational Geometry, pages 112–115,1996.

[7] M. Gopi and D. Eppstein. Single-strip triangulation ofmanifolds with arbitrary topology. In Computer Graph-ics Forum (EUROGRAPHICS), volume 23, 2004.

[8] F. Harary and A. Schwenk. Trees with hamiltoniansquare. Mathematika, 18:138–140, 1971.

[9] M. Isenburg. Triangle strip compression. In Proceed-ings of Graphics Interface 2000, pages 197–204, Mon-treal, Quebec, Canada, May 2000.

[10] G. Taubin and J. Rossignac. Geometric compressionthrough topological surgery. ACM Transactions onGraphics, 17(2):84–115, 1998.

31

Trees on Tracks∗

Justin Cappos and Stephen Kobourov

Department of Computer ScienceUniversity of Arizona

justin,[email protected]

1. Introduction

Simultaneous embedding of planar graphs is related tothe problems of graph thickness and geometric thickness.Techniques for simultaneous embedding of cycles havebeen used to show that the degree-4 graphs have geomet-ric thickness at most two [3]. Simultaneous embeddingtechniques are also useful in visualization of graphs thatevolve through time.

The notion of simultaneous embedding is related tothat of graph thickness. Two vertex-labeled planar graphson n vertices can be simultaneously embedded if thereexist a labeled point set of size n such that each of thegraphs can be realized on that point set (using the vertex-point mapping defined by the labels) with straight-lineedge segments and without crossings. For example, anytwo paths can be simultaneously embedded, while thereexist pairs of outerplanar graphs that do not have a simul-taneous embedding.

In this paper we present new results about embeddinglabeled trees and outerplanar graphs on labeled tracks,as well as related results on simultaneous embeddingof tree-path pairs. In particular, we show that labeledtrees cannot be embedded on labeled parallel straight-linetracks, but they can be embedded on labeled concentriccircular tracks; see Fig. 1. The results generalize to out-erplanar graphs as well. We also show that tree-path pairscan be simultaneously embedded when edges of the pathare represented by circular arcs. Finally, we show how toembed a straight-line tree and a path with O(log n)-bendsper edge, where n is the number of vertices.

1.1. Related Work

The existence of straight-line, crossing-free drawings fora single planar graphs is well known [5]. The exis-tence of simultaneous geometric embeddings for pairs ofpaths, cycles, and caterpillars is shown in [1]. Counter-examples for pairs of general planar graphs, pairs ofouter-planar graphs, and triples of paths are also pre-sented there. It it not known whether tree-tree or tree-pathpairs allow simultaneous geometric embeddings. If the

∗This work is partially supported by the NSF under grant ACR-0222920 and by ITCDI under grant 003297.

straight-line edge condition is relaxed, it is known how toembed tree-path pairs using one bend per tree edge andhow to embed tree-tree pairs using at most 3 bends peredge [4].

A related problem is the problem of graph thick-ness [7], defined as the minimum number of planar sub-graphs into which the edges of the graph can be parti-tioned into. Geometric thickness is a version of the thick-ness problem where the edges are required to be straight-line segments [2]. Thus, if two graphs have a simultane-ous geometric embedding, then their union has geomet-ric thickness two. Similarly, the union of any two planargraphs has graph thickness two. Simultaneous geometricembedding techniques are used to show that degree-fourgraphs have geometric thickness two [3].

Simultaneous drawing of multiple graphs is also re-lated to the problem of embedding planar graphs on afixed set of points in the plane. Several variations of thisproblem have been studied. If the mapping between thevertices V and the points P is not fixed, then the graphcan be drawn without crossings using two bends per edgein polynomial time [6]. However, if the mapping betweenV and P is fixed, then O(n) bends per edge are necessaryto guarantee planarity, where n is the number of verticesin the graph [8].

1.2. Our Contributions

We begin with results on track embeddability. Given aset of labeled parallel lines (tracks) Li, 1 ≤ i ≤ n and atree T = (V, E) with n vertices labeled with the numbers1 though n, it is not always possible to obtain a straight-line crossings-free drawing of T such that vertex vi is ontrack Li; see Fig. 1(a). However, if the tracks are con-centric circles, such drawings are always possible andwe describe a linear time algorithm for obtaining suchdrawings; see Fig. 1(b). The algorithm easily generalizesto outerplanar graphs as well. Thus, parallel line tracksdo not allow tree or outerplanar embeddings on predeter-mined tracks, while circular tracks do. Tracks defined bycircular arcs, stairs, sin-waves also suffice; see Fig. 2.

Our motivation for the problem of track embeddingscomes from two open problems in simultaneous geomet-ric embedding. Formally, in the problem of simultane-

32

0

1

2

3

4

5

6

7

8

9

7

6

54

3

2

10

98

Figure 1: A tree that cannot be drawn on line tracks without crossings but which can be drawn on circular tracks.

5

24 3

1 8 0 7 6

Figure 2: A tree drawn on various staircases: (a) staircase; (b) sin(x); (c) x sin(x); (d) x − bxc.

ous geometric embedding we are given two planar graphsG1 = (V, E1) and G2 = (V, E2) and we would like tofind plane straight-line drawings D1 and D2 such that forall vertices v ∈ V the location of the corresponding ver-tices in D1 and D2 is the same (i.e., Di(v) = Dj(v)).While path-path, cycle-cycle, caterpillar-caterpillar pairscan be simultaneously embedded, it is not known whethertree-tree or tree-path pairs have such embeddings.

The circular track layout of trees and outerplanargraphs can be used to obtain simultaneous embeddingsof tree-path pairs so that the tree edges are straight-lineand crossings-free and the path edges are crossings-freecircular-arc segments. Moreover, the staircase layout oftrees can be used to obtain simultaneous embeddings oftree-path pairs so that the tree edges are straight-line andcrossings-free and the path edges are crossings-free andhave at most lo g n bends per edge.

2. References

[1] P. Brass, E. Cenek, C. A. Duncan, A. Efrat, C. Erten, D. Is-mailescu, S. G. Kobourov, A. Lubiw, and J. S. B. Mitchell.On simultaneous graph embedding. In 8th Workshop onAlgorithms and Data Structures, pages 243–255, 2003.

[2] M. B. Dillencourt, D. Eppstein, and D. S. Hirschberg. Ge-ometric thickness of complete graphs. Journal of GraphAlgorithms and Applications, 4(3):5–17, 2000.

[3] C. A. Duncan, D. Eppstein, and S. G. Kobourov. Thegeometric thickness of low degree graphs. In 20th An-nual ACM-SIAM Symposium on Computational Geometry(SCG), pages 340–346, 2004.

[4] C. Erten and S. G. Kobourov. Simultaneous embedding ofplanar graphs with few bends. In 12th Symposium on GraphDrawing (GD). To appear in 2004.

[5] I. Fary. On straight lines representation of planar graphs.Acta Scientiarum Mathematicarum, 11:229–233, 1948.

[6] M. Kaufmann and R. Wiese. Embedding vertices at points:Few bends suffi ce for planar graphs. Journal of Graph Al-gorithms and Applications, 6(1):115–129, 2002.

[7] P. Mutzel, T. Odenthal, and M. Scharbrodt. The thicknessof graphs: a survey. Graphs Combin., 14(1):59–73, 1998.

[8] J. Pach and R. Wenger. Embedding planar graphs at fi xedvertex locations. Graphs and Combinatorics, 17:717–728,2001.

33

On the Non-Redundancy of Split Offsets in Degree Coding

Martin [email protected]

Jack [email protected]

University of North Carolina at Chapel Hill

AbstractThe connectivity coder by Touma and Gotsman encodesa planar triangulation through a sequence of vertex de-grees and occasional “ split” symbols that have an asso-ciated offset value. We show that the split offsets of theTG coder are not redundant by giving examples of degreesequences that have two different decodings if the splitoffsets are not specified. Surprisingly, such examples arerare and a large number of encodings remain unique.

1 Introduction

Recent years have seen a number of schemes that com-pactly encode triangle mesh connectivity by a sequenceof symbols that specify how to grow a “ compressionboundary” enclosing an already encoded region, one tri-angle at a time. A popular scheme is the Triangle MeshCompression method by Touma and Gotsman [7], or TGcoder for short. For planar triangulations, the TG codergenerates a sequence of vertex degrees that usually con-tains a few “ split” symbols with associated offset values.

There has been speculation that it might be possible tomodify the TG coder to operate without explicitly stor-ing the offsets values. The Cut-border Machine [3] andDual-Graph Method [5] explicitly include split offsets,but the otherwise identical Edgebreaker [6] and Face-Fixer [4] schemes avoid them, getting by only with the“ end” symbols in the code sequence. However, the TGcoder does not store explicit “ end” symbols. It maintainsmore state information on the compression boundary thanEdgebreaker or Face Fixer that—together with explicitoffsets—makes “ end” symbols implicit. But if we omitthe offsets we can find sequences with two valid decod-ings even if we add explicit “ end” s to the code sequence.

2 Connectivity coding with the TG Coder

To encode a triangulation, the TG coder [7] grows anencoded region, maintaining one or more compressionboundaries into which it includes triangle after triangle.It usually includes the triangle adjacent to the gate edge,which advances in clockwise order around the focus ver-tex. However, it immediately includes any triangle thatshares two edges with the compression boundary.

4

3

34 4

34

3

… S 4 3 E 4 3 Ei1 = 2

d1 = 7i2 = 2

d2 = 7

6 6 5 6 6 5 6 … 5 7 5 6 6 6 5 … 4 8 5 6 7 5 5 …

b = 6 s = 16

gate

focus

i = 4 d = 14

t = 12

Figure 1: The smallest scenarios where an offset-less encodingwith “ split” and “ end” symbols is not unique occur in triangula-tions of 11 vertices. The top right example, however, is unique.

The encoder starts with an compression boundary oflength two around an arbitrary edge, records the degree ofthe two initial boundary vertices, and sets their slot countto be one less than their degree. Whenever all boundaryvertices have a slot count of one or higher, the triangleadjacent to the gate shares only one edge with the com-pression boundary. Usually the third vertex of the trian-gle at the gate is a previously unprocessed vertex, andthe encoder simply “ adds” this vertex as a new bound-ary vertex and records its degree. Occasionally, the thirdvertex of this triangle is already on the boundary, and theencoder splits the compression boundary into two loops,temporarily stores one on a stack, and continues encodingon the other. Here the encoder records a split symbol andoffset, which is the number of slots that are on the bound-ary part that is stored on the stack, from which it can de-rive how many slots are clockwise along the boundarybetween the gate and the split vertex.

In the full paper we consider four offset-less encod-ings, each with less information about the split operationsthat occur during the encoding process. The strongest isif we omit the offset, but still record “ end” symbols thatmark the completion of a boundary loop. Figure 1 illus-trates the smallest examples for which this encoding isnon-unique: in triangulations with 11 vertices there are

34

two possible ways of splitting the boundary of length 6

that has its 1 6 slots distributed in this particular configura-tion and where both resulting boundary parts still enclosetwo unprocessed vertices of degree 3 and 4. However,there are not always two valid decodings in this scenario.Sometimes the focus is already connected to other ver-tices of the boundary through previously decoded trian-gles. This places additional constraints on the possibili-ties for splitting the boundary. The top-right triangulationfrom Figure 1, for example, has only one valid decoding.

Theorem. For the TG coder without split offsets, but withend symbols, there exist two different triangulations thathave the same offset-less encoding.

3 Searching for valid decodings

In order to find all valid decodings of an offset-less en-coding we search through all possibilities of performingsplit operations. For each attempt it recursively starts todecode the first boundary part and in case this is success-ful does the same for the second. Only if both recursionsare successful it returns a success, otherwise it tries outthe next possibility or returns a failure if there are noneleft. A few observations help us to immediately eliminatesome splits from further consideration.

Initial experiments seemed to indicate that the split off-sets of the TG coder might in fact be replaced by “ end” s.On our standard set of example meshes the search forsplit offsets would find the correct answer every run wetried. Table 1 shows that non-unique encoding are sur-prisingly rare. The full paper has further experimentsshowing that only a small fraction of random triangula-tions have non-unique encodings.

4 Closing discussion

There have been attempts to establish a guaranteed boundon the coding costs of the TG coder. However, the infre-quently occuring “ split” symbols and their offsets madethis a difficult task. Our work shows that these split off-sets are not completely redundant. There remains thetask of determining if any degree-based coder can avoidoffsets. Alliez and Desbrun [1] suggested an adaptivetraversal heuristic that lowered the number of split oper-ations and the remaining number of “ splits” seemed neg-ligibly small. Therefore the authors restricted their worstcase analysis to the vertex degrees. But Gotsman [2] hasshown that the entropy analysis of Alliez and Desbrun in-cludes many degree distribution that do not correspond toactual triangulations, and that there are fewer valid per-mutations of degrees than triangulations and that addi-tional information is necessary to distinguish between.So split information does contribute a small but neces-sary fraction to the encoding.

meshes splits non-uniquename vertices encodings min max avg encodings

cow 2,904 17,412 13 22 16.8 0fandisk 6,475 38,838 0 12 3.3 0horse 48,485 290,898 7 29 15.4 9dinosaur 56,194 337,152 27 56 40.4 10rabbit 67,039 402,222 0 27 9.0 56armadillo 172,974 1,037,832 36 76 55.2 146

Table 1: We show several meshes with their histograms ofsplits. The table lists numbers of vertices and encodings, splitstatistics, and number of non-unique encodings for each mesh.

References[1] P. Alliez and M. Desbrun. Valence-driven connectivity en-

coding for 3D meshes. In Proceedings of Eurographics’01,pages 480–489, 2001.

[2] C. Gotsman. On the optimality of valence-based connec-tivity coding. Computer Graphics Forum, 22(1):99–102,2003.

[3] S. Gumhold and W. Strasser. Real time compression of tri-angle mesh connectivity. In Proceedings of SIGGRAPH’98,pages 133–140, 1998.

[4] M. Isenburg and J. Snoeyink. Face Fixer: Compressingpolygon meshes with properties. In Proceedings of SIG-GRAPH’00, pages 263–270, 2000.

[5] J. Li and C. C. Kuo. A dual graph approach to 3D triangularmesh compression. In Proceedings of ICIP’98, pages 891–894, 1998.

[6] J. Rossignac. Edgebreaker: Connectivity compression fortriangle meshes. IEEE Transanctions on Visualization andComputer Graphics, 5(1):47–61, 1999.

[7] C. Touma and C. Gotsman. Triangle mesh compression. InProceedings of Graphics Interface’98, pages 26–34, 1998.

35

Fast almo st-linear-sized nets fo r b o x es in the plane

Herv e B ro nnima nn∗

Jo na tha n L enchner

CIS, P o ly technic Univ ersity , Six metro tech, B ro o k ly n, NY 1 1 2 0 1 , USA.

1 Intro ductio n

Let B b e any set of n ax is-aligne d b ox es in Rd, d ≥

1. F or any p oint p, w e d e fi ne th e su b se t Bp of B asBp = B ∈ B : p ∈ B. A b ox B in Bp is saidto b e stabbed b y p. A su b set N ⊆ B is a (1/c)-net

for B if Np 6= ∅ for any p ∈ Rd su ch th at | Bp | ≤ n/c.

T h e nu m b e r of d istinct su b se ts Bp is O((2n)d), so th ese t sy ste m d e scrib e d ab ov e h as so-calle d fi nite V C -d im ension d. T h is ensu res th at th e re alw ay s e x ists(1/c)-nets of siz e O(dc log(dc)), and th at th e y can b efou nd in tim e Od(n)cd, u sing q u ite general m ach inery(se e for e x am p le th e b ook s b y M atou se k [3] or b yP ach and Agarw al [7]). F or som e set sy ste m s, su chas h alfp lanes in R

2 and translates of a sim p le c lose dp oly gon, it w as sh ow n th at th e re e x ist (1/c)-nets ofsiz e O(c) [4]. T h is w as e x tend e d to h alfsp ac es in R

3

and p se u d o-d isk s1 in R2 [2].

In th is p ap er, w e inv estigate a fast, O(n log c)-tim econstru c tion of (1/c)-nets of siz e O(c) for any v alu e1 < c ≤ n and d = 2. Until righ t b e fore J C DC G , Ith ou gh t I cou ld p rov e th e follow ing (w h ich u nfortu -nate ly re m ains a conje c tu re):

Conjecture 1 L et B be a set of n axis-aligned boxes

in R2 and c be any param eter 1 ≤ c ≤ n. T hen there

exists a (1/c)-net N for B of size O(c).

W e can p rov e th is resu lt and also p rov id e algorith m sth at ru n in tim e O(n log c) only for sp e c ial cases: seg-m ents on th e real line (th e one-d im ensional case),q u ad rants of th e form (−∞, x] × (−∞, y ] in R

2, andu nb ou nd e d b ox es of th e form [x1, x2]×(−∞, y ] (w h ichw e call a sky line). F or th e general case of b ox es, w ecan p rov e a siz e b ou nd of O(c log log c). B u t th e con-je c tu re still stand s.

∗Resea rch o f th is a u th o r h a s b een su p p o rted b y NS F C A-

RE E R G ra n t C C R-0 1 3 3 5 9 9 .1In th is c o n tex t, a c o llec tio n o f sh a p es is c a lled a p seu d o -

d isk set sy stem if g iv en a n y th ree p o in ts, th ere is a t m o st o n e

sh a p e in th e c o llec tio n w h o se b o u n d a ry p a sses th ro u g h th ese

th ree p o in ts.

2 Interv als o n the line

W e fi rst p rov e th at it is easy to fi nd sm all nets forinterv als on th e line , th e one -d im ensional case of th ep rob le m ab ov e .

Theorem 2 L et B be a set of n intervals on the real

line R and c be any param eter 1 < c ≤ n. T here

exists a su bset N of at m ost 2dc − 1e boxes in B that

is a (1/c)-net for B. S u ch a set can be fou nd in O(n+n log c) tim e.

3 Rectang les in the plane

W e generaliz e th e m e th od of th e p re v iou s p aragrap hto th e p lane . W e b egin w ith th e easier p rob le m w h enall th e b ox es are sou th -w e st q u ad rants, i.e . th e y con-tain th e p oint (−∞,−∞).

Theorem 3 L et B be a set of n qu adrants w ith the

sam e orientation in R2, and c be any param eter 1 <

c ≤ n. T hen there exists a (1/c)-net N for B of size

dc − 1e. S u ch a net can be fou nd in tim e O(n log c).

Let u s ad d one m ore sid e to th e q u ad rants: a sky line

is a set of b ox es th at all interse c t a com m on line . W eare only intereste d in w h at h ap p ens on one sid e ofth at line , so w e can consid e r u nb ou nd e d b ox es of th eform [x1, x2] × (−∞, y ]. W e can ex tend th e p re v iou sresu lt to a sk y line .

Theorem 4 L et B be a set of n axis-aligned boxes,

all u nbou nded in som e com m on direction, and c be

any param eter 1 < c ≤ n. T hen there exists a (1/c)-net N for B of size at m ost d2c − 1e. S u ch a net can

be fou nd in tim e O(n log c).

Using th is resu lt, w e can now solv e th e general p rob -le m for b ox es. Unfortu nate ly , w e cannot solv e th econje ctu re , b u t w e can p rov e :

36

Theorem 5 Let B be a set of n axis-aligned boxes inR

2 and c be any param eter 1 < c ≤ n. T hen thereexists a (1/c)-net N for B of size O(c log log c). S u cha net can be fou nd in tim e O(n log c).

4 k-o riented o b je cts

A natu ral generalization of b ox es in R2 is th at of a k-

oriente d conv e x p oly gon [6], w h ich is sim p ly a conv e xp oly gon w h ose sid e s are constraine d to b e p aralle l toa set of k fi x e d d ire c tions (k = 2 for b ox es). O u rp roof e x tend s th e re as w e ll. In fact, w e su sp e c t th atany resu lt for b ox es w ou ld e x tend to k-oriente d p oly -gons w h e re th e constants in th e O() notations b e com efu nctions of k. B u t w e h av e no p roof of su ch a generalstate m ent.

5 O rthants in hig he r dim ensio n

As sh ow n in th e p re v iou s se c tion, th e k e y p rob le m isth at for th e generaliz e d orth ants, w h ich w e call or-th ants for sh ort. W e are now intereste d in th is p rob -le m for any d im ension. W e fi rst p rov e th at fi nd ingnets for orth ants d oes ind e e d h e lp for b ox es.

Theorem 6 A ssu m e that there exists ε-nets of sizes(ε) for any set of orthants in R

d, and that s() is non-decreasing and has polynom ial grow th (w hich im pliess(O(x)) = O(s(x))). Let B be a set of n boxes in R

d

and c be any param eter 1 ≤ c ≤ n. T hen there existsa (1/c)-net N for B of size Od(s(1/c)). S u ch a netcan be fou nd in tim e O(n log c).

Now w e sh ow h ow sm all nets w e can fi nd e ffi c ientlyfor orth ants. T h e b ou nd is far from op tim al for d i-m ensions greater th an 2, as nets of siz e Od(c log c)e x ist b u t tak e m u ch longer to com p u te (se e th e intro-d u c tion).

Theorem 7 Let B be a set of n orthants w ith thesam e orientation in R

d and c be any param eter 1 ≤c ≤ n. T hen there exists a (1/c)-net N for B of sizeOd(c

d−1). S u ch a net can be fou nd in tim e O(n log c).

F or p oints and h alfsp ac es in Rd, M atou se k , S e id e l,

and W e lz l [4] h av e sh ow n th at th ere e x ist ε-nets ofsiz e O(1/ε). T h e y also sh ow th at it su ffi c e s to restric tto p oints in conv e x p osition, alb e it b y h av ing netsb igger b y a factor of d. W e p rov e an analog resu lt fororth ants, w ith ou t th e b low u p factor. T h e analogu eof conv e x p osition for orth ants is m ax im al p osition,as d e fi ne d in [8].

L em m a 8 S u ppose there exists a ε-net of size s(ε)for any set of orthants in R

d in m axim al position.T hen there exists an ε-net for any set of orthants inR

d of size s(ε).

6 C o nclusio n

T h is sh ow s anoth er set sy ste m w h e re th e generalb ou nd O(c log c) for a (1/c)-net cou ld b e im p rov e dto O(c), and m ore e ffi c ient algorith m s can b e fou nd .K om los, P ach and W oeginger [2] h av e sh ow n th atth e re e x ist set sy ste m s for w h ich (1/c)-nets m u st h av esiz e Ω (c log c).T h is also p oses th e analog p rob le m of fi nd ing goodap p rox im ations, in th e sense th at not only d oes p h itfe w b ox es if it m isses N , b u t th e nu m b e r of h its in Nre fl e c ts th e nu m b e r of h its in B (scale d b y | N | / | B| ).T h e ap p roach ab ov e se e m s to collap se b e cau se noth -ing gu arante es th e re p re sentativ ity of N .

Refe rence s

[1] J. P ach and P .K . Agarw al. C om binatorial G e-om etry . J. W ile y , New Y ork , 1995.

[2] J. K om los, J. P ach , G .J. W oeginger. Alm ostT igh t B ou nd s for e p silon-Nets. Discrete & C om -pu tational G eom etry 7:163–173, 1992.

[3] J. m atou se k . Lectu res on Discrete G eom etry .S p ringer, B e rlin, 2002.

[4] J. M atou se k , R . S e id e l and E . W e lz l. How tonet a lot w ith little : sm all ε-nets for d isk s andh alfsp ac es. In P roceedings of the S ixth A nnu alS y m posiu m on C om pu tational G eom etry , p .16-22, Ju ne 07-09, 1990, B e rk e le y , C alifornia.

[5] F rank Nie lsen, F ast S tab b ing of B ox es in HighDim ensions. T heoretical C om pu ter S cience, E l-se v ie r S c ienc e , V olu m e 246, Issu e 1-2, 2000.

[6] F . Nie lsen. O n P oint C ov ers of c -O riente d P oly -gons. T heoretical C om pu ter S cience, E lse v ie rS c ienc e , V olu m e 265, Issu e 1-2, p p . 17-29, 2001.

[7] J. P ach and P .K . Agarw al. C om binatorial G e-om etry . J. W ile y , New Y ork , 1995.

[8] F . P re p arata and M .I. S h am os. C om pu tationalG eom etry . S p ringer, New Y ork , 1985.

37

The Metric TSP and the Sum of its Marginal V alues

Moshe Dror∗

Y usin Lee†

James B. Orlin ‡

August 31, 2004

Abstract

This paper examines the relation between the length of an optimal Traveling Salesman tourand the sum of its nodes’ marginal values (a node’s marginal value is the difference between thelength of an optimal TSP tour over a given node set and the length of an optimal TSP tour overthe node set minus the node). To our knowledge, this problem has not been studied previously. W efind that in metric spaces L1,L4/3,L2,L4,L∞, the event in which the sum of TSP marginal valuesis greater than the length of the optimal tour is very rare. W e present a number of cases for whichthe sum of marginal values is never greater that the optimal tour. W e establish a worst case ratioof 2 for any metric TSP. In addition, for 6 node TSPs we determine the worst ratio for L1,L∞

norms, triangular inequality, and symmetric distance, of 10/9, 10/9, 1.2, and 1.5 respectively, bysolving the appropriate mixed integer programming problems.

∗MIS Department, College of Business and Public Administration, University of Arizona, T ucson, Arizona 85721,

and Operations Research Center, Massachusetts Institute of T echnology , 77 Massachusetts Avenue, Cambridge,

Massachusetts 02139, U.S.A.†Operations Research Center, Massachusetts Institute of T echnology , 77 Massachusetts Avenue, Cambridge,

Massachusetts 02139, U.S.A. and Department of Civil Engineering, National Cheng Kung University , T ainan, T aiw an.‡Sloan School of Management, Massachusetts Institute of T echnology , 77 Massachusetts Avenue, Cambridge,

Massachusetts 02139, U.S.A.

38

Optima l A re a Tria n g u la tio n w ith A n g u la r C o n stra in ts

J. M a rk K e il, T zveta lin S . Va ssilev

D e p a rtm ent o f C o m p u te r S c ienc e , U nive rsity o f S a ska tch e w a n

5 7 C a m p u s D rive , S a ska to o n, S a ska tch e w a n, S 7 N 5 A 9 C a na d a

e -m a ils: ke il@ c s.u sa sk.c a , tsv5 5 2 @ m a il.u sa sk.c a

1 8 O c to b e r 2 0 0 4

Extended A b stra c t

1 In tro d u c tio n

We stu d y triangu lations of planar sets of points. T he problem of interest is to optim ize the area of theind ivid u al triangles in the triangu lation, or to fi nd the M inM ax and M axM in area triangu lations of the givenpoint set. T hese two problem s ad m it polynom ial tim e algorithm s in the case when the point set is a convexpolygon. In the general case, the problem s are of u nknown com plexity. T he problem of fi nd ing the M inM axarea triangu lation is m entioned as a hard open problem in E d elsbru nner’s book [2]. In this paper we d iscu ssan approach for approxim ating these two optim al area triangu lations with a triangu lation that has angu larrestrictions im posed . T his resu lts in a su bcu bic algorithm for fi nd ing the approxim ating triangu lation. T heapproxim ation ratio d epend s on angu lar param eters, analytical resu lts on this are shown.

2 A n g u lar re stric tio n s an d fo rb id d e n z o n e s

S u ppose the optim al area triangu lation (either M axM in or M inM ax area) contains sm all angles, which isknown to be u nsu itable for practical pu rposes. D enote the sm allest angle in the optim al triangu lation (whichwe d on’t know) as β. We call triangu lation that has all of its angles larger than β a β -triangu lation. We wantto constru ct an α-triangu lation, which is ”fatter” (α > β), that approxim ates the optim al with a practicalcoeffi cient. G iven the fact that all of the angles of the α-triangu lation are going to be larger than α, we cand efi ne a region su rrou nd ing each ed ge of the triangu lation, called forbid d en zone. T he forbid d en zone of aned ge is by d efi nition a region that is em pty of points of the original point set if the ed ge is in the triangu lation.In these circu m stances, the forbid d en zone is a polygonal region, recu rsively d efi ned by ad d ing to the ed geisosceles triangles with a base the ed ge itself, and base angles of α, and continu ing this process ou tward s ofthe alread y tiled area infi nitely. F irst fou r steps are shown in F igu re 1. T he param eters of the forbid d enzone are fu lly d eterm ined by the length of the ed ge a and the angle α. T he forbid d en zone entirely contains atrapezoid with the given ed ge as a base, base angles of 3α and heighth of (a/2) tanα. T he zone also entirelycontains a circle su rrou nd ing each of the end points of the ed ge. P lease refer to F igu re 2 for an illu stration.

F igu re 1: Recu rsive constru ction of the forbid d en zone

39

Figu re 2: The bord er of the forbid d en zone u p to third ord er

3 Pe rfe c t m atch in g s an d bo u n d s

To obtain the bou nd s for approximation factors, we u se the perfect matching between the β -triangu lationand the α-triangu lation, as d escribed in A ichholzer’s paper [1]. We stu d y all the possible cases of matchedtriangles and positions of the points with respect to the forbid d en zones. B ased on this we d erive the followingbou nd s of the approximation factors for the M axM in area triangu lation:

f1 = m a x

(

1

tanα tan2 β tan β

2

,1

sinα tan2 β,2(1 + k)(1 + 2k)

tanα,

k2

sinα

)

and for the M inM ax area triangu lation:

f2 = m a x

(

1

tanβ tan2 α tan α2

,1

sinβ tan2 α,

1

2(1 − k)(1 − 2k) sinβ tan α2

,1

k2

1sin 2β

)

where k, k1 and n are the following parameters of the forbid d en zone:

k =tanα

2 sin 3αk1 = 1

2n co sn αn =

⌈

180

2α−

1

2

⌉

The approximation factor f1 shows how many times the smallest area triangle in the approximating α -triangu lation is smaller than the smallest area triangle in the optimal (M axM in area) triangu lation. S imilarly,f2 gives the ratio of the largest area triangle in the approximating triangu lation, compared to the largestarea triangle in the optimal (M inM ax area) triangu lation.

4 A lg o rith m ic re su lts an d sam p le valu e s

B ased on the fact that we can compu te the optimal 30-triangu lation (if it exists) by mod ifi ed K lincsek ’salgorithm, or we can relax D elau nay by area eq u aliz ing fl ips, we achieve a su bcu bic algorithm that approx-imates the optimal area triangu lations, by the above given factors. The valu e of α can be chosen frompractical consid erations. H ere are some sample resu lts, su mmarized in a table:

α 30 30 25 25 20 20 15β 25 20 20 15 15 10 10f1 35.930 74.149 91.807 226.87 290.66 1010.1 1372.0f2 24.010 30.716 56.994 77.418 311.28 455.06 7900.1

Table 1: S ample valu es for f1 and f2

Refe re n c e s

[1] O . A ichholzer, F. A u renhammer, S .-W. C heng, N . K atoh, G . Rote, M . Taschwer, and Y .-F. X u . Trian-gu lations intersect nicely. Discrete a n d C o m p u ta tio n a l G eo m etry, 16(4):339–359, 1996.

[2] H . E d elsbru nner. G eo m etry a n d To po logy fo r M esh G en era tio n . C ambrid ge U niversity P ress, U K , 2001.

40

Detectin g Du p lica tes Am o n g Sim ila rBit V ecto rs

(o f co u rse, w ith g eo m etric a p p lica tio n s)

Bo ris Aro n o v1 a n d J o h n Ia co n o 2

Abstra c t We sh ow h ow to d etect d u p licates in a

sequ ence of k n-bit vectors p resented as a list of

single-bit ch anges betw een consec u tive vectors, in

O((n + k) log n) tim e .

Proble m We are given a sequ ence S = v1, . . . ,vkof k n-bit vectors, presented as follow s: The first bit

vector is all zeros and each su bsequ ent vector vi is

obtained from the previous vector vi−1 by flipping a

single b it in p osition bi, 0 ≤ bi < n. S is re p re sente d

as b2,b3, . . . ,bk. The problem is to detect duplicates

in the seq uence v1,v2, . . . ,vk. More formally , we seek

a labeling S → 1, . . . ,k, vi 7→ ci, su ch th at ci = cj

iff vi = vj .

Solution Without loss of generality in the remaind er

of this note we assume that n is a p ower of two. L et T

b e the p erfectly b alanced b inary tree on n leaves. We

numb er the leaves of T from 0 to n − 1 and assoc iate

each with a b it p osition. Each interior nod e x of T is

similarly associated with a block B(x) of consecutive

bit positions corresponding to the leaves of the sub-tree rooted at x. F or a bit vector vi, let vi(x) be itssubstring in B(x). The idea behind our data struc-ture is simple: each node x has an associated datastructure that stores implicitly the set

⋃k

i=1vi(x).The data structure stored at node x consists of two

array s Dx an d Fx th at store th e follow in g d ata:

• Dx[1, . . . , d x] con tain s th e sorte d se t in c lu d in g 1and all distinct values i, 1 < i ≤ k, such thatvi−1(x) 6= vi(x).

• Fx[1, . . . , d x] contains integers in the range

1, . . . , d x with the property that Fx[i] = Fx[j]iff vDx[i](x) = vDx[j](x).

1Research su p p o rted in p art by NS F IT R Gran t CCR-00-

819 64 and b y a grant from U S-Israe l B inational Sc ienc e F oun-

dation; part of w ork has b een carried out w hile v isiting M ax -

Planck-Institut fur Informatik. D ep artment of C omp uter and

Information Science, P oly technic U niversity , 5 M etroTech Cen-

ter, Bro o k ly n , NY 1 1 2 0 1 U S A; h ttp :/ / c is.p o ly .e d u / ˜ a ro n o v .2Research su p p orted in p art by NSF g rant CCF-0430849.

Department of C omputer and Information Science, P oly technic

University, 5 M etroTech Center, Brook lyn, NY 11201 USA;

h ttp :/ / jo h n .p o ly .e d u .

We now complete th e d escription of our algorith m,

by explaining h ow to initialize Dz and Fz for all

leaves z ∈ T and h ow to compute Dx and Fx from

Dl,Fr,Dl,Fr for any internal node x with ch ildren l

an d r. Froot d e scrib e s th e d e sire d lab e lin g of S, sin ce

Droot con tain s all th e n u m b e rs 1, . . . , k .

If on e store s all of th e leave s z in an array in n u m eri-

cal ord er, a lin ear scan of th e se q u e n ce b2,b3, . . . ,bk of

bit up dates allows one to initialize arrays Dz and Fz,

for all z. S p ec ifically , w e store th e c u rren t bit vector

vi−1 explicitly in a bit array V [0, . . . , n − 1]. Since

bi = j ind icates a bit fl ip in position z = j (recallthat bit p ositions, and thus leaves are identified with

integers 0, . . . , n − 1), we fl ip the valu e of V [j], add j

to Dz, and d epend ing on th e resulting value of V [j],set th e n ex t en try in Fz to zero or on e.

Algorithm 1 The pseu docode for com pu ting Dx,Fx

from Dl,Fl,Dr,Fr.

1: i← j ← k ← 12 : Dl[dl + 1]← DR[dr + 1]←∞3 : rep eat

4 : Px[k]← (Fr(i),Fl(j),k,0)5 : if Dl[i] < Dr[j] then

6 : Dx[k]← Dl[i]; i← i + 17 : else if Dl[i] = Dr[j] then

8 : Dx[k]← Dl[i]; i← i + 1; j ← j + 1;9 : else if Dl[i] > Dr[j] then

10 : Dx[k]← Dr[j]; j ← j + 1;11: end if

12 : k ← k + 1;13 : until i = dl + 1 and j = dr + 114 : dx ← k − 1 . dx is th e len gth of Px an d Dx

15 : Sort Px lexicograph ically on th e first tw o field s,b y rad ix sort

16 : for k ← 2 to dx do

17 : if Px[k − 1][1] = Px[k][1] and Px[k − 1][2] =Px[k][2] then

18 : Px[k][4]← Px[k − 1][4]19 : else

2 0 : Px[k][4]← Px[k − 1][4] + 12 1: end if

2 2 : end for

2 3 : for k ← 1 to dx do

2 4 : Fx[Px[k][3]]← Px[k][4]2 5 : end for

N ow, we describ e, for an internal node x of T withchildren l and r, how to constru ct Dx,Fx from arrays

Dl,Dr,Fl,Fr; see Algorithm 1. The new sorted arrayDx[1, . . . ,dx] is built by merging th e arrays Dl and

Dr, eliminating any d u p licates, in time O(dl + dr) =

41

O(dx). Simultan eously, w e create an aux iliary arrayPx[0, . . . ,dx] th at records h ow th e merge step h as

proceeded; it con sists of quadruples of items. R eferto lin e s 1–13. W e c learly h av e

Lem m a 1. The array Dx as co nstructed is co rrect.

The first two colum ns of the array Px contain pairs of

integers in the range 1, . . . ,dx with the property that

(Px[i][1],Px[i][2]) = (Px[j][1],Px[j][2]) iff vDx[i](x) =vDx[j](x). T he third colu m n ju st n u m bers the row s of

Px con secu tively .

The array Px contains all the inform ation w e need,

in a sense, b u t not in the right ord er. At this p oint we

radix -sort Px according to the first tw o fields, in tw o

passes. As Px is of size dx and each of these fields is

a p ositiv e integer no larger than dx, this tak es O(dx)time. In lines 16–22, w e nu mb er th e rearranged lines

of Px consec u tiv ely, ignoring d u p licate p airs in the

first tw o colu mns. T hese integers w ill b e u sed to fill

in Fx and are gu aranteed to b e in the range 1, . . . ,dx.

This tak es time O(dx). W e finally fill the array Fx

u sing the d ata in Px, as d etailed in lines 23–25. A

few m inu tes of contem p lation will conv ince the read er

th at th e follow in g le m m a h old s.

Lem m a 2. The array Fx co ntains integers in the

ran ge 1, . . . ,d with the property that Fx[i] = Fx[j] iff

vDx[i](x) = vDx[j](x).

Application Suppose one is interested in com puting

an arrangement of n simple shapes (such as disks,triangles, or halfplanes) in R

2. T here are plenty of al-gorithms, inc luding deterministic ones, that can solvethis problem in O(n2 log n) time for a variety of shapes.But what if, in addition to the face structure of the

arrangement, one is interested in labeling the faces

with the bit vector indicating which of the objectseach face belongs to? Clearly, an explicitly storedlab e lin g is too e x p e n siv e , re q u irin g Θ (n3) b its in th e

worst case. H owever, traversing the arrangement byan Eulerian path of the face incidence graph allows

one to encode the bit vectors using single-bit fl ips

between consecutive vectors along the path. In par-ticular using the algorithm described above, we can

detect which faces correspond to identical vectors and

thus are contained in identical sets of shapes. T he

process tak es O(n2 log n) time. An entirely analogous

process can process an arrangement of n objects inany dimension, traversing k cells in O((n + k) log n)time, provided adjacent cells diff er only in a single con-tainment and an adjacency structure encoding local

b it d iff e re n c e s is av ailab le .Note th at th e assu m p tion th at th e fi rst bit v ector

in the seq uence is all zeros can be dropped withoutchanging our algorithm. Also observe that our algo-rithm can be used with slight modifi cations to de-tect duplicates among bit vectors coming from severalsequ en ces—on e ju st n eed s to artificially con caten ate

the seq uences together by using dummy intermediatevectors, if the number of seq uences is small. T his willresult in an additional O(n log n) cost per concate-nation. A less brute-force approach to dealing withmultiple seq uences which results in a O(n) concatena-tion cost will be described in the full version of thisp ap e r.

A geometric application with two sequences of bit

v e c tors w as p re se n te d in [1].

Refe re n c e

[1] P.K. Agarwal, B. Aronov, V. Koltun, “Effi-cient Algorithms for Bichromatic S eparability,”man u scrip t, 2004. Pre limin ary v e rsion ap p eare d

in Proceedin gs 15th An n u al ACM-SIAM Sym po-

sium on Discrete Algorithm s (SODA 2004), 2004,

pp. 675–683.

42

Approx ima tion Alg orith ms for T wo O ptima l Loc a tion Prob lems in

S e n sor N e twork s

Alon Efra t Sa riel H a r-Peled J oseph S. B . Mitchell

Abstract

This paper stu d ies two problem s that arise in optim ization of sensor network s: F irst, wed evise provable approx im ation schem es for locating a base station and constru c ting a networkam ong a set of sensors each of which has a d ata stream to g et to the base station. S u bject topower constraints at the sensors, ou r g oal is to locate the base station and establish a networkin ord er to m ax im ize the lifespan of the network .

S econd , we stu d y optim al sensor placem ent problem s for q u ality coverag e of g iven d om ainsc lu ttered with obstac les. U sing line-of-site sensors, the g oal is to m inim ize the nu m ber of sensorsreq u ired in ord er to have each point “ well covered ” ac cord ing to prec ise c riteria (e.g ., that eachpoint is seen by two sensors that form at least ang le α, or that each point is seen by three sensorsthat form a triang le containing the point).

1 Intro d u c tio n

Consider a wireless sensor network with a larg e nu m ber of deployed sensors, each captu ring dataon a continu ou s basis. T he sensors m ay be captu ring video data, au dio data, environm ental data,etc . T here is a base station that collects all of the data stream s from all of the sensors. E ach sensorpasses data packets along som e rou te in a network, from sensor to sensor, so that all data arrivesat the base station. S ince each sensor is g enerally powered by som e form of battery, the du rationof the sensor node is determ ined in larg e part by its power dissipation rate and energ y provision.A fu ndam ental issu e assoc iated with wireless sensor networks is m ax im iz ing their u sefu l lifetim e,g iven their power constraints. T his can be sig nifi cantly aff ected by the location of the base stationas well as the forwarding protocols u sed (i.e. which sensor forwards packag es of which other sensor)in establishing the network. H ou 1 has su g g ested the u se of the leng th of tim e u ntil the fi rst sensorex hau sts its battery as a defi nition of the lifespan of the system . In the paper, we show how tofi nd a location of the base station su ch that the lifespan of the system is optim ized to within anydesired approx im ation bou nd. S pec ifi cally, we g ive a m ethod for locating the base station thatprovably obtains a lifespan of at least (1 − ε) tim es that of the optim al lifespan, where ε > 0 isany pre-determ ined fi x ed valu e. T he alg orithm is based on solving O(n ε−4 log 2(n / ε)) instances ofa linear prog ram m ing problem . It is sim ple and easy to im plem ent.

Th eorem 1.1 G iven a set of sensors S = s1, . . . , sn and a param eter ε > 0, one can com pu te

a location of the base station and a transm ission schem e su ch that the netw ork lifespan is at least

(1 − ε)to pt, w here to pt is the lifespan of an optim al transm ission schem e for S. T his algorithm s

requ ires M = O(n ε−4 log 2(n / ε)) preprocessing tim e, and needs to solve M instances of linear

program m ing.

1Tho m a s H o u, perso na l c o m m unic a tio n, 2 0 0 3 .

43

We also stu d y another optim al location problem in sensor network s: H ow d oes one chooselocations of sensors for line-of-sig ht coverag e of a g iven reg ion, u nd er the assu m ption that a point isonly “ covered ” if it is “ well seen” ? We consid er two d efi nitions of “ well seen” : A point p is well seen(or robu stly covered) if either (a) there are two sensors that see p, and these sensors are separatedby ang le at least α with respect to p; or (b) there are three sensors that see p and they form atriang le that contains p. T he objective is to m inim ize the nu m ber of sensors to achieve robu stcoverag e. O u r resu lts on this problem g ive effi cient approx im ation alg orithm s for m inim iz ing thenu m ber of sensors.

Th eorem 1.2 G iven P and Q as above, an angle α, and a grid Γ of edge-length δ inside P , w e

can fi nd a set G of sensors in P su ch that G 2-gu ards Q at angle α/2, and |G| = O(kopt log kopt),w here kopt is the cardinality of smallest set of vertices of Γ that 2-gu ard Q at angle α. T he ru nning

time of the algorithm is O(nk4opt log 2 n log m), w here m is the nu mber of vertices of Γ ∩ P .

Th eorem 1.3 D eciding if k sensors w ithin P su ffi ce to triangle-gu ard Q is N P -hard.

Th eorem 1.4 G iven P and Q as above, and a grid Γ of edge-length δ, w e can fi nd a set G of sensors

in P , that triangle-gu ard Q, w ith |G| = O(kopt log kopt), w here kopt is the cardinality of smallest set

of vertices of Γ that triangle-gu ard Q. T he ru nning time of the algorithm is O(nk2opt log 2 n log m),

w here m is the nu mber of vertices of Γ ∩ P .

T his second problem is a variant of the classical art g allery problem , in which one is to place thefewest sensors (“ g u ard s” ) to see all points of a certain g eom etric d om ain. Art g allery problem s havebeen stu d ied ex tensively; see, e.g ., [K ei0 0 , U rr0 0 ] for recent su rveys. T he alg orithm ic problem ofcom pu ting a m inim u m nu m ber of g u ard s is k nown to be N P -hard , even if the inpu t d om ain, D, is asim ple polyg on. T hu s, eff orts have concentrated on approx im ation alg orithm s for optim al g u ard ingproblem s. R ecently, researchers [E H 0 2, G L0 1 ] have applied set cover m ethod s that ex ploit fi nitenessof VC -d im ension. In particu lar, E frat and H ar-P eled [E H 0 2] obtain an O(log k∗)-approx im ationalg orithm for g u ard ing a polyg on with vertex g u ard s, u sing tim e O(n(k∗)2 log 4 n), where k∗ is theoptim al nu m ber of vertex g u ard s. C heong et al. [C E H 0 4 ] have recently shown how to com pu tek g u ard s in ord er to optim ize (approx im ately) the total area seen by the g u ard s. T he triang le-g u ard ing coverag e problem we stu d y is related to recent work of S m ith and E vans [S E 0 3 ].

Refe re n c e s

[CEH04] O . Cheo ng, A. Efrat, and S. Har-Peled. O n finding a guard that se es mo st and a sho p that se lls mo st.In Proc. 1 5 th AC M -SIAM Sy m pos. D iscrete Algorithm s, pages 1 09 1 – 1 1 00, 2 004.

[EH02 ] A. Efrat and S. Har-Peled. L o cating guards in art gallerie s. 2 nd IFIP Interna t. C onf. T heo. C om p . Sci.,pages 1 8 1 – 1 9 2 , 2 002 .

[G L 01 ] H. G o nz ale z -B ano s and J .-C. L atombe . A randomized art-gallery algo rithm fo r senso r plac ement. InProc. 1 7 th Annu. AC M Sy m pos. C om p ut. G eom ., pages 2 3 2 – 2 40, 2 001 .

[K e i00] J . M . K e il. Po lygo n dec o mpo sitio n. In J .-R . Sack and J . U rrutia, edito rs, H a ndbook of C om p uta tiona l

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46

Computational Geometric Aspects of Musical Rhythm

Godfried Toussaint∗

McGill University

Montreal, Quebec, Canada

For our purpose a rhythm is represented as acyclic binary string. Consider the following three12/8 time ternary rhythms expressed in box-like nota-tion: [x . x . x . x . x . x .], [x . x . x x . x . x . x]and [x . . . x x . . x x x .]. Here “x” denotes thestriking of a percussion instrument, and “.” denotesa silence. It is intuitively clear that the first rhythmis the most even (well spaced) of the three. Tradi-tional rhythms have a tendency to exhibit such prop-erties of evenness. Therefore mathematical measuresof evenness find application in the new field of math-ematical ethnomusicology [2], [17], where they mayhelp to identify, if not explain, cultural preferences ofrhythms in traditional music.

Clough and Duthett [3] introduced the notion ofmaximally even sets with respect to pitch scales rep-resented on a circle. Block and Douthett [1] went fur-ther by constructing several mathematical measuresof the amount of evenness contained in a scale. One oftheir measures simply adds all the interval arc-lengths(geodesics along the circle) determined by all pairs ofpitches in the scale. However, this measure is toocoarse to be useful for comparing rhythm timelinessuch as those studied in [13] and [15]. Using inter-val chord-lengths (as opposed to geodesic distances),proposed by Block and Douthet [1], yields a more dis-criminating measure, and is therefore a function thatreceives more attention. In fact, this problem hadbeen investigated by Fejes Toth [12] some forty yearsearlier without the restriction of placing the points onthe circular lattice. He showed that the sum of thepairwise distances determined by n points on a circleis maximized when the points are the vertices of aregular n-gon.

One may also examine the spectrum of the fre-quencies with which all the durations are presentin a rhythm. In music theory this spectrumis called the interval vector (or full-interval vec-tor) [7]. For example, the interval vector for the claveSon pattern [x . . x . . x . . . x . x . . .] is given by[0,1,2,2,0,3,2,0].

∗This research was partially supported by NSERC and

FCAR. e-mail: [email protected]

Examination of such rhythm histograms leads toquestions of interest in a variety of fields of en-quiry: musicology, geometry, combinatorics, andnumber theory. For example, David Locke [9]has given musicological explanations for the char-acterization of the Gahu bell pattern, given by[x . . x . . x . . . x . . . x .], as “rhythmically po-tent”, exhibiting a “tricky” quality, creating a “spi-ralling effect”, causing “ambiguity of phrasing” lead-ing to “aural illusions.” Comparing the full-intervalhistogram of the Gahu pattern with the histograms ofother popular 4/4 time traditional clave-bell rhythmsleads to the observation that the Gahu is the onlypattern that has a histogram with a maximum heightof 2, and consisting of a single connected componentof occupied histogram cells.

In 1989 Paul Erdos [5] asked whether one couldfind n points in the plane (no three on a line and nofour on a circle) so that for every i, i = 1, ...n − 1there is a distance determined by these points thatoccurs exactly i times. Solutions have been found for2 ≤ n ≤ 8. A musical scale whose pitch intervalsare determined by points drawn on a circle, and thathas the property asked for by Erdos is known in mu-sic theory as a deep scale [7]. We will transfer thisterminoly from the pitch domain to the time domainand refer to cyclic rhythms with the Erdos propertyas deep rhythms.

The analysis of cyclic rhythms suggests yet anothervariant of the question asked by Erdos. From themusicological point of view it is desirable (especiallyin African timelines) not to allow empty semicircles.Such constraints suggest the following problem. Is itpossible to have k points on a circular lattice of n

points so that for every i, i = ks, ks+1, ..., kf (s andf are pre-specified) there is a geodesic distance thatoccurs exactly i times, with the further restrictionthat there is no empty semicircle?

These problems are closely related to the generalproblem of reconstructing sets from interpoint dis-tances: given a distance multiset, construct all pointsets that realize the distance multiset. This problemhas a long history in crystallography [8], and morerecently in DNA sequencing [11]. Two noncongruent

47

sets of points are called homometric if the multisetsof their pairwise distances are the same [10].

The preceeding suggests that it would be desirableto be able to eficiently generate rhythms that containprescribed histogram shapes, (such as deep rhythms)and to find approximations when such rhythms do notexist.

The problem of comparing two binary strings of thesame length with the same number of one’s suggestsan extremely simple edit operation called a swap. Aswap is an interchange of a one and a zero that areadjacent to each other in the binary string. The swapdistance between two rhythms is the minimum num-ber of swaps required to convert one rhythm to theother.

Consider two n-bit (cyclic) binary strings, A and B,represented on a circle (necklace instances). Let eachsequence have the same number k of 1’s. We are in-terested in computing the necklace-swap-distance be-tween A and B, i.e., the minimum number of swapsneeded to convert A to B, minimized over all rota-tions of A. This distance may be computed in O(n2)time by solving a linear time problem in each of the n

rotated positions. The open problem is whether theO(n2) may be improved. In contrast, the necklace-

Hamming-distance may be computed in O(n log n)time using the Fast Fourier Transform [6].

For additional discussion of the preceeding topicsthe reader is referred to [13], [15], [14], [17], [16], [4],and the references therein.

References

[1] Steven Block and Jack Douthett. Vector prod-ucts and intervallic weighting. Journal of Music

Theory, 38:21–41, 1994.

[2] M. Chemillier. Ethnomusicology, ethnomathe-matics. The logic underlying orally transmittedartistic practices. In G. Assayag, H. G. Fe-ichtinger, and J. F. Rodrigues, editors, Math-

ematics and Music, pages 161–183. Springer,2002.

[3] J. Clough and J. Douthett. Maximally even sets.Journal of Music Theory, 35:93–173, 1991.

[4] Miguel Dıaz-Banez, Giovanna Farigu, FranciscoGomez, David Rappaport, and Godfried T. Tou-ssaint. El compas flamenco: a phylogenetic anal-ysis. In Proc. BRIDGES: Mathematical Connec-

tions in Art, Music and Science, SouthwesternCollege, Kansas, July 30 - August 1 2004.

[5] Paul Erdos. Distances with specified multiplici-ties. American Math. Monthly, 96:447, 1989.

[6] D. Gusfield. Algorithms on Strings, Trees, and

Sequences: Computer Science and Computa-

tional Biology. Cambridge University Press,Cambridge, 1997.

[7] Timothy A. Johnson. Foundations of Diatonic

Theory: A Mathematically Based Approach to

Music Fundamentals. Key College Publishing,Emeryville, California, 2003.

[8] Paul Lemke, Steven S. Skiena, and Warren D.Smith. Reconstructing sets from interpoint dis-tances. Tech. Rept. DIMACS-2002-37, 2002.

[9] David Locke. Drum Gahu: An Introduction to

African Rhythm. White Cliffs Media, Gilsum,New Hampshire, 1998.

[10] Joseph Rosenblatt and Paul Seymour. The struc-ture of homometric sets. SIAM Journal of Alge-

braic and Discrete Methods, 3:343–350, 1982.

[11] S. S. Skiena and G. Sundaram. A partial digestapproach to restriction site mapping. Bulletin of

Mathematical Biology, 56:275–294, 1994.

[12] L. Fejes Toth. On the sum of distances deter-mined by a pointset. Acta. Math. Acad. Sci.

Hungar., 7:397–401, 1956.

[13] Godfried T. Toussaint. A mathematical analysisof African, Brazilian, and Cuban clave rhythms.In Proc. of BRIDGES: Mathematical Connec-

tions in Art, Music and Science, pages 157–168,Towson University, MD, July 27-29 2002.

[14] Godfried T. Toussaint. Algorithmic, geometric,and combinatorial problems in computationalmusic theory. In Proceedings of X Encuentros de

Geometria Computacional, pages 101–107, Uni-versity of Sevilla, Sevilla, Spain, June 16-17 2003.

[15] Godfried T. Toussaint. Classification and phylo-genetic analysis of African ternary rhythm time-lines. In Proceedings of BRIDGES: Mathematical

Connections in Art, Music and Science, pages25–36, Granada, Spain, July 23-27 2003.

[16] Godfried T. Toussaint. A comparison of rhyth-mic similarity measures. In Proc. 5th Inter-

national Conference on Music Information Re-

trieval, pages 242–245, Barcelona, Spain, Octo-ber 10-14 2004. Universitat Pompeu Fabra.

[17] Godfried T. Toussaint. A mathematical measureof preference in African rhythm. In Abstracts of

Papers Presented to the American Mathematical

Society, volume 25, page 248, Phoenix, January7-10 2004. American Mathematical Society.

48

Provably Better Moving Least Squares

Ravikrishna Kolluri James F. O’Brien Jonathan R. ShewchukUniversity of California, Berkeley

1 Introduction

We analyze a variant of the implicit moving least squares(MLS) algorithm proposed by Shen, O’Brien, andShewchuk [4]. We show that under certain samplingconditions the surface reconstructed by the MLS algorithmis geometrically and topologically correct.

The input to the MLS algorithm is a set of sample pointsS near a surface F , with approximate normals. For eachsample s ∈ S we define a linear point function thatapproximates the signed distance function of F in the localneighborhood of s. These functions are blended togetherusing Gaussian weight functions yielding a smooth functionI whose zero set U is the reconstructed surface. We provethat I is a good approximation to the signed distance functionof the sampled surface F , and that U is homeomorphic to Fand geometrically close to F .

Shen, O’Brien, and Shewchuk originally proposed theirMLS construction with different weight functions for build-ing manifold surfaces from polygon soup. Kolluri [3]showed that for reconstructing surfaces from points sets, avariant of this algorithm is geometrically and topologicallycorrect under uniform sampling conditions. In this work weextend the analysis to handle adaptively sampled point datain which the sampling density is proportional to the localsurface complexity. Our sampling requirements definedin Section 2, are similar to the sampling requirements ofDelaunay-based algorithms like Crust [1].

2 Sampling Requirements

The local feature size (lfs) at a point p ∈ F is the distancefrom p to the nearest point of the medial axis of F , as shownin Figure 1. S is an ε-sample of the surface F if the distancefrom any point p ∈ F to its closest sample in S is less thanε lfs(p). Our results are valid for values of ε ≤ 0.01.

Amenta and Bern [1] show that the function lfs is 1-Lipschitz. We extend the definition of the function lfsbeyond the points on the surface F . This extension isused in defining our sampling requirements and our MLSconstruction. We define the extended local feature size of apoint p as

e lfs(x) = m inp∈F

lfs(p) + d(x,p) − |φ(x)| .

Here, φ(x) is the signed distance from x to the surface F andd(x,p) is the distance between point x and point p. It is easyto show that the function e lfs is 1-Lipschitz and reduces tothe function lfs for points on the surface.

Observation 1 For any two points, p and q, |e lfs(p) −e lfs(q)| ≤ d(p,q). For any point p ∈ F , e lfs(p) = lfs(p).

Figure 1: A closed curve along with its medial axis. Thelocal feature size of p is the distance to the closest point x onthe medial axis.

Our sampling requirements allow for noisy data when theamount of noise in the sample coordinates is small comparedto the sample spacing. We assume that for each samples, the distance to its closest surface point p ∈ F is lessthan ε2e lfs(s). We also allow a small amount of noise inthe estimated sample normal. Consider a sample r withestimated normal ~nr, as shown in Figure 1, whose closestpoint in F is q with true normal ~nq. The angle between ~nr

and ~nq should be less than ε.Our MLS construction builds the function I by blending

together functions associated with each sample point. Hencearbitrary oversampling in one region of the surface candistort the value of the function in other regions. To prohibitsuch oversampling, we require that local changes in thesampling density be bounded. Let α be the number ofsamples inside a ball of radius ε e lfs(p) centered at a pointp. If α > 0, the number of samples inside a ball of radius2ε e lfs(p ) at p is at most 8α.

3 Surface Definition

The input to the MLS algorithm is a set of sample pointsS near the surface F . Each sample s ∈ S has an associatedvector ~ns that approximates the outside normal of the surfacenear s.

We build a point function for each sample s ∈ S thatapproximates the signed distance function of F near s. Thepoint function Ps(x) of sample point s with normal ~ns is thesigned distance from x to the tangent plane at s, Ps(x) =(x− s) ·~ns. A weighted average of the point functions givesthe function I whose zero set is the implicit surface we seek.

I(x) =

∑s∈S Ws(x)((x − s) · ~ns)

∑s∈S Ws(x)

.

The weight functions are Gaussian functions modified by a

49

B2

F

xB1

p(a) (b)

Figure 2: (a)The function I at point x is mostly determinedby the point functions of the samples inside the thin shellbounded by B1 and B2. (b) The offset curves Fin and Fo u t

of a curve F .

normalization factor associated with each sample point.

Ws(x) = e−‖x−si‖2/e lfs2(x)/As.

The normalization factor associated with each sample point saccounts for oversampling near s. Let α > 0 be the numberof samples inside a ball Bε of radius ε e lfs(s) centered atsample s, including s itself. The value of As is given by

As =α

ε3e lfs3(s).

4 Results

Consider a point x whose closest point on the surface isp as shown in Figure 2(a). Let B1(x) be a ball of radius|φ(x)| centered at x. Consider a second ball B2(x), that isslightly bigger than B1(x), also centered at x. The radiusof B2(x) is |φ(x)| + τ e lfs(x). Here τ = 2ε is a constantthat depends on the sampling density. Our results are basedon the observation that the value of the function at point xis mostly determined by the samples inside the thin shellbounded by B1(x) and B2(x).

Let Fo u t be the τ -offset surface outside of F that isobtained by moving each point p ∈ F along the normalat p by a distance τ · e lfs(p). Similarly, let Fin be the τ -offset surface inside of F as shown in Figure 2(b). The τ -neighborhood is the region bounded by the inside and theoutside offset surfaces. Our first geometric result is that thezero set U of I is inside the τ -neighborhood of F .

Theorem 2 For each point x outside Fo u t , I(x) > 0 and foreach point y inside Fin , I(y) < 0.

Theorem 2 proves that the function I does not have anyspurious zero crossings far away from the sample points. Oursecond geometric result is about the gradient of I at pointsin the zero set of I .

Theorem 3 Let x be a point in the τ -neighborhood of F andlet p be the point on F closest to x. Let ~n be the normal of p.Then, ~n · ∇I(x) > 0.

Theorem 3 proves that the gradient can never be zero insidethe τ -neighborhood. From Theorem 2, the zero set of I is

Figure 3: MLS reconstruction of the Stanford Dragon modelfrom raw data.

inside the τ -neighborhood of F . Hence, from the implicitfunction theorem [2], zero is a regular value of I and thezero set U is a compact, two-dimensional manifold.

We use these geometric results to define a homeomor-phism between F and U . As F and U are compact, a one-to-one, onto, and continuous function from U to F defines ahomeomorphism.Definition: Let Γ : IR3 → F map each point q ∈ IR3 to itsclosest point on F .

Theorem 4 The restriction of Γ to U defines a homeomor-phism from U to F .

5 Discussion

Our sampling requirement that ε ≤ 0.01 is probably anartifact of our proof technique. The MLS algorithm worksquite well on data obtained from laser range, for which ε ismuch larger than 0.01 as shown in Figure 3.

Our definition of the MLS surface requires knowledge ofthe e lfs(x), function which is unknown. In our analysis,e lfs can be replaced by any 1-Lipschitz function f such thatf(x) ≤ e lfs(x) at all points x, and the input sample is anεf -sample for ε ≤ 0.01. We can relax our requirements andassume that the e lfs function is known only at the samplepoints. A 1-Lipschitz function function f(x) can now bedefined as

f(x) = m ins∈S

d(x, s) + elfs(s) − d(x, n n (x)),

where n n (x) is the sample nearest x in S.

References

[1] Nina Amenta and Marshall Bern. Surface Reconstructionby Voronoi Filtering. Discrete & Computational Geometry,22:481–504, 1999.

[2] Jules Bloomenthal, editor. Introduction to Implicit Surfaces.Morgan Kaufman, 1997.

[3] Ravikrishna Kolluri. Provably Good Moving Least Squares. InACM-SIAM Symposium on Discrete Algorithms, 2005.

[4] Chen Shen, James F. O’Brien, and Jonathan R. Shewchuk. In-terpolating and Approximating Implicit Surfaces from PolygonSoup. ACM Transactions on Graphics, 23(3):896–904, 2004.

50

Contour Tree Simplification With Local Geometric Measures

Hamish CarrDept. of Computer Science

Univ. College Dublin

Jack SnoeyinkDept. of Computer Science

UNC Chapel Hill

Michiel van de PanneDept. of Computer ScienceUniv. of British Columbia

rendered isosurface

current isovalue

brain

nasalcavity

nasalseptum

lower jaw?

iso

va

lue

fea

ture

siz

e

Simplification

tree size

Contour TreeData Display

bloodvessels

brain

bloodvessels

eyeball

eyeballs ventricle

eye socket

eyesockets

skull

nasalseptum

nasal cavity

currentlevel of simpli-fication

ventricle

Figure 1: Left, an isosurface of the UNC Head (109×256×256 MRI) shows mostly the skull: the contour tree is unmanageable (1,573,373edges). Right, contour surfaces chosen using a simplified contour tree. (Annotation and colour chosen to emphasize the structure of the data.)

Abstract

The contour tree, an abstraction of a scalar field that encodesthe nesting relationships of isosurfaces, has several potential ap-plications in scientific and medical visualization, but noise inexperimentally-acquired data results in unmanageably large trees.We attach geometric properties of the contours to the branches ofthe tree and apply simplification by persistence to reduce the sizeof contour trees while preserving important features of the scalarfield.

Keywords: Isosurfaces, contour trees, topological simplification

1 Introduction

The contour tree is a topological abstraction of a scalar field used inscientific and medical visualization [BPS97; vKvOB+97; PCM02;CSvdP04]. It represents changes in isosurface connectivity. In thispaper, we simplify the contour tree using geometric properties ofcontours, permitting online simplification of the contour tree.

Figure 1 shows a conventional isosurface and a flexible isosur-face [CS03] extracted from the same data set after contour tree sim-plification. On the left, we see that the outermost surface (the skull)occludes other surfaces, making it difficult to study structures in-side the head, and the contour tree has too many edges to be useful.On the right, we see the result of using the simplified contour treeas an interface tool to enable a user to explore, color, and annotatethe contours – the structures inside the head can be seen in relationto each other.

2 Related Work

The contour tree, a special case the Reeb graph [Ree46], is the re-sult of contracting each contour in a scalar field to a single point;it tracks how contours, connected components of isosurfaces of adata set, appear, merge, split, and vanish as we vary the chosen iso-value. Efficient algorithms for constructing the contour tree have

been reported for various meshes and interpolants [vKvOB+97;CSA03; PCM02; CLLR02; TNTF04]; the contour tree has applica-tions ranging from fast isosurface extraction [vKvOB+97; CS03]and volume rendering [TNTF04] to mesh simplification, abstractrepresentation of scalar fields [BR63; BPS97] and contour manip-ulation [CS03]. Unfortunately, noise in the input data can createmany new contours by creating local minima and maxima. Fornoisy experimentally-acquired data such as the UNC head data setshown in Figure 1, contour trees commonly have millions of edges– too many to serve as a visual representation for the input data.

To simplify the contour tree, we would like to assign an importanceto each edge and collapse edges of lower importance. This is a sim-ple case of the ideas of topological persistence [EHZ03; ELZ02;BEHP03] applied to trees. Two works have applied persistence tothe isovalues: [HSKK01] simplify the Reeb graph using hierarchi-cal quantization of the data values, which can introduce errors atedges that span the quantization boundaries, and [TNTF04] sim-plify the contour tree using data values. We allow any geometricproperty to guide simplification (and are most efficient with decom-posable properties, such as volume and surface area.)

3 Contour Tree Simplification

Given a contour tree and a scalar field, we simplify the contour treewith graph operators, then reflect the simplication back to the inputdata or use the simplified contour tree directly to extract individualcontours from the simplified data set.

To compute geometric measures for individual contours, we re-place the single isovalued sweep in [BPS97] with multiple separatesweeps of individual contours corresponding to sweeping individ-ual points through the tree. Doing this requires combining partialsweep results whenever a saddle in the tree is swept past. In threedimensions, we can compute surface area, volume or hypervolume:isovalue integrated inside the contour.

To simplify the contour tree, we then choose a leaf edge that cor-responds to contours for which the chosen geometric property issmall and prune the leaf from the tree. By removing only leaves,

51

we guarantee that the structure remains a tree and corresponds to asubregion of the scalar field in which isovalued contour sweeps canstill be performed without discontinuous jumps.

Leaf pruning can result in a redundant vertex in the tree, as in Fig-ure 2. We remove such vertices with no loss of topological infor-mation in the tree. Since there is no geometric cost to doing so, weprefer these vertex simplications where available and also preferleaf prunes that maximize the number of future vertex simplifica-tions.

Removing a leaf of the tree corresponds to flattening a local ex-tremum of the data set as shown in Figure 2. By minimizing thegeometric cost of our simplification, we are able to achieve simpli-fication of the tree by 4 orders of magnitude without causing signifi-cant errors in the underlying field being represented, and preservingdetails of the contours that remain.

0

90

0

90

0

90

50

0

90

50

0

90

50

80

0

90

50

80

Leaf 80 is pruned Vertex 50 is reduced

Figure 2: Leaf Pruning Levels Extrema; Vertex Reduction LeavesScalar Field Unchanged

4 Results and Discussion

We have tested this form of simplification on a variety of data setsusing the flexible isosurface interface [CS03]. In Figure 1, we showa typical result using hypervolume as the importance measure. Con-tours for the skull were not selected because they occlude the inter-nal organs.

embryo gut?

lungs

eyesbrain

windpipe?shoulderblades

breastbone

Figure 3: A Pregnant Rat MRI (240×256×256). Despite low qual-ity data, simplifying the contour tree from 2,943,748 to 125 edgesallows identification of several anatomical features.

Similarly, Figure 3 shows the result of a similar exploration of a240× 256× 256, low-quality MRI scan of a rat from the WholeFrog Project at http://www-itg.lbl.gov/ITG.hm.pg.docs/Whole.Frog/Whole.Frog.html. Again, simplifica-tion reduces the contour tree to a useful size. After using thedot tool from the graphviz package (http://www.research.att.com/sw/tools/graphviz/) to lay out the contour tree,these images took less than 10 minutes to explore and annotate. Theresult is purely a function of the topology of the isosurfaces of theinput data, and uses no special constants.

5 Conclusions and Future Work

We have presented a novel algorithm for the simplification of con-tour trees based on local geometric measures. The algorithm is on-line, meaning that simplifications can be done and undone at anytime. This addresses the scalability problems of the contour treein exploratory visualization of 3D scalar fields. The simplificationcan also be reflected back onto the input data to produce an on-linesimplified scalar field. The algorithm is driven by local geometricmeasures such as area and volume, which make the simplificationsmeaningful. Moreover, the simplifications can be tailored to a par-ticular application or data set.

Future directions of research include extension to vectors of geo-metric measures, user-directed local simplification of the contourtree, utilization of the contour tree as a query structure for geomet-ric properties, application of similar methods to volume renderingand to non-isovalue segmentation, extension to time-varying datasets, parallelization and improvements to contour tree layout algo-rithms.

6 Acknowledgements

Acknowledgements are due to NSERC, NSF and IRIS for funding,and to the sources of volumetric data used.

ReferencesPeer-Timo Bremer, Herbert Edelsbrunner, Bernd Hamann, and Valerio Pascucci. A

Multi-resolution Data Structure for Two-dimensional Morse-Smale Functions. InProceedings of IEEE Visualization 2003, pages 139–146, 2003.

Chandrajit L. Bajaj, Valerio Pascucci, and Daniel R. Schikore. The Contour Spectrum.In Proceedings of IEEE Visualization 1997, pages 167–173, 1997.

Roger L. Boyell and H. Ruston. Hybrid Techniques for Real-time Radar Simulation.In Proceedings of the 1963 Fall Joint Computer Conference, pages 445–458. IEEE,1963.

Yi-Jen Chiang, Tobias Lenz, Xiang Lu, and Gunter Rote. Simple and Output-SensitiveConstruction of Contour Trees Using Monotone Paths. Technical Report ECG-TR-244300-01, Institut fur Informatik, Freie Universtat Berlin, 2002.

Hamish Carr and Jack Snoeyink. Path Seeds and Flexible Isosurfaces: Using Topol-ogy for Exploratory Visualization. In Proceedings of Eurographics VisualizationSymposium 2003, pages 49–58, 285, 2003.

Hamish Carr, Jack Snoeyink, and Ulrike Axen. Computing Contour Trees in All Di-mensions. Computational Geometry: Theory and Applications, 24(2):75–94, 2003.

Hamish Carr, Jack Snoeyink, and Michiel van de Panne. Progressive Topological Sim-plification Using Contour Trees and Local Spatial Measures. Accepted to IEEEVisualization 2004.

H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse-Smale complexesfor piecewise linear 2-manifolds. Discrete Comput. Geom., 30:87–107, 2003.

H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and sim-plification. Discrete Comput. Geom., 28:511–533, 2002.

M. Hilaga, Yoshihisa Shinagawa, T. Kohmura, and Tosiyasu L. Kunii. Topologymatching for fully automatic similarity estimation of 3d shapes. In SIGGRAPH2001, pages 203–212, 2001.

Valerio Pascucci and K. Cole-McLaughlin. Efficient Computation of the Topology ofLevel Sets. In Proceedings of IEEE Visualization 2002, pages 187–194, 2002.

Georges Reeb. Sur les points singuliers d’une forme de Pfaff completement integrableou d’une fonction numerique. Comptes Rendus de l’Academie des Sciences deParis, 222:847–849, 1946.

Shigeo Takahashi, Gregory M. Nielson, Yuriko Takeshima, and Issei Fujishiro. Topo-logical Volume Skeletonization Using Adaptive Tetrahedralization. In GeometricModelling and Processing 2004, 2004.

Marc van Kreveld, Rene van Oostrum, Chandrajit L. Bajaj, Valerio Pascucci, andDaniel R. Schikore. Contour Trees and Small Seed Sets for Isosurface Traversal.In Proceedings of the 13th ACM Symposium on Computational Geometry, pages212–220, 1997.

52

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54

Parallel Guaranteed Quality Planar Delaunay

Mesh Generation by Concurrent Point Insertion∗

Extended Abstract

Andrey N. Chernikov and Nikos P. Chrisochoides

Computer Science Department

College of William and Mary

Williamsburg, VA 23185

Abstract We develop a theoretical framework forconstructing parallel guaranteed quality Delaunay planarmeshes using commercial off-the shelf software (COTS).We call two points Delaunay-independent if they can be in-serted concurrently without destroying the conformity andDelaunay properties of the mesh. First, we present a suf-ficient condition of Delaunay-independence. It is based onthe distance between points, can be verified very efficientlyand used in practice. Second, we show that a simple blockmesh decomposition can be utilized in order to guarantee a-priori Delaunay-independence of points in certain regions.Third, we derive an expression which relates three meshquality and size parameters that allow to conduct the pre-processing step of our approach using a sequential Delau-nay refinement algorithm. We conclude with our currentwork in progress that includes extending the presented ap-proach to generate nonuniform graded meshes.

Introduction

Nave, Chrisochoides, and Chew [7] presented a practicalprovably-good parallel mesh refinement algorithm for polyhe-dral domains. The approach in [7], due to absence of sequentialcode reuse, as well as intensive unpredictable communicationand setbacks, is labor intensive. In this paper, we develop anapproach which allows to use COTS, requires only structuredbulk communication, and eliminates setbacks by ensuring thatthe inserted points are Delaunay-independent.

Linardakis and Chrisochoides [6] described a Parallel Do-main Decoupling Delaunay method for 2-dimensional domains,which is capable of leveraging the serial meshing codes. How-ever, it is based on the Medial Axis which is very expensive anddifficult to construct for 3-dimensional geometries. The ap-proach developed in the present work is domain decompositionindependent, i.e. it does not require an explicit construction ofinternal boundaries.

Blelloch, Hardwick, Miller, and Talmor [2] describe a divide-and-conquer projection-based algorithm for constructing De-launay triangulations of pre-defined point sets in parallel. Ourgoal, though, is to refine an existing mesh by inserting trianglecircumcenters, i.e., the set of points in the final mesh is notknown in advance.

Kadow in [5] extended [2] for parallel mesh generation. Theprincipal difference between [7] and [5] is that in [5] the need

∗This work was supported by NSF grants: CCR-0049086, ACI-

0085969, EIA-9972853, EIA-0203974, and ACI-0312980

to construct an initial mesh sequentially is eliminated.Edelsbrunner and Guoy [4] define the points x and y as in-

dependent if the closures of their prestars (or cavities [7]) aredisjoint. We start with proving a similar condition of pointindependence [3]. Our formulation is less restrictive: it allowsthe cavities to share a point. However, computing the cavitiesand their intersections for all candidate points is very expen-sive. That is why we do not use coloring methods that arebased on the cavity graphs and we prove a theorem, which al-lows to use only the distance between the points for checkingtheir Delaunay-independence. The minimum separation dis-tance argument in [4] is used to derive the upper bound on thenumber of inserted vertices and prove termination, but not toensure point independence.

Spielman, Teng, and Ungor [9] presented the first theoreti-

cal analysis of the complexity of parallel Delaunay refinement

algorithms. However, the assumption is that the global mesh

is completely retriangulated each time a set of independent

points is inserted [11]. In [10] the authors developed a more

practical algorithm which takes O (logm) time (i.e. number of

parallel iterations) using m processors, where m is the size of

the output. In contrast, our approach [3] uses only four par-

allel refinement iterations with a fixed number of processors,

where each iteration on a single processor is performed by a

sequential mesher [8]. The present work is an extension of the

work we presented in [3].

The Theoretical Framework

Sequential Delaunay refinement algorithms are based on in-serting circumcenters of triangles which violate the requiredbounds, e.g. the upper bound ρ on circumradius-to-shortestedge ratio, and the upper bound ∆ on triangle area. Let thecavity CM(p) of point p with respect to mesh M be the set oftriangles in M, whose open circumdisks include p. We expectour parallel Delaunay refinement algorithm to insert multiplecircumcenters concurrently in such a way that at every itera-tion the mesh will be both conformal and Delaunay. Figure 1illustrates how the concurrently inserted points can violate oneof these conditions.

Theorem 1 Let r be the upper bound on triangle circumradius

in the mesh and pi, pj ∈ Ω ⊂ R2. Then if ‖pi − pj‖ ≥ 4r, then

independent insertion of pi and pj will result in a mesh which

is both conformal and Delaunay.

To show that Theorem 1 is applicable throughout therun of the algorithm, we prove that the execution of the

55

p9

p

p

pp

p2 4

5

7

p1

6

p3

p8

p10

p

p

pp

p2 4

5

7

p1

6

p3

p8

(a) (b)

Figure 1: (a) If 4p3p6p7 ∈ C(p8) ∩ C(p9), then concurrentinsertion of p8 and p9 yields a non-conformal mesh. Solid linesrepresent edges of the initial triangulation, and dashed lines —edges created by the insertion of p8 and p9. Note that the inter-section of edges p8p6 and p9p7 creates a non-conformity. (b) Ifedge p3p6 is shared by C(p8) = 4p1p2p7, 4p2p3p7, 4p3p6p7and C(p10) = 4p3p5p6, 4p3p4p5, the new triangle 4p3p10p6

can have point p8 inside its circumdisk, thus, violating the De-launay property.

# of Pipe with holes Unit squarepro- Time, # of elmnts, Time, # of elmnts,

cessors sec. ×106 sec. ×106

4 179.1 14.6 293.7 23.8

64 273.6 233.3 300.1 470.7

121 212.7 441.1 293.7 873.5

Table 1: Scaled workload, the area bound is inversely propor-tional to the number of processors.

Bowyer/Watson kernel [7], either sequentially or in parallel,does not violate the condition that r is the upper bound ontriangle circumradius in the entire mesh.

Theorem 2 The condition that r is the upper bound on trian-

gle circumradius in the entire mesh holds both before and after

the insertion of a point.

In order not to check the independence condition for everypair of candidate points, we utilize a coarse-grained domaindecomposition scheme. A coarse uniform lattice is overlappedover the triangulation domain in such a way that any pair ofpoints in non-adjacent cells are guaranteed to be no less that4r apart. To enforce the r circumradius bound in the mesh wederive the following relation which allows to use the standardsequential Delaunay refinement algorithms for preprocessing:

Theorem 3 If ρ and ∆ are upper bounds on triangle

circumradius-to-shortest edge ratio and area, respectively, then

r = 2(ρ)3/2√∆ is an upper bound on triangle circumradius.

Some results for shared and distributed memory implemen-

tations1 are shown in Tables 1 and 2. Table 1 also indicates

that there is potential for improvement by using the Load Bal-

ancing Library [1].

1This work was performed using computational facilities at the

College of William and Mary which were enabled by grants from

Sun Microsystems, the National Science Foundation, and Virginia’s

Commonwealth Technology Research Fund.

# of Time, sec. Time, sec. # of elmnts,processors MPI OpenMP ×106

4 220.3 214.1 14.6

Table 2: Pipe cross-section, distributed (MPI) and shared(OpenMP) memory implementations.

Figure 2: Graded mesh of Jonathan Shewchuk’s key. Theparallel refinement is guided by a quadtree.

Conclusions and Work In Progress

The approach we developed allows the use of sequentialCOTS for guaranteed quality parallel meshing.

Currently, we are working on extending our results to graded

meshes like the one shown in Fig. 2 by using a quadtree instead

of a uniform lattice, and to 3 dimensions.

References

[1] K. Barker, A. Chernikov, N. Chrisochoides, and K. Pin-gali. A load balancing framework for adaptive and asyn-chronous applications. IEEE TPDS, 15(2):183–192, Feb.2004.

[2] G. E. Blelloch, J. Hardwick, G. L. Miller, and D. Tal-mor. Design and implementation of a practical parallelDelaunay algorithm. Algorithmica, 24:243–269, 1999.

[3] A. N. Chernikov and N. P. Chrisochoides. Practical andefficient point insertion scheduling method for parallelguaranteed quality Delaunay refinement. Proc. 18th Intl.

Conf. on Supercomputing, 48–57. ACM Press, 2004.[4] H. Edelsbrunner and D. Guoy. Sink-insertion for mesh

improvement. Proc. 17th Simp. on Comp. Geom., 115–123. ACM Press, 2001.

[5] C. Kadow. Adaptive dynamic projection-based parti-tioning for parallel Delaunay mesh generation algorithms.SIAM Workshop on Combinatorial Sci. Comp., Feb. 2004.

[6] L. Linardakis and N. Chrisochoides. Parallel domain de-coupling Delaunay method. SIAM J. Sci. Comp., underrevision, 2004.

[7] D. Nave, N. Chrisochoides, and L. P. Chew. Guaranteed–quality parallel Delaunay refinement for restricted poly-hedral domains. Comp. Geom.: Theory and Applications,28:191–215, 2004.

[8] J. R. Shewchuk. Triangle: Engineering a 2D quality meshgenerator and Delaunay triangulator. Proc. 1st Workshop

on Appl. Comp. Geom., 123–133, 1996.[9] D. A. Spielman, S.-H. Teng, and A. Ungor. Parallel De-

launay refinement: Algorithms and analysis. Proc. 11th

Intl. Meshing Roundtable, 205–217, 2001.[10] D. A. Spielman, S.-H. Teng, and A. Ungor. Time com-

plexity of practical parallel Steiner point insertion algo-rithms. Proc. 16th Simp. on Parallelism in alg. and arch.,267–268. ACM Press, 2004.

[11] S.-H. Teng. Personal Communication. SIAM PP’04, Feb.2004.

56

Tightening: Curvature-Limiting Morphological Simplification

Jason Williams and Jarek Rossignac

Given a planar set S of arbitrary topology and a radius r, we define an r-tightening of S,

which is a set that has a radius of curvature everywhere greater than or equal to r and that

only differs from S in a morphologically-defined tolerance zone. This zone, which we

call the mortar, contains only the details of S, such as high curvature portions of its

boundary, thin gaps and constrictions, and small holes and connected components. We

describe how to approximately compute r-tightenings for shapes represented as binary

images using constrained, level set curvature flow.

Our work addresses a formulation of the shape smoothing problem different from

those given in most prior art. The energy minimization-based fairing methods in the

CAD/CAM literature (such as [3]) and the various polygon mesh smoothing techniques

in the graphics literature (such as [2]) typically do not guarantee a bound on curvature or

confine shape changes to a tolerance zone like the mortar.

The mortar, which we introduced in [7], is defined in terms of the morphological

operations of rounding and filleting, which are described in detail in [4]. S rounded by r,

denoted Rr(S), is the union of disks of radius r contained in S, while S filleted by r,

denoted Fr(S), is the complement of the union of disks contained in the complement of S.

The mortar is Fr(S)-Rr(S). It is empty away from thin and high-curvature regions of S,

and around those regions it is a subset of all points within a distance r of the boundary of

S.

We define a simple closed curve C lying in a set T as tight with respect to T if it

locally minimizes length, so that there exists a ! such that for all t and all " ! !,

d(C(t),C(t+")) = ", where d(A,B) is the length of the shortest path connecting A and B in

T and C is parameterized by arclength. We define a point on the boundary of a shape as

concave if the line segment connecting the intersections of a small circle centered on the

point with the boundary lies completely outside the shape. Tight loops through a set

consist of concave portions of its boundary connected by tangent line segments. Because

concave portions of the boundary of the mortar have a radius of curvature greater than or

equal to r, if we define an r-tightening of S as a set T, Rr(S) # T # Fr(S), such that the

bounding loops of T are tight with respect to the mortar of S, it follows that the boundary

of an r-tightening also has a radius of curvature greater than or equal to r.

When Rr(S) and the complement of Fr(S) each consist of a single connected

component, the tightening is unique, and its boundary is the shortest loop around Rr(S)

lying in Fr(S). In this case the tightening corresponds to the relative convex hull or

minimum perimeter polygon [6] of Rr(S) in Fr(S). When Rr(S) and Fr(S) have more

complex topologies, there may be several different tightenings, each of which may have

holes and multiple connected components

We conjecture that for shapes of arbitrary topology represented as binary images,

level-set curvature flow [5] constrained to the mortar always converges to a tightening,

which includes as a corollary that a tightening always exists. In our implementation of

curvature flow, we initialize the level set function $ to be the signed Euclidean distance

to the boundary of the core of the input shape, approximately computed using

57

DanielssonÕs vector propagation algorithm [1], which we also use for implementing the

morphological operations. At each iteration, we compute $t = -F |%$|, where F is the

velocity of the level set, which is equal to the curvature in the mortar and zero outside the

mortar. The curvature is given by

! = " #"$

"$=$ xx$ y

2% 2$ x$ y$ xy + $ yy$ x

2

$ x

2+ $ y

2( )3 / 2

Where |%$| = ($x2+$y

2)1/2, and the partial derivatives are computed using finite

differences. We then compute $ (t+"t, x, y) = $ (t, x, y) + "t ¥ $t(t, x, y), where "t is

inversely proportional to the maximum value of F at time t, so that the level set crosses at

most one pixel each iteration. During most iterations, we only update $ in a narrow band

of pixels around the zero level set.

Curvature flow converges slowly where the radius of curvature spans several pixels.

We therefore downsample the image representation of the core by a factor of two until r

corresponds to 1-2 pixels. We perform the flow at this coarse resolution, then iteratively

upsample by a factor of two and re-perform the flow. We find we need less than 100

iterations at each level of resolution. We anticipate adapting this technique to generate

three-dimensional results using mean curvature flow.

References

[1] P. Danielsson. 1980. Euclidean distance mapping. Computer Graphics and Image

Processing 14, 227-248.

[2] M. Desbrun, M. Meyer, et. al. Implicit Fairing of irregular meshes using diffusion

and curvature flow. SIGGRAPH 99, 317-324.

[3] H. Moreton and C. Sequin. 1992. Functional optimization for fair surface design.

SIGGRAPH 92, 167-176.

[4] J. Rossignac. 1985. Blending and offsetting solid models. Ph.D. thesis, University

of Rochester.

[5] J. Sethian. 1999. Level Set Methods and Fast Marching Methods. Cambridge

University Press.

[6] J. Sklansky, R. Chazin, and B. Hansen. 1972. Minimum-perimeter polygons of

digitized silhouettes. IEEE Transactions on Computers, 21, 3, 260-268.

[7] J. Williams and J. Rossignac. 2004. Mason: morphological simplification. GVU

Technical Report 04-05, http://www.cc.gatech.edu/gvu/research/techreports.html.

Submitted for publication.

58

Compact Data Representations and their

Applications

Moses Charikar

Princeton University

Several algorithmic techniques have been devised recently to deal with largevolumes of data. At the heart of many of these techniques are ingenious schemesto represent data compactly. This talk will present some constructions of suchcompact representation schemes (also referred to as sketches) for estimating dis-tances between sets, vectors, and distributions on an underlying metric (wheredistance is measured by the Earth Mover Distance (EMD)). The constructionof these compact representation schemes is motivated by techniques used inapproximation algorithms to round solutions of LP and SDP relaxations foroptimization problems. There are interesting connections between such sketch-ing methods and low distortion embeddings of metric spaces into `1. Suchcompact representation directly lead to efficient approximate nearest neighborsearch algorithms. We will also see how such schemes lead to efficient, one-passalgorithms for processing large volumes of data (streaming algorithms).

Finally, I will talk about some recent work applying these ideas to designingcompact data structures for content-based image retrieval systems. The mainchallenge here is to achieve high-quality similarity searches while using verycompact meta-data. We adapt the ideas from the sketch constructions for EMD,as well as other ideas from embeddings of normed spaces to produce compactsketches for images. I will discuss results from a prototype implementationon a database with 10,000 images. Our results show that our method canachieve more effective similarity searches than previous approaches with meta-data significantly smaller than previous systems.

The work on content-based image retrieval is joint work with Qin Lv andKai Li at Princeton.

59

NEARPT3 — Nearest Point Query in E3 with a Uniform Grid

[Extended Abstract]

W. Randolph FranklinECSE Dept, 6026 JEC, RPI, Troy NY 12180

[email protected], http://wrfranklin.org

1. INTRODUCTIONWe present Ne a r p t3, an algorithm and prelim inary im ple-m entation to preprocess a large set of fix ed points and thenperform nearest point q ueries against them . With fix ed andq uery points draw n from the sam e distribution, Ne a r p t3’sex pected preprocessing and q uery tim e are Θ (N ), w ith av ery sm all constant factor. Ne a r p t3, designed for largedatasets, has been tested on the largest datasets in the G eor-gia T ech L arge G eom etric Models A rchiv e, [7]. T herefore,processing tens of m illions of points is q uite feasible.

T he prior art inc ludes v arious data structures and algo-rithm s for v ariants of nearest neighbor searching. T he costof a V oronoi diagram , [6], in E3 is data dependent, am druns from Ω (N log N ) to O(N2) in tim e and space for pre-processing, w ith each q uery costing θ(log N ). R ange trees,[6], cost θ(N log N ) tim e to preprocess, w ith each q uery alsocosting θ(log N ). A N N (A pprox im ate N earest N eighbors),[2], is a C ++ library for approx im ate and ex act nearestneighbor searching in Ed, allow ing a v ariety of m etric s, im -plem ented w ith sev eral diff erent data structures, based onk d-trees and box -decom position trees. A ll these algorithm sand data structures are m ore general, hence bigger, thanNe a r p t3, w hich is optim ized spec ifically for the L2 m etricin E3, although its ideas w ould generalize.

Ne a r p t3 appears to be the only m ethod that enthusias-tically rejects hierarchical data structures and search tech-niq ues. T rees and subdiv ision searching are m ore robustagainst adv ersarially chosen input. H ow ev er, w e believ e,based on tests on real data, that they are often suboptim al inpractice. T his is true ev en w hen the real data is m oderatelyunev enly distributed. T he ex trem e data unev enness thatw ould destroy Ne a r p t3’s perform ance w ould also force hi-erarchical data structures to hav e m any lev els. In that case,w here the hierarchies w ould then be faster than Ne a r p t3,though not fast, a shallow hierarchy w ould perhaps be theleast slow .

Ne a r p t3 has three stages, as follow s. T he data structure isa uniform grid, [1, 4].

Prep rep ro cess: T his step, w hich does not depend on thedata, need be perform ed only once. H ence it is ex -c luded from the tim e statistic s, just as the com pilationtim e is also ex c luded. Indeed, this prepreprocessingcould be forced into the C ++ com pilation step us-ing the tem plate spec ialization fac ilities, though that

w ould be silly .

1. G enerate the coordinates (x,y,z) of all grid cellsw ith 0 ≤ x ≤ y ≤ z ≤ R for som e fix ed R.

2. Sort them byp

x2 + y2 + z2.

3. P ass dow n the list in order. F or each cell c1, findthe last cell, c2, w hose c losest point to the originis at least as c lose as the farthest point of c1. C allc2 the sto p cell.

Since the stop cells are m onotonically increasing,all this req uires only one pass dow n the cell list.

T he point is that if a point has been found in c1,w e hav e to continue searching through c2 to besure of finding any c loser points.

4. Write the sorted list of cells and stop cells to afile.

Prep ro cess: H ere the fix ed points are built into the datastructure.

1. C om pute a uniform grid resolution, G from thenum ber of fix ed points, Nf or get it from the user.A reasonable v alue is G = r 3

p

Nf , for 1/2 ≤ r ≤

2.

2. A llocate a uniform grid w ith one w ord per cell, tostore a count of the num ber of points in each cell.

3. R ead the fix ed points, determ ine w hich cell of theuniform grid each w ould fall in, and update thecounts.

4. A llocate a ragged array for the uniform grid, w ithjust enough space in each cell for the points inthat cell.

A ragged array contains storage for the pointsplus a dope v ector pointing to the first point ofeach cell. T he total v ariable storage is one w ordper cell, plus the storage for the points.

5. P rocess the fix ed points again, com puting for asecond tim e the cell that each falls into. T histim e, store each point in its proper cell.

T he goal is to m inim ize both the storage usedand the num ber of storage reallocations. Storagereallocations becom e espec ially costly as the pro-gram ’s m em ory w ork ing set approaches the com -puter’s av ailable real m em ory .

A possible alternativ e w ould be to use a link edlist for the points in each cell. H ow ev er, the space

60

used for the pointers would be significant, and thepoints in each cell would be scattered throughoutthe memory, which might reduce the cache per-formance.

A nother alternativ e would be to use a C + + ST Lv ector, which reallocates its storage as it grows.O ur ex perience finds this to be v ery suboptimal.

Qu ery : T his reports the c losest fix ed point to a q uerypoint.

1. Determine which cell, c, contains the q uery point.

2. U sing the sorted cell list computed in the prepre-processing step, spiral out from c until a cell withat least one point is found. O ften this is c itself.

For each cell with coordinates (x ,y,z) in thesorted cell list, up to 47 other reflected and ro-tated cells are deriv ed, such as (−x , z,y). If anyordinate is zero, or any two are eq ual, there willbe fewer other cells.

It would be possible to do this reflection, rota-tion, and duplicate deletion in the prepreprocess-ing stage. T his would cause a much larger sortcell list. H owev er it would reduce the q uery timebecause that code would hav e fewer conditionals,which should mak e it more optimizable.

3. C ontinue spiralling out until c’s stop cell to findany c loser points, if one ex ists.

T his spiralling process is conserv ativ e since it ig-nores the location of the q uery point inside c.

O n the av erage 200 cells are searched for eachq uery, but check ing each cell is v ery fast.

2. TESTSNe a r p t3’s performance is data dependent. A n improperchoice of input, such as q uery points that are v ery far fromall the fix ed points, will be intolerably slow. N ev ertheless,all the data sets tested so far perform q uite well, inc ludingthese:

D ata

set

n ame

S o u rce

#

fi x ed

p o in ts

#

q u eries

C PU

time,

secs

Bunny G IT 17973 17974 1.9Bone6 G IT 284818 284818 28Dragon G IT 218882 218883 21H and G IT 163661 163662 16U niformrandom

generated 1M 1M 128

T he env ironment is a 2002-v intage IBM T 30 T hink pad lap-top computer with 768 MB of memory, a 1600 MH z P entium4 Mobile C P U , and Intel’s icpc 8.1 C + + compiler, with alloptimizations enabled. T he times inc lude reading the dataand writing the results. T hese ex periments also v alidatethat the cost is linear. T he preprocessing cost is Θ (N ).E ach q uery may cost O(N ), but typically costs Θ (1).

Ne a r p t3’s cost is aff ected by the grid resolution, howev erv alues within a factor of two of the optimum typically changethe time less than a factor of 2.

3. EXTENSIONSNe a r p t3 could return approx imate nearest matches inmuch less time since the spiral search could stop sooner.In Ed for other d, the cost of searching is ex ponential in d,as for any search procedure.

Ne a r p t2 is a simplified v ersion for preprocessing andsearching for points in E2. We tested 1M q ueries against1M fix ed points, both sets randomly generated, usingNe a r p t2, C G A L 3.0.1’s Ne a r e st n e igh bo r se a r ch in g,[3], and ANN 0 .2, [5]. A proper choice of compiler flagswould probably speed both C G A L and A N N , but not re-duce their storage cost. N one of these tests req uired anydata I/O since the input was randomly generated and theoutput not written. P erforming 1M q ueries against 1M fix edpoints cost as follows.

Pro g ram Time S to rag e

Ne a r p t2 9.4 46MBCG AL NNS 41 120MBANN 41 128MB

We then tried 10M fix ed and 10M q uery points but C G A Land A N N req uired too much memory. Ne a r p t2 used458MB and 98 seconds.

4. SUMMARYT he general lesson of Ne a r p t3 is that simple data struc -tures lik e the uniform grid can be q uite effi c ient in both timeand space in E3.

5. ACKNOWLEGEMENTST his research was supported by N SF grant C C R -0306502.

6. REFERENCES[1] A k man, V ., W. R . Frank lin, M. K ank anhalli, and

C . N arayanaswami. G eometric computing and theuniform grid data techniq ue. C omputer A ided Design2 1 (7), (1989), 410–420.

[2] A rya, S. and D. M. Mount. A pprox imate nearestneighbor q ueries in fix ed dimensions. In Proc. 4th

AC M -SIAM Sy m pos. Discrete Algorithm s. 271–280.

[3] C G A L . T he C G A L home page.http://www.cgal.org/, 2003.

[4] Frank lin, W. R . and M. K ank anhalli. P arallelobject-space hidden surface removal. In Proceed ings of

SIGGR APH’9 0 , v olume 24. 87–94.

[5] Mount, D. and S. A rya. A N N : library for approx imatenearest neighbor searching v ersion 0.2 (beta release).http://www.cs.umd.edu/~mount/ANN/, 1998.

[6] P reparata, F. P . and M. I. Shamos. C om p uta tiona l

Geom etry : An Introd uction. Springer-V erlag, N ewY ork , N Y , 1985.

[7] T urk , G . and B. Mullins. L arge geometric modelsarchiv e, 2003. U R Lhttp://www.cc.gatech.edu/projects/large_models/.

61

Algebraic Number Comparisons for Robust Geometric Operations

John Keyser, Koji Ouchi

Department of Computer Science

Texas A&M University

In this talk, we describe a method for exact comparison of algebraic numbers, with

application to geometric modeling.

M otivation: Computations with algebraic numbers are of key importance in several

geometric computations. Algebraic numbers arise as solutions to systems of

polynomials a common operation in many geometric applications, particularly those

involving curved objects. Polynomials regularly describe the relationships between even

basic geometric objects, for example the (squared) distance between two points. For more

complex curved geometric objects, polynomials are often used to describe the actual

shapes. Finding solutions to systems of polynomials thus becomes a key operation in

numerous geometric applications.

Unfortunately, the robustness issues well-known in traditional computational geometry

become even more significant when dealing with algebraic numbers and curved

geometry. Bounding and handling the numerical error that arises in these computations

becomes more problematic, as does the detection and elimination of degeneracy

problems. For reliable computation on curved geometry, we therefore need robust

operations on algebraic numbers. W e choose to use an exact-computation approach,

achieving robustness by eliminating numerical error, while supporting straightforward

operations even in the presence of degeneracies.

Our work is particularly motivated from the field of computer-aided geometric design.

Finding intersections of geometric objects involves solving systems of polynomials,

usually of moderate degree in a few variables. Our methods apply, however, to a far

wider range of problems.

Background: Our earlier work focused on techniques for exact manipulation of

algebraic curves and 2D points [MAPC00], and applied these to solid modeling,

producing the first exact boundary evaluation system [ESOLID04]. Although this work

yielded greater robustness by eliminating numerical error, degeneracies could not be

handled effectively.

Computations with algebraic numbers has been a topic of recent research interest among

a variety of other researchers. LEDA supports a limited set of constructions for algebraic

numbers, though it does not solve polynomial systems [LEDA]. The Core library

supports a wider variety of number types, including real algebraic numbers [CORE].

Such exact computation approaches have been incorporated in larger projects, such as

Exacus [EXACUS] and CGAL [CGAL]. Recently, Emiris and Tsigaridas have

developed an approach for exact comparison of algebraic numbers of relatively small

degree (at most 4) that is asymptotically faster than an expliit solution [ET04].

62

Major Results: W e describe a method for comparing complex algebraic numbers

exactly. More precisely, one can know whether or not the real and imaginary parts of

given a pair of complex algebraic numbers are identical. In particular, one can test

whether or not the real and imaginary parts of a given complex algebraic number vanish.

Our method is based on the rational univariate reduction (RUR). The RUR computes

common roots of systems of multivariate polynomials with rational coefficients. The

roots are represented in terms of a set of univariate polynomials with rational coefficients.

These polynomials, when evaluated at the roots of another univariate polynomial, yield

the coordinates of the common roots of the original system. The RUR can be computed

exactly, i.e. the coefficients of these polynomials can be computed to full precision.

As a key advantage over our earlier methods, this RUR method works even in the

presence of degenerate situations. For example, the RUR approach handles roots of

high multiplicity, finds roots at singularities, and works even when the underlying set of

roots is positive dimensional. As such, it offers a general method for achieving robust

calculations with algebraic numbers.

Our RUR implementation gives an exact representation of points with algebraic

coordinates. This allows us to exactly determine geometric predicates such as whether a

point lies on a curve or surface. W e can also determine how surfaces intersect, e.g.

whether the surfaces meet in general position. Thus, our representation is very general,

and can serve as a single representation for all such algebraic points. Although this point

representation is more robust than our earlier approach that could not represent

degeneracies, it is also less efficient. W e therefore propose the use of the RUR

computation in a hybrid fashion, using it in cases where there are likely to be difficulties

due to degeneracies.

W e describe several applications, with special emphasis on geometric modeling. In

particular, we describe a new implementation that has been used successfully on certain

degenerate boundary evaluation problems. W e describe how the RUR can be used to

detect when degeneracies occur, and how it can then be combined with a numerical

perturbation scheme to achieve an overall more robust boundary evaluation..

References:[CGAL] Computational Geometry Algorithms Library: http://www.cgal.org

[CORE] Core library: http://www.cs.nyu.edu/exact/core

[ESOLID04] John Keyser, Tim Culver, Mark Foskey, Shankar Krishnan, Dinesh Manocha, ESOLID: A

System for Exact Boundary Evaluation, Computer Aided Design, vol. 36, no. 2, pp. 175-193, 2004.

[ET04] Ioannis Emiris, Elias Tsigaridas, Comparing Real Algebraic Numbers of Small Degree,

Proceedings of European Symposium on Algorithms, 2004.

[EXACUS] Efficient and Exact Algorithms for Curves and Surfaces: http://www.mpi-

sb.mpg.de/projects/exacus/

[LEDA] Library for Efficient Data Types and Algorithms: http://www.algorithmic-

solutions.com/enleda.htm

[MAPC00] John Keyser, Tim Culver, Dinesh Manocha, Shankar Krishnan, Efficient and Exact

Manipulation of Algebraic Points and Curves, Computer Aided Design. Vol 32, No. 11. pp. 649-662.

2000. Special Issue on Robustness.

63

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65

Analysis of L ayered H ierarchy for N eck laces

Kath ry n B ean∗ S ergey B ereg∗

1 Introduction

M olec u lar confi gu rations can b e m od eled as sets ofsp h eres in 3D. B y im p osing geom etric constraintson th e sp h eres b ased on m olec u lar b iology onecan ap p ly effi c ient tech niq u es [3], for ex am p le, toanaly ze th e com p lex ity of m olec u lar su rface [1].G u ib as et al. [2] introd u ced a m od el called n eck-

lace th at fi nd s ap p lications in com p u ter grap h ic s,com p u ter v ision, rob otics, geograp h ic inform ationsy stem s and m olec u lar b iology . T h ey stu d ied tw od ata stru c tu res, wrapped hierarchy and layered hier-

archy, for rep resenting neck laces and for p erform ingcollision tests.

G u ib as et al. [2] p rov ed th at th e w rap p ed h ierar-ch y ad m its a sep arating fam ily of size O(n2−2/d).T h is is th e fi rst su b q u ad ratic b ou nd p rov ed for col-lision d etection u sing p red efi ned h ierarch ies. Al-th ou gh th e lay ered h ierarch y can b e u sed for colli-sion d etection, “no su b q u ad ratic b ou nd on th e sizeof a sep arating fam ily b ased on th e lay ered h ierar-ch y is c u rrently k now n” [2]. T h e m ain resu lt of th isp ap er is th at th e sam e u p p er b ou nd h old s for th elay ered h ierarch y .

O ne of th e ad v antages of lay ered h ierarch y ov erw rap p ed h ierarch y is th e “local” d efi nition of th ecages. W e p rop ose a m od ifi cation of th e lay eredh ierarch y so th at som e d eform ations of a neck lacecan b e m aintained effi c iently . In p artic u lar, w e sh owth at rigid-body con form ation al chan ges can b e h an-d led in O(log n) tim e only .

2 N eck laces and B ounding-V olum e H ierarchies

W e d efi ne th e notion of neck lace in a sligh tly m oregeneral form 1.

Definition 1 (Necklace) A neck lace is a se-

qu en ce of beads N=〈B1, B2, . . . , Bn〉 in Rd space

that has the followin g properties:

1. T he radiu s of each bead is in the in terval

∗Dep a rtm e n t o f C o m p u te r S c ie n c e , Un iv e rsity o f T e x a s a t

Da lla s, Bo x 8 3 0 6 8 8 , R ich a rd so n , T X 7 5 0 8 3 , US A.1W e d o n o t re q u ire th a t a n y tw o c o n se c u tiv e b e a d s a lo n g

th e n e ck la c e h a v e a p o in t in c o m m o n .

[ρm in, ρm a x] where ρm in, ρm a x are positive con -

stan ts.

2. T he distan ce between the cen ters of two adja-

cen t beads is bou n ded by a con stan t δ.

C ollision d etection p rob lem arises in su ch ap p li-cations as p rotein fold ing and p rotein d ock ing. Ab ou nd ing v olu m e h ierarch y is a com m on ap p roachth at m od els and d etects collisions and self-collisionsof d iff erent geom etrical sh ap es inc lu d ing neck laces.T h is ap p roach red u ces com p u tational tim e since itsu ffi ces to test collisions am ong b ou nd ing v olu m es.

Definition 2 (Hierarchies) F or a n ecklace N ,

let T (N ) den ote the balan ced bin ary tree defi n ed re-

cu rsively so that, for an in tern al n ode v, its left

su btree con tain s bm/2c leaves where m is the n u m -

ber of leaves below v. Both wrapped an d layered

hierarchies have a cage C(v) associated with a n ode

v of T (N ) an d the cages of leaves correspon d to

the beads, C(v) = Bi for i-th leaf v where i =1, 2, . . . , n. F or an in tern al n ode v, the cage is de-

fi n ed diff eren tly for two hierarchies:

• W rap p ed h ierarch y . T he cage C(v) is the

sm allest en closin g ball of beads correspon din g

to the leaves below v.

• Lay ered h ierarch y . T he cage C(v) is the sm all-

est en closin g ball of the cages correspon din g to

the children of v.

T h e lay ered h ierarch y h as m any ad v antages ov erth e w rap p ed cou nterp art. F or a neck lace size n, th elay ered h ierarch y can b e constru c ted in linear tim esince th e cage of each nod e can b e com p u ted in O(1)tim e. T h e w rap p ed h ierarch y can b e constru c tedin O(n log n) tim e b y u sing linear-tim e algorith m[4] for com p u ting th e m inim u m enc losing sp h ereM E S . A sim p ler algorith m for com p u ting M E S inex p ected O(n) tim e can b e u sed . Note th at th ealgorith m for th e lay ered h ierarch y is ev en sim p ler.

3 N eck lace D eform ation w ith L ayered H ierarchy

M od eling conform ation ch anges is req u ired in p ro-tein d ock ing, rob otics and com p u ter grap h ic s. O ne

66

type of local deform ation - rigid-body conforma-tional change [5] - can b e defi ned as follow s.

Definition 3 (C onform ational C hange) L etN = 〈B1,B2, . . . ,Bn〉 be a necklace and let i be anindex from 1 to n − 1. L et M be a rigid motion inR

3 (the composition of a translation and a rota-tion). If N ′ = 〈B1, . . . ,Bi,M(Bi+1), . . . ,M(Bn)〉is a necklace then N ′ is a rigid-body conformationalchange of N .

W e sh ow h ow to m odify th e layered h ierarch yso th at a rigid-b ody conform ational ch ange can b edone effi c iently. W e store a rigid m otion M(v) as-soc iated w ith a v ertex v of T (N ). E ach rigid m o-tion M(v) is a com position of a 3D rotation R(v)and a translation T (v). T h e rotation can b e rep-resented as a q u aternion or a rotation m atrix . W eassu m e th at th e inv erse rigid m otion M−1(v) canb e com pu ted in O(1) tim e. W e call th e h ierarch yau gm ented w ith rigid m otions as au gmented layeredhierarchy.

T h e au gm ented layered h ierarch y defi nes th e po-sition of a b ead Bi in th e space as follow s. Let v1 =vro o t,v2, . . . ,vk b e th e path from th e root of th e lay-ered h ierarch y to th e v ertex w ith th e b ead Bi andlet ci b e th e center of th e b ead stored in vk. T h enth e real position of Bi is M1(M2(. . .Mk(ci) . . .))w h ere Mj = M(vj),j = 1, . . . , k . Let I denoteth e identity transformation, i.e. I(p) = p for anyp ∈ R

3.

Theorem 1 T he au gmented layered hierarchy canbe maintained in O(log n) time if a rigid-body con-formational change is applied.

4 Cages of L ayered H ierarchy

Alth ou gh th e layered h ierarch y can su pport rigid-b ody conform ational ch anges effi c iently (if au g-m ented as in prev iou s section), th e w rapped h ierar-ch y oc c u pies sm aller space since its cages are alw aysno larger th at th e corresponding cages of th e layeredh ierarch y. Despite th is ev idence w e sh ow th at th ecages of th e layered h ierarch y h av e th e sam e u pperb ou nd as th e corresponding cages of th e w rappedh ierarch y.

Let Tu denote th e su b tree rooted at a v ertex u ofT (N ) and let nl(u) b e th e nu m b er of leav es in Tu.

Theorem 2 T he radiu s of any cage C(v) of layeredhierarchy T (N ) is at most δ(nl(v) − 1)/2 + ρm a x,w here nl(v) is the nu mber of leaves for a tree isrooted at v.

5 Collision D etection for L ayered H ierarchy

An u sefu l tool for th e collision testing is a separatingfamily [2].

Definition 4 (Separating F am ily) A separat-ing fam ily Σ = (u,v) is a set of pairs of nodesof a volu me bou nding hierarchy satisfying thefollow ing properties:

1. If (u,v) ∈ Σ then C(u) and C(v) are disjointed.

2. F or any tw o non-adjacent beads Bp,Bm ∈ N ,there is a pair (u,v) ∈ Σ su ch that Bp ⊆ C(u)and Bm ⊆ C(v).

T o deriv e a b ou nd for th e layered h ierarch y w eanalyze a separating fam ily Σ b u ilt b y th e follow -ing algorith m for collision detection. T h e algorith mstarts w ith th e pair Q = (r o o t, r o o t) and, for apair (u,v) ∈ Q su ch th at C(u)∩C(v) 6= ∅, replacedit b y at m ost 4 pairs of ch ildren of u and v. O u ranalysis is b ased on th e analysis of th e w rapped h i-erarch y b y G u ib as et al. [2].

L em m a 3 If (C(u),C(v)) ∈ Σ then the cage C(v)is contained in K − C(u) w here K is the ball con-centric w ith C(u) and of radiu s 4(δnl(u) + ρm a x).

Theorem 4 L et HL be the layered hierarchy for anecklace N in R

d, d ≥ 3 w ith n beads. Algorithm2 bu ilds the u niqu e separating family Σ for N ofsize O(n2−2/ d). T his bou nd is asymptotically tightin the w orst case.

R eferences

[1] Y .-E . A . Ban, H . E delsbrunner, and J. R udolph. In-terface surfaces for protein-protein com plex es. InProc. of th e 8 th Annua l Interna t. C onf. on R e-sea rch in C om p uta tiona l M olecula r B iology (R E -C OM B ’0 4 ), pp. 205–212, 2004.

[2] L . J. G uibas, A . N guy en, D. R ussel, and L . Zhang.C ollision detec tion for deform ing neck laces. C om -p ut. Geom . T h eory Ap p l., 28(2-3):137–163, 2004.

[3] D. H alperin and M. H . O v em ars. Spheres, m olecules,and hidden surface rem ov al. C om p ut. Geom . T h eoryAp p l., 11(2):83–102, 1998.

[4] N . Megiddo. L inear-tim e algorithm s for linear pro-gram m ing in R

3 and related problem s. In Proc. 2 3 rdAnnu. IE E E Sym pos. F ound . C om p ut. Sci., pp. 329–338, 1982.

[5] P . Som pornpisut, Y . L iu, and E . P erozo. C alcu-lation of rigid-body conform ational changes usingrestraint-driv en cartesian transform ations. B iop h ysJourna l, 81(5):2530–2546, 2001.

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Author Index

Aronov, Boris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Asano, Tetsuo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Balkcom, Devin J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Bean, Kathryn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Benbernou, Nadia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Bereg, Sergey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Bronnimann, Herve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Burr, M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Burrowes-Jones, Jason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Cahn, Patricia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Cappos, Justin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Carr, Hamish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Cary, Matthew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Charikar, Moses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chazal, Frederic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chernikov, Andrey N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chrisochoides, Nikos P. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Daescu, Ovidiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 45

Dalal, Sourav . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Damian, Mirela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Demaine, Erik D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 16

Demaine, Martin L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 16

Devai, Frank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Dror, Moshe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Efrat, Alon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Franklin, W. Randolph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Glass, Julie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Gotsman, Craig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Har-Peled, Sariel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Hoffmann, Michael . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Iacono, John . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Isenburg, Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Keil, J. Mark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Keyser, John . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Kobourov, Stephen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Kolluri, Ravikrishna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Langerman, Stefan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Lee, Yusin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Lenchner, Jonathan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Lieutier, Andre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Lindy, Jeffrey F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Luo, Jun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Meijer, Henk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Mitchell, Joseph S. B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

O’Brien, James F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

O’Rourke, Joseph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 18

Orlin, James B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Ouchi, Koji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Panne, Michiel van de . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Rafalin, Eynat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 3

Rahman, Md Mizanur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Rossignac, Jarek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53, 57

Rudra, Atri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Rus, Daniela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Sabharwal, Ashish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Serfling, Robert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Shewchuk, Jonathan R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Snoeyink, Jack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 34, 51

Souvaine, Diane L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 3, 16

Steiger, William . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Taslakian, Perouz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Tokuyama, Takeshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Toth, Csaba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Toussaint, Godfried . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30, 47

Vassilev, Tzvetalin S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Vee, Erik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Williams, Jason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Xu, Guang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Xu, Jinhui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Zhong, Jianyuan K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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