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Conflict-free Chromatic Art Gallery Covering with Vertex Guards

Zarei Moradi, Somayeh | 2017

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 50213 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Ghodsi, Mohammad
  7. Abstract:
  8. The visibility is one of the most important problems in computational geometry. Two points are said to be visible to each other, if the line segment that joins them does not intersect any obstacles. The art gallery problem is a well-studied visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. In many of applications, the “guards” are “landmarks” deployed in an environment to help provide navigation and localization service to mobile robots. The mobile device communicates with these landmarks through wireless, or other “line-of-sight” signaling mechanisms. F or the signaling mechanism to work correctly, different landmarks visible to the robot at any position must operate on different frequencies. This motivates a “chromatic” version of the art gallery theorem, where the goal is not to optimize the number of guards, but rather the number of distinct colors needed to distinguish the guards. W e consider a weaker chromatic condition, which is sufficient for the original robotics application of interference-free communication with a guard at all locations. A placement of 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. Thus, for any placement of the robot in the polygon, there is at least one guard that can communicate with the robot without interference. The goal is to determine the smallest number k(n) of colors that ensure a conflict-free coloring of some guard set in all n-vertex polygons. We call this the conflict-free chromatic art gallery problem. In this thesis, We consider a variant of the problem in which the guard positions are restricted to vertices of the polygon. For this case of vertex guards, we show that the conflict-free chromatic guard number has an upper bound of O((log(n))^2) for simple polygons
  9. Keywords:
  10. Art Gallery Problem ; Conflict-Free-Coloring ; Chromatic Art Gallery ; Vertex Guard

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