Handouts: HW1
2/4/08: More on polygons, visibility, and art galleries


Statement of art gallery theorem (G(n)=floor(n/3) guards are sufficient to guard an n-gon and sometimes necessary) and proof by Fisk: (1) triangulate, (2) 3-color the triangulation graph, (3) place guards at the smallest color class of vertices (which must be of size at most floor(n/3), and these vertices see all of P, since every triangle has a corner of each of the 3 colors.). Recall: Chvatal's comb example to show necessity.
Do examples (e.g., convex polygon, with a "bad" choice of triangulation) showing that Fisk's proof gives a method yielding at most floor(n/3) guards, BUT this is not necessarily the optimal number of guards (even if "redundant" guards are deleted)
Discussion of ordinary (usual) visibility versus clear visibility.

I discussed briefly the fact that computing g(P) exactly is known to be "hard" (it is NP-complete to decide if g(P) is at most k).
Discussion of the (trivial) convex polygon case (one guard is needed), star-shaped polygons, and the "twisted hexagon" ("Godfried's favorite polygon") polygon, for which we need 2 guards, but there is no pair of independent witness points (a slightly more involved argument is needed in order to claim that 2 guards are needed).

Open challenge: Can you devise an efficient heuristic method for finding a "small" (not necessarily optimal) set of guards that cover P? A Java applet would be ideal!