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Art Gallery Theorems and Art Gallery Theorems and Algorithms Daniel G. Aliaga Computer Science Department Purdue University

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Page 1: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Art Gallery Theorems andArt Gallery Theorems and Algorithmsg

Daniel G. Aliaga

Computer Science DepartmentPurdue University

Page 2: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Art Gallery

Problem: determine the minimum number Problem: determine the minimum number of guards sufficient to cover the interior of an n-wall art galleryf w g y

Victor Klee, 1973Vasek Chvatal, 1975,

Main reference for this material: Art Gallery Theorems and Algorithms, Joseph O’Rourke, Oxford

University Press, 1987

Page 3: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Contents

Interior Visibility G ll P blArt Gallery Problem

OverviewFisk’s ProofReflex VerticesReflex VerticesConvex PartitioningOrthogonal Polygons

Mobile GuardsMiscellaneous Shapes

Star, Spiral, MonotoneExterior Visibility

F rtr ss Pr bl mFortress ProblemPrison Yard Problem

Minimal Guards

Page 4: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Contents

Interior Visibility G ll P blArt Gallery Problem

OverviewFisk’s ProofReflex VerticesReflex VerticesConvex PartitioningOrthogonal Polygons

Mobile GuardsMiscellaneous Shapes

Star, Spiral, MonotoneExterior Visibility

F rtr ss Pr bl mFortress ProblemPrison Yard Problem

Minimal Guards

Page 5: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Definitions

P is a simple polygon (i.e., does not cross over p p yg ( ,itself)Point x ∈ P “covers” a point y ∈ P if xy ⊆ PL t G(P) b th i i b k f i t f P Let G(P) be the minimum number k of points of P, such that for any y ∈ P, some x=x1…xk covers yLet g(n) be the max(G(P)) over all polygons of n Let g(n) be the max(G(P)) over all polygons of n vertices

Thus, g(n) guards are occasionally necessary and always sufficientsufficient

Page 6: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Guard Placement

1 Can we just place one guard on every 3rd1. Can we just place one guard on every 3vertex?

03

96

Page 7: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Guard Placement

1 Can we just place one guard on every 3rd1. Can we just place one guard on every 3vertex? – No!

x

03

x is not visible96

Page 8: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Guard Placement

1 Can we just place one guard on every 3rd1. Can we just place one guard on every 3vertex? – No!

x2

69

15 21

x1 x0

0312

15 21

182724

one of xi is not visible

Page 9: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Guard Placement

2 If guards placed so they can see all the 2. If guards placed so they can see all the walls, does that imply they can see all the interior?

No!

x

x is not visible

Page 10: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Guard Placement

3 If we restrict guards to vertices is 3. If we restrict guards to vertices, is gv(n) = g(n)?

In general, yes, equal for g(n) = max(G(P))In general, yes, equal for g(n) max(G(P))

x

one point guard x

or

two vertex guards?

Page 11: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Art Gallery

Theorem: floor(n/3) guards are Theorem: floor(n/3) guards are occasionally necessary and always sufficient to cover a polygon of n verticesff p yg f

“Chvatal’s Art Gallery Theorem”“Watchman Theorem”

Page 12: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fisk’s Proof

g(n) = floor(n/3)g(n) floor(n/3)Published in 1978 (three years are Chvatal’s original proof, but it is much more compact)g p , p )

NecessityNecessityg(n) ≥ floor(n/3) are sometimes necessary

Sufficiencyg(n) ≤ floor(n/3) are always sufficientg(n) ≤ floor(n/3) are always sufficient

Page 13: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Necessity: Base Cases

n=3 n=5n 3 n 5

n=4 n=6

Page 14: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Necessity: Base Cases

n≥6n≥6

g(n) ≥ floor(n/3)g(n) ≥ floor(n/3)

Page 15: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Sufficiency: Fisk’s Proof

Step 1 of 3Step 1 of 3Triangulate the polygon P by adding only internal diagonalsg

f

h

e

fg

i

a

b

cd

j

Page 16: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Triangulation Theorem

A polygon of n-vertices may be partitioned A polygon of n vertices may be partitioned into n-2 triangles by the addition of n-3 internal diagonalsg

Page 17: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Sufficiency: Fisk’s Proof

Step 2 of 3Step 2 of 3Perform a 3-coloring of the triangulation graph

Using three colors, no two adjacent nodes have same g , jcolor

1

3

1

12

2

1 2

33 3

Page 18: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Four Color Theorem

Problem stated in 1852 by Francis Guthrie Problem stated in 1852 by Francis Guthrie and Augustus De Morgan

“Given a map on a flat plane, what is the Given a map on a flat plane, what is the minimum number of colors needed to color the different regions of the map in such a way that

t dj t i h th l “no two adjacent regions have the same color.“

Page 19: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Four Color Theorem

Several attempted proofs and algorithmsSeveral attempted proofs and algorithmsKempe (1879), Tait (1880), Birkhoff (1922), …

Appel and Haken - first complete proof Appel and Haken - first complete proof (1976)Robertson Sanders Seymour and Thomas Robertson, Sanders, Seymour, and Thomas - second more compact proof (1994)

Page 20: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Four Color Theorem

The proof creates a large number of cases The proof creates a large number of cases (~1700 for Appel-Haken and ~600 for Robertson et al.).)A computer is used to rigorously check the casescasesSolution is (still) controversial because of the use of a computerthe use of a computer

Page 21: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Sufficiency: Fisk’s Proof

Step 3 of 3Step 3 of 3Note that one of three colors must be used no more than floor(1/3) of the time( )

Let a,b,c be # of nodes of each colora ≤ b ≤ c and n = a + b + cIf /3 h ( b ) If a > n/3, then (a+b+c) ≥ nThus a ≤ floor(n/3)Since each triangle is a complete graph each triangle Since each triangle is a complete graph, each triangle has a node of color ‘a’Since each triangle is convex and the triangles partition all of P at most ‘a’ guards are necessary!partition all of P, at most a guards are necessary!

Page 22: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fisk’s Proof

NecessityNecessityg(n) ≥ floor(n/3) are sometimes necessary

Sufficiencyg(n) ≤ floor(n/3) are always sufficient

Thus, g(n) = floor(n/3)

O(nlogn) overall algorithm

Page 23: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Reflex Vertices

We wish to investigate the art gallery We wish to investigate the art gallery question as a function of r (the number of reflex vertices of a polygon)f f p yg )

reflex vertex r ≤ (n-3)

Page 24: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Reflex Vertices

NecessityNecessityHow many reflex-vertex guards are necessary?

1 needed r needed

Page 25: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Reflex Vertices

NecessityNecessityr guards are sometimes necessary

SufficiencySufficiencyPlace 1 guard at each reflex vertex

Proved via a convex partitioning of the polygon PProved via a convex partitioning of the polygon PAny polygon P can be partitioned into at most r+1 convex pieces

Page 26: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Reflex Vertices

NecessityNecessityr guards are sometimes necessary

SufficiencySufficiencyPlace 1 guard at each reflex vertex

TheoremTheoremr guards are occasionally necessary and always sufficient to see the interior of a n-gon of r ≥ 1 sufficient to see the interior of a n gon of r ≥ 1 reflex vertices

Page 27: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Convex Partitioning

Naïve Algorithm (Chazelle 1980)Naïve Algorithm (Chazelle 1980)

Because at most two reflex vertices can be Because at most two reflex vertices can be resolved by a single cut, the minimum number of pieces is m=ceil(r/2)+1pThis approach achieves no more than r+1≤2m in O(rn)=O(n2) time

Page 28: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Convex Partitioning

A fast algorithm: O(n log log n)A fast algorithm: O(n log log n)Any triangulation can be divided into 2r+1 convex pieces by removing diagonalsp y g g

Page 29: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Convex Partitioning

Chazelle 1980Chazelle 1980O(n3) optimal algorithm using dynamic programmingp g g(description is 97 pages long)

Page 30: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Orthogonal Polygons

Kahn Klawe Kleitman 1980Kahn, Klawe, Kleitman 1980Floor(n/4) guards are occasionally necessary and always sufficientyBased on convex quadrilateralization

Any orthogonal polygon P is convexly d il i bl ( h )quadrilaterizable (theorem)

Page 31: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Orthogonal Polygons

NecessityNecessityg(n) ≥ floor(n/4)

Sufficiency Four-colorable, and thus:thus:

g(n) ≤ floor(n/4)

Theorem: g(n)=floor(n/4) g( ) f ( )

Page 32: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Orthogonal Polygons

In an orthogonal polygonIn an orthogonal polygonn verticesc interval vertices with π/2c interval vertices with π/2r interval vertices with 3π/2n = c + rn c rsum of internal angles (n-2)πyields n=2r+4y

Theorem restated as g(n)=floor(r/2)+1Theorem restated as g(n) floor(r/2) 1

Page 33: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Quadrilateralization

Sacks’s AlgorithmSacks s AlgorithmO(nlogn)

Lubiw’s AlgorithmLubiw s AlgorithmO(nlogn)

Page 34: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Mobile Guards

TheoremTheorem

Shape Stationary Mobile

General floor(n/3) floor(n/4)

Orthogonal floor(n/4) floor((3n+4)/16)

In general, only ¾ as many mobile guards are needed

g ( ) (( ) )

as stationary guards

Page 35: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Mobile Guards

General PolygonsGeneral Polygons

Vertex guards

Ed dEdge guards

Diagonal guards

Page 36: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Mobile Guards

Goal of the proofGoal of the proofGiven a triangulation graph T

Vertex guard = nodegEdge guard = adjacent arcDiagonal guard = any arc

The analog of covering is dominationA collection of guards C={g1,…,gk} dominates triangulation graph T if every face has at least triangulation graph T if every face has at least one of its three nodes in some gi ∈C.

Page 37: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Mobile Guards

NecessityNecessity

Polygon that requires floor(n/4) edge diagonal (or line) guards

Sufficiency: a little more complicated…

Polygon that requires floor(n/4) edge, diagonal (or line) guards

ff y p

Page 38: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Miscellaneous Shapes

(General polygon convex orthogonal)(General polygon, convex, orthogonal)Star, spiral, monotone

Page 39: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Star Shape

A star polygon P is a polygon that may be A star polygon P is a polygon that may be covered by a single point guard

Page 40: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Star Shape

Toussaint’s TheoremToussaint s TheoremA star polygon P requires floor(n/3) vertex guardsg

Page 41: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Star Shape

Toussaint’s TheoremToussaint s TheoremA star polygon P requires floor(r/2)+1 reflex guardsg

Page 42: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Star Shape

Toussaint’s TheoremToussaint s TheoremA star polygon P requires at least floor(n/5) edge guardsg g

Page 43: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Star Shape

Toussaint’s TheoremToussaint s TheoremFor a star polygon P

Unrestricted patrol, one line guard is neededp , gRestricted to diagonal lines, two are needed

Page 44: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Spiral Polygon

A spiral polygon is a polygon with at most A spiral polygon is a polygon with at most one chain of reflex vertices

Page 45: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Monotone Polygon

A polygon with no “doubling back” with A polygon with no doubling back with respect to a line

Page 46: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Spiral and Monotone Polygons

Aggarwal’s TheoremAggarwal s Theoremfloor(n/3) vertex guards are neededfloor(r/2)+1 reflex-vertex guards are neededfloor(r/2) 1 reflex vertex guards are neededfloor((n+2)/5) diagonals guards are needed

Page 47: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Contents

Interior Visibility G ll P blArt Gallery Problem

OverviewFisk’s ProofReflex VerticesReflex VerticesConvex PartitioningOrthogonal Polygons

Mobile GuardsMiscellaneous Shapes

Star, Spiral, MonotoneExterior Visibility

F rtr ss Pr bl mFortress ProblemPrison Yard Problem

Minimal Guards

Page 48: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Exterior Visibility

“Fortress Problem”Fortress Problem“Prison Yard Problem

(independently stated by Derick Wood and Joseph Malkelvitch early 1980s)Joseph Malkelvitch, early 1980s)

Page 49: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

How many vertex guards are needed to see How many vertex guards are needed to see the exterior of a polygon P?

Simplex convex polygon

ceil(n/2) vertex guardsceil(n/2) vertex guards

Page 50: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygon

ceil(n/2) vertex guards?

Page 51: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygon

Page 52: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygon

v∞

x

Page 53: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygon

v∞

x’x’’

x

Page 54: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygonThree-color the resulting triangulation graph T (of n+2 nodes)( )

Page 55: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygonIf least frequently used color is red and v∞ is not red then,,

floor((n+2)/3) vertex guards are needed

Page 56: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygonArbitrary polygonIf least frequently used color is red and v∞ isred then, ,

No guard can be placed at v∞ because it’s not part of original polygonThus place guards at second least frequently used Thus, place guards at second least frequently used colora ≤ b ≤ c and a + b + c = n + 2a ≥ 1 and b + c ≤ n + 1b ≤ floor((n+1)/2)=ceil(n/2) vertex guards are needed

Page 57: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Arbitrary polygon (Summary)Arbitrary polygon (Summary)1. Triangulate the convex hull of the polygon P2. Add edges from all exterior vertices to new vertex v∞3. Split a vertex x into x’ and x’’4. Open-up the convex hull, straighten the lines to v∞, and form a triangulation graph T of (n+2) nodesform a triangulation graph T of (n 2) nodes5. Three-color graph T6. Use least or second least frequently used color

At most ceil(n/2) vertex guards are needed

Page 58: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygon

Page 59: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygon

Page 60: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygon

Page 61: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygon

l

Orthogonal polygon

Solution A

Solution B

ceil(n/4)+1 vertex guards necessary

Page 62: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygon

e

Page 63: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygon

Q

e

Page 64: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygonInterior of new polygon P’ coincides with the immediate exterior of P, except for Q which is , p Qexterior to both

Page 65: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Orthogonal polygonOrthogonal polygonFor P’ of n+4 vertices, floor(r/2)+1 or floor((n+4)/4) vertex guards suffice to cover (( ) ) gthe interior

None of the new vertices of P’ are reflex verticesN d dditi l f QNeed an additional one for Q

Thus, floor(n/4)+2 vertex guards are sufficientFor n mod 4=0 ceil(n/4)+1For n mod 4=0, ceil(n/4)+1

Page 66: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Guards in the planeGuards in the planeNecessity

ceil(n/3) point guards needed

(n=3k+4 and k+2 guards)(n=3k+4 and k+2 guards)

Page 67: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Guards in the planeGuards in the planeSufficiency a’ a’’

λ1 λ22

Page 68: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Guards in the planeGuards in the planeNew triangulated polygon P’ of n+3 vertices

floor((n+3)/3) = ceil((n+1)/3) point guards(( ) ) (( ) ) p g

Page 69: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress Problem

Guards in the planeGuards in the planeMore lengthy proof to remove “1/3 of a guard”

Add only 2 guards and 3-color triangulationy g gIf even hull vertices, trivialIf odd hull vertices, need some extra work

Result: ceil(n/3) point guards are necessary to cover the exterior of a polygon P of n verticescover the exterior of a polygon P of n vertices

Nice duality with floor(n/3) for the interior

Page 70: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Prison Yard Problem

How many vertex guards are needed to How many vertex guards are needed to simultaneously see the exterior and interior of polygon P?f p yg

Page 71: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Prison Yard Problem

General PolygonsGeneral Polygons

Worst case is a convex polygonWorst-case is a convex polygonceil(n/2) vertex guards needed

Multiply-connected polygonsmin(ceil(n/2), floor((n+ceil(h/2))/2), floor(2n/3))

Page 72: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Prison Yard Problem

Orthogonal PolygonsOrthogonal Polygons

floor((7n/16)+5) vertex guards are neededfloor((7n/16)+5) vertex guards are needed

Page 73: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Fortress/Prison Yard Problem

Problem Techniques Guards TimeProblem Techniques Guards Time

Fortress

General Triangulation, 3-coloring ceil(n/2) O(T)

Orthogonal L-shaped partition ceil(n/4)+1 O(T)

Prison Yard

General Exterior ceil(n/2)+r O(T)

Triangulation, 4-coloring floor((n+ceil(h/2))/2) O(n2)

Triangulation, 4-coloring floor(2n/3) O(n2)

Exterior triangulation 3 coloring floor(2n/3+1) O(T)Exterior, triangulation, 3-coloring floor(2n/3+1) O(T)

Orthogonal Exterior, quad., 4-coloring floor((7n/16)+5) O(T)

Page 74: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Contents

Interior Visibility G ll P blArt Gallery Problem

OverviewFisk’s ProofReflex VerticesReflex VerticesConvex PartitioningOrthogonal Polygons

Mobile GuardsMiscellaneous Shapes

Star, Spiral, MonotoneExterior Visibility

F rtr ss Pr bl mFortress ProblemPrison Yard Problem

Minimal Guards

Page 75: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Minimal Guard Coverage

Seek the placement of a minimal number of Seek the placement of a minimal number of guards that cover a polygon P

In general, a NP-complete problemIn general, a NP complete problem

Page 76: Art Gallery Theorems andArt Gallery Theorems and Algorithms · Art Gallery Problem: determine the minimum number of guards sufficient to cover the interior of an n-wwgyall art gallery

Minimal Guard Coverage

Polygon Cover PartitionPolygon Cover Partition

w. Steiner w/o Steiner w. Steiner w/o Steiner

Simple NP-hard NP-complete ? O(n7logn)pPolygons

p ( g )

Polygons with Holes

NP-hard NP-complete NP-hard NP-complete