Conflict-free Chromatic Art Gallery Coverage

Abstract

We consider a chromatic variant of the art gallery problem, where each
guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen
by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that
ensure a conflict-free covering of all n-vertex polygons? We call this
the conflict-free chromatic art gallery problem. The problem is motivated by applications in distributed robotics and wireless sensor
networks where colors indicate the wireless frequencies assigned to a
set of covering "landmarks" in the environment so that a mobile robot
can always communicate with at least one landmark in its line-of-sight
range without interference.
	
Our main  result shows that  k(n) is O(log  n) for orthogonal  and for
monotone polygons, and O(log^2 n) for arbitrary simple  polygons. By
contrast, if  all guards  visible from each point must have distinct
colors, then k(n)is Omega(n) for arbitrary simple polygons and Omega(sqrt(n)) for orthogonal polygons, as shown by Erickson and LaValle [Proc. of RSS 2011].