Art Gallery Theorem

Computational Geometry, WS 2006/07Lecture 8, Part 1

Prof. Dr. Thomas Ottmann

Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 2

Agenda

• Motivation: Guarding art galleries

• Art gallery theorem for simple polygons

• Partitioning of polygons into monotone pieces

• Triangulation of y-monotone polygons

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 3

Guarding art galleries

Problem Definition

Imagine an art gallery room whose floor plan can be modeled by a polygon of n vertices.

Victor Klee asked (1973): How many stationary guards are needed to guard the room?

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 4

Guarding art galleries

The gallery is represented by a simple polygon A guard is represented by a point within the polygon Guards have a viewport of 360°, and of course cannot see through a wall A polygon is completely guarded, if every point within the polygon is guarded by at least one of the watchmen

Visibility polygon: The visibility polygon of a polygon P is defined by the set of all points that are visible from a base point p.

Demo

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Guarding art galleries

We are NOT interested in the minimum number of guards for a specific polygon, but rather want to determine the number of guards that suffice for an arbitrary polygon with n vertices.

Even if two polygons have the same number of vertices, one may be easier to guard than the other.

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 6

Guarding art galleries

If the polygon is complex, it is not obvious to see how many gurads are needed.

Idea: Divide the polygon into pieces that are easy to guard

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 7

Guarding a triangulated polygon

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Triangulation of simple polygons

Does every simple polygon admit a triangulation?If yes, what is the number of triangles?

Does any triangulation of a polygon P lead to the same number of triangles?

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 9

Theorem

Theorem: Every simple polygon admits a triangulation.

Proof: By induction on n. Let n>3, and assume theorem is truefor all m<n. Let P be polygon with n vertices. We first prove the existence of a diagonal in P. Let v be leftmostvertex of P. Let u and v be two neighboring vertices of v.If uw lies in the interior of P we have found a diagonal. Else, there are one or more vertices inside the triangle defined by u, v, and w. Let v´ be the farthest vertex from uw. The segment connecting v´ to v cannot intersect an edge of p (contradicts the definition of v´). Hence vv´ is a diagonal.

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 10

Case 2: uw not completely in P

Proof of Diagonal

Consider leftmost vertex v of P

Case 1: uw completely in P

u

v

w

u

v

w

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 11

Diagonals and Triangulation

So a diagonal exists. Any diagonal cuts P in two simple sub-polygons P1 and P2. Let m1 be the number of vertices of P1and m2 the number of vertices of P2. Both m1 and m2 mustbe smaller than n, so by induction hypothesis P1 and P2 can be triangulated so P can be triangulated as well.

Back to the questionsDoes any triangulation of a polygon P lead to the same number of triangles?If yes, what is the number of triangles?

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 12

Theorem

Theorem: Any triangulation of P contains n-2 triangles.

Proof: By induction on n. n=3 trivial. Let n>3, and assume theorem is true for all m<n. Consider an arbitrary diagonal in some triangulation Tp.This diagonal cuts P into 2 subpolygons with m1 and m2 vertices: m1 + m2 = n + 2. So by induction hypothesis any triangulation of Pi contains mi - 2 triangles (m1 - 2) + (m2 - 2) = n - 2 triangles.

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 13

Upper and lower bounds for the number of guards

We know that for any simple polygon with n vertices n-2 guards are always enough.

But can we do better?

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 14

Upper and lower bounds for the number of guards

Idea: Compute a 3-coloring of the vertices and place guards on nodes with the least-frequently used color.

No two nodes connected by an edge are assigned the same color.

-> Every triangle has all 3 colors. Hence every triangle is watched. Hence the entire polygon is watched.

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 15

Example

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3- coloring

Theorem: The triangulation graph of a polygon P is 3-colorable.

Proof: Induction on n. Clearly, a triangle can be 3-colored. Let n>3, and assume theorem is true for all m<n. By Meister’s Two Ears Theorem, P has an ear abc.

Corollary: n/3 guards are always sufficient to guard a simplePolygon with n vertices.

cb

a

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Art gallery theorem

For a simple polygon with n vertices, n/3 cameras are occasionally necessary and always sufficient to have every point in the polygon visible from at least one of the cameras.

Worst-case example?

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 18

Worst-case example

Demo