1Institute of Information Science Academia Sinica

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Introduction to Geomtric Introduction to Geomtric ComputingComputing

D. T. LeeInstitute of Information Science

Academia Sinica

October 14, 2002

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What is Computer Science about?What is Computer Science about? Not about programming But about problem solving Basic disciplinary areas include• Software & Programming-related• Hardware-related• Computer systems• Theory • Mathematics of computer science• Applications• etc.

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MinMax Problem MinMax Problem 44

Min&Max Problem

Input: A set S of N numbers on the real line

Output: The Minimum and Maximum of S• Simple method: Finding the min. or max. of N

numbers requires N-1 comparisons.• Reduce Min&Max to 2 Min or Max problems!

2N-3 comparisons

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min

min

maxSimple comparator

MIN min

MinMax Problem MinMax Problem 33

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MinMax Problem MinMax Problem 22

But finding both Maximun and Minimum can be done with 3N/2 2 comparisons

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min

max

MIN min

MAX max

MinMax Problem MinMax Problem 11

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Sorting ProblemSorting Problem Sorting N numbers• Simple method: find Min repeatedly

(N-1) + (N-2) + … + 2 + 1 = (N-1)N/2• Selection method:

–Smallest can be found in (N-1) comparisons

– 2nd smallest found in log2 N additional comp.

– 3rd smallest found in log2 N additional comp.

(N-1)+ (N-1)log2 N

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min

1st round 3rd round2nd round

Total number of matches for 8 competitors is 7N/2 + N/4 + … + 4 + 2 + 1 = N-1

Total number of rounds is log2 N

Knockout TournamentKnockout Tournament

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Geometric ComputingGeometric Computing Design and Analysis of Algorithms for

geometric problems Theory: Computational Complexities• Time, Space, Degree• Lower and Upper Bounds of Computation

Applications: • VLSI CAD, Computer-Aided Geometric Design, Computer

Graphics, Operations Research, Pattern Recognition, Robotics, Geographical Information Systems, Statistics, Cartography, Metrology, etc.

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Theory of ComputationTheory of Computation

Models of Computation Real arithmetic, RAM Computation Tree Model - Unit Cost Off-line, On-line, Real-time Parallel Computation vs Serial Computation Problem Complexity vs Algorithm Complexity Decision vs Computation Problems• Output Sensitive Algorithms

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Checking if a Polygon is Convex Checking if a Polygon is Convex 44

A polygon P is represented by a sequence of points v0,v1,v2, …,vn-1 where (vi, vi+1) is an edge, i=0,1,..,n-1, vn = v0.

Theorem: A polygon P is convex if segments connecting any two points inside the polygon always lie inside P.

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Problem: There are infinitely many possibleSegments to check!

Checking if a Polygon is Convex Checking if a Polygon is Convex 33

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Checking if a Polygon is Convex Checking if a Polygon is Convex 22

Theorem: A polygon is convex if every diagonal lies inside the polygon.• It takes O(N3) time to check O(N2)

diagonals against every edge of P.• Very inefficient!

Theorem: A polygon is convex if every interior angle of each vertex is less than 180o.

•This result would imply an O(N) time algorithm!

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Consider three vertices vi,vj,vk and determinant

det(vi,vj,vk ) = xi yi 1xj yj 1xk yk 1

where (xi yi ) are the x- and y-coordinates of v i

det(vi,vj,vk ) = 0 when vi,vj,vk are collinear

det(vi,vj,vk ) < 0 when vi,vj,vk form a right turn

det(vi,vj,vk ) > 0 when vi,vj,vk form a left turn

vi

vj

vk

left turn

Checking if a Polygon is ConvexChecking if a Polygon is Convex11

vi

vj

vk

right turnvi

vj

vk

collinear

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Checking if a Polygon is SimpleChecking if a Polygon is Simple

v1

v2

v3

v4

v0

v5

How do you check if the input polygon is simple??

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Art Gallery Problem Art Gallery Problem

Consider an art gallery room whose floor plan is modeled by a simple polygon with N sides.

How many stationary guards are needed to guard the room?

Assumption: Each guard is considered to have 360o unlimited visibility.

Two points p and q are said to be visible to each other, if the line segment (p,q) does not intersect any edge of the polygon.

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Art Gallery ProblemArt Gallery Problem

p and q are visible

q

r

r and q are not visible

pq

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Art Gallery ProblemArt Gallery Problem

Visibility polygon of q & r

q r

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Art Gallery TheoremArt Gallery Theorem Theorem: Given a simple polygon with

N sides, stationary guards are sufficient, and sometimes necessary to cover the entire polygon.

N/3

Theorem: Given a simple polygon with N sides, the problem of finding a minimumnumber of stationary guards to cover the entire polygon is intractable.

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Lower Bound of Art Gallery ProblemLower Bound of Art Gallery Problem

A comb with 8 prongs (N=24) requiresN/3 = 8 guards

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Upper Bound of Art Gallery ProblemUpper Bound of Art Gallery ProblemTriangulation of a Simple PolygonTriangulation of a Simple Polygon

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Coloring of a GraphColoring of a Graph

Coloring of a graph: an assignment of colors to nodes of the graph such that no two adjacent nodes are assigned the same color.

Chromatic number: The minimum number of colors required to color a graph.

Theorem: Every triangulation of a simple polygon is 3-colorable.

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N = 14

Art Gallery Problem: ExampleArt Gallery Problem: Example

Triangulation

Select red color to place guards

3 N/3 = 4

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Art Gallery TheoremArt Gallery Theorem

Theorem: In a 3-coloring of a triangulation of a simple polygon with N vertices, there exists a color that is used no more than N/3 times.

Proof: Pigeon-hole principle• If n pigeons are placed into k pigeon-holes,

at least one of the holes must have no more than n/k pigeons.

• Since N is an integer, there exists at least one color that is used in no more than N/3 times.

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Variations of Art Gallery ProblemVariations of Art Gallery Problem

Shapes of Art Gallery:• Rectilinear polygon• Polygon with holes

Capabilities of Guard• Mobile guards moving along a track• Guards with limited visibility (angle and dist

ance)

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Rectilinear PolygonRectilinear Polygon

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Polygons with HolesPolygons with Holes

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Guard CapabilitiesGuard Capabilities

Mobile GuardLimited visibility

Mobile guard with limited visibility

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Map Coloring Map Coloring 22

Given a planar graph (planar map), find a coloring of the regions so that no two adjacent regions are assigned the same color.

How many colors are sufficient?• 4-color theorem!

Find an algorithm that colors the map.

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Map Coloring Map Coloring 11

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ProximityProximity

Closest Pair Problem: Consider N points in the plane, find the pair of points whose Euclidean distance is the smallest.• Brute force: Enumerate all N2 pairwise

distances and find the minimum.• Can one do better?

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Closet Pair on the Real LineCloset Pair on the Real Line

Consider N numbers x0, x1, x2, …, xn-1 on the real line, find the two numbers xi and xj such that |xi – xj | is the smallest.

Lemma: The two numbers xi and xj such that |xi – xj | is the smallest must be adjacent to each other, when these numbers are sorted.

xi+1xi

Closest pair

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Closest Pair in the Plane Closest Pair in the Plane

2

1

= 2

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Voronoi DiagramVoronoi Diagram

Delaunay TriangulationClosest pair must beVoronoi neighbors

Circumcircle of a triangle

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Closest Pair ProblemClosest Pair Problem

Theorem: Given N points in k-dimensional space, the closest pair of points can be determined in O(N log N) time, which is asymptotically optimal.

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Geometric OptimizationGeometric Optimization Shortest Path Problem• Given a polygonal domain (obstacles),

and two points s and t, find a shortest path between s and t avoiding these obstacles.

Homotopic Routing• Given a layout and the homotopy of k 2-

terminal nets, find a detailed routing of minimum wire length.

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Homotopic RoutingHomotopic Routing

Input: A sketch S=(F, M, W), a planar drawing of a physical layout of modules M in the grid and a finite set W of nets connecting point features F, each represented by a vertex-disjoint simple path.

Output: A detailed routing on the grid without overlapping of these nets of minimum total length among those that are homotopic to the given sketch.

Net p is homotopic to net q, if they have the same terminals, and one can be transformed continuously into the other without crossing features.

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AfterCompaction

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Topological Via Minimization Topological Via Minimization 44

Given a set of n 2-terminal nets in a two-layer bounded routing region, find a layer assignment of wire segments so as to minimize the number of vias used.

layer1

layer2

via

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Layer 1

Layer 2

Via

Layer 1

Layer 2

Via

# of Vias = 4

# of Vias = 1

Topological Via Minimization Topological Via Minimization 33

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Layer 1

Layer 2

Via

# of Vias : 1

Topological Via Minimization Topological Via Minimization 22

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Topological Via Minimization Topological Via Minimization 11

Lemma: There exists an optimal solution to the two-layer bounded region TVM problem such that each net uses at most one via.

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Definition: A set of chords is said to be 2-independent if it can be partitioned into two subsets of independent chords

Circle Graph Circle Graph 22

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Theorem: Given n chords in a circle and an integer k, the problem of deciding if there exists a 2-independent set of chords of size k or more (2-independent set in circle graphs) is NP-complete.

Proof (Sketch): The 2-independent set problem is in NP. Planar 3SAT is polynomially reducible to

the 2-independent set problem.

Circle Graph Circle Graph 11

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Permutation graphs:

Given n chords in a two-sided channel in which each chord connects one point in each side, find a 2-independent set of chords of maximum cardinality.

2-Independent Set Problem2-Independent Set Problem

Circle graphs:

Given n chords in a circle, find a 2- independent set of chords of maximum cardinality.

NP-hard

O(n log n) time

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Permutation GraphPermutation Graph

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Geometric OptimizationGeometric Optimization

Minimum Covering• Given a set of N points, find the smallest di

sk covering the entire set.• Smallest enclosing circle, rectangle, squar

e, annulus, parallel strip, etc.• P-center problem• minimum covering with fixed-size disk,

square,etc. • weighted version

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Minimum Covering Minimum Covering 33

1-center problem (smallest circle)

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Minimum Covering Minimum Covering 22

1-center problem (smallest rectangle)

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Minimum Covering Minimum Covering 11

p-center problem (fixed-size disk covering)

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Matching Matching 22

Pattern matching: Given a target pattern P, and A test pattern Q, find if Q is a sub-pattern in P.

PQ

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Matching Matching 11

P

Q

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Inexact MatchingInexact Matching

PQ

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Bipartite MatchingBipartite Matching Bi-chromatic matching: Given two sets,

P and Q of points, find a one-to-one mapping such that the sum of distances of matched pairs of points is minimized.

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SearchingSearching

Searching Problems:• Given a set S of objects (points, line segments,

etc.), determine if there exists any object with certain properties.

• Given a set S of objects, report objects that satisfy certain properties.

• With preprocessing allowed, how fast can we find objects in S satisfying certain properties?

• What if S is subject to updates?• Memory requirement, query time, update time

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Searching and Retrieval Searching and Retrieval Exact Match (membership problem) Given a set S of objects pi(x1,x2,,…, xn), is

(q1,q2,,…, qn) in S? Range Search Given a set S of objects pi(x1,x2,,…, xn), find

those whose xk lies within a range interval [qk1,qk2] for each k=1,2,..,n.• In 2D, the problem is solvable in O(log2 N + K)

optimal time;• In higher dimensions, it is solvable in O((log2 N)d-1

+ K) time, where K is the output size.

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Example of Range SearchExample of Range Search

salary

Date of birth

19,500,000 19,559,999

3,000

4,000

G. Ometer

born: Aug 19, 1954

Salary: 3500

19,500,000

19,559,999

3,000

4,000

24

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ConclusionConclusion

Computer Science is a discipline addressing areas ranging from theory to applications.

Mathematics is a fundamental to theoretical computer science.

Geometric computing is a very rich area with many applications

Cryptography (one-way function, security) Number theory, combinatorics Packing Problem, Trapezoid Problem, etc.

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Trapezoid Construction ProblemTrapezoid Construction Problem

c

a

bde f

Given d, b, e, f construct a trapezoid as shown

Given a,c, e, f construct a trapezoid as shown:Solved.

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The EndThe End

Thank you for your attention!