G UARDING A RT G ALLERY By: Venkatsai Bhimanthini Fall’2014

A GENDA What is Guarding Art Gallery ?? Problem statement Graph Construction Graph triangulation Problem solution References

G UARDING AN A RT G ALLERY ?? Art galleries have to guard their collections carefully. This has to be done by video cameras which are usually hung from the ceiling. The images from the cameras are sent to TV screens in the office. o It is easier to keep an eye on few TV screens rather than on many, the num- ber of cameras should be as small as possible.

G UARDING AN A RT G ALLERY ?? An additional advantage of a small number of cameras is that the cost of the security system will be lower o Here we need to find minimum number of cameras required, such that whole gallery must cover.

5 R EAL W ORLD P ROBLEM : Problem: Given the floor plan of an art gallery (= simple polygon P in the plane with n vertices). Place (a small number of) cameras/guards on vertices of P such that every point in P can be seen by some camera.

6 G UARD U SING T RIANGULATIONS Decompose the polygon into shapes that are easier to handle: triangles A triangulation of a polygon P is a decomposition of P into triangles whose vertices are vertices of P. In other words, a triangulation is a maximal set of non-crossing diagonals. diagonal

7 G UARD U SING T RIANGULATIONS A polygon can be triangulated in many different ways. Guard polygon by putting one camera in each triangle: Since the triangle is convex, its guard will guard the whole triangle.

G RAPH C OLORING : Graph coloring mean coloring the vertices of the graph such a way that no adjacent vertices will have the same color and should be colored with minimum number of colors. The minimum number of colors used to color the graph is called chromatic number

9 3-C OLORING A 3-coloring of a graph is an assignment of one out of three colors to each vertex such that adjacent vertices have different colors.

10 3-C OLORING L EMMA Lemma: For every triangulated polgon there is a 3- coloring. Proof: Consider the dual graph of the triangulation: –vertex for each triangle –edge for each edge between triangles

11 3-C OLORING L EMMA Lemma: For every triangulated polgon there is a 3- coloring. The dual graph is a tree (acyclic graph; minimally connected): Removing an edge corresponds to removing a diagonal in the polygon which disconnects the polygon and with that the graph.

12 3-C OLORING L EMMA Lemma: For every triangulated polgon there is a 3- coloring. Traverse the tree (DFS). Start with a triangle and give different colors to vertices. When proceeding from one triangle to the next, two vertices have known colors, which determines the color of the next vertex.

13 A RT G ALLERY T HEOREM Theorem 2: For any simple polygon with n vertices guards are sufficient to guard the whole polygon. There are polygons for which guards are necessary. n3n3    n3n3    Proof: For the upper bound, 3-color any triangulation of the polygon and take the color with the minimum number of guards. Lower bound: n3n3    spike s Need one guard per spike.

P ROBLEM S OLUTION :

R EFERENCES f_Graph_Coloring_in_Modern_Computer_Science ook_HowToGuardAnArtGallery_Michael/Sample _Chapter_Art_Gallery_Michael.pdf

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