ABSTRACT
In chromatic variants of the art gallery problem, simple polygons are guarded with point guards that are assigned one of k colors each. We say these guards cover the polygon. Here we consider the conflict-free chromatic art gallery problem, first studied by Bärtschi and Suri (Algorithmica 2013): A covering of the polygon 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. We are interested in the smallest number k(n) of colors that ensure such a covering for every n-vertex polygon.
It is known that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. Our main contribution in this paper is an improvement of the upper bound on k(n) to O(log n) for simple polygons.
The bound is achieved through a partitioning of the polygon into weak visibility subpolygons, which is known as a window partition. In a weak visibility polygon, there is a boundary edge e such that each point of the polygon is seen by some point on e. We show for the first time for this special class of polygons an upper bound of O(log n). For the subpolygons of the window partition we prove a novel concept of independence that allows to reuse colors in independent subpolygons. Combining these results leads to the upper bound of O(log n) for arbitrary simple polygons.
- D. Avis and G. T. Toussaint. An optimal algorithm for determining the visibility of a polygon from an edge. IEEE Trans. Comput., C-30(12):910--914, Dec 1981. Google ScholarDigital Library
- A. Bärtschi and S. Suri. Conflict-free chromatic art gallery coverage. Algorithmica, 68(1):265--283, Jan 2014.Google ScholarCross Ref
- P. Bose, A. Lubiw, and J. I. Munro. Efficient visibility queries in simple polygons. Computational Geometry, 23(3):313--335, Nov 2002. Google ScholarDigital Library
- P. Cheilaris. Conflict-free coloring. PhD thesis, The City University of New York, Graduate Faculty in Computer Science, 2009. Google ScholarDigital Library
- P. Cheilaris, L. Gargano, A. A. Rescigno, and S. Smorodinsky. Strong conflict-free coloring for intervals. In K.-M. Chao, T.-s. Hsu, and D.-T. Lee, editors, Algorithms and Computation, volume 7676 of Lecture Notes in Computer Science, pages 4--13. Springer Berlin Heidelberg, 2012.Google ScholarCross Ref
- P. Cheilaris and S. Smorodinsky. Conflict-free coloring with respect to a subset of intervals. CoRR, Apr 2012.Google Scholar
- K. Chen, A. Fiat, H. Kaplan, M. Levy, J. Matoušek, E. Mossel, J. Pach, M. Sharir, S. Smorodinsky, U. Wagner, and E. Welzl. Online conflict-free coloring for intervals. SIAM Journal on Computing, 36(5):1342--1359, Dec 2006. Google ScholarDigital Library
- S.-H. Choi, S. Y. Shin, and K.-Y. Chwa. Characterizing and recognizing the visibility graph of a funnel-shaped polygon. Algorithmica, 14(1):27--51, Jul 1995.Google ScholarDigital Library
- V. Chvátal. A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B, 18(1):39--41, Feb 1975.Google ScholarCross Ref
- P. Colley, A. Lubiw, and J. Spinrad. Visibility graphs of towers. Computational Geometry, 7(3):161--172, Feb 1997. Google ScholarDigital Library
- L. Erickson and S. LaValle. An art gallery approach to ensuring that landmarks are distinguishable. Proceedings of Robotics: Science and Systems, VII:81--88, Jun 2011.Google Scholar
- G. Even, Z. Lotker, D. Ron, and S. Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J. Comput., 33(1):94--136, Nov 2003. Google ScholarDigital Library
- S. K. Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007. Google ScholarCross Ref
- J. E. Goodman and J. O'Rourke, editors. Handbook of Discrete and Computational Geometry, chapter 33.4, 47.1, pages 624, 864. CRC Press, 1997.Google Scholar
- M. J. Katz, N. Lev-Tov, and G. Morgenstern. Conflict-free coloring of points on a line with respect to a set of intervals. Computational Geometry, 45(9):508--514, Nov 2012. Google ScholarDigital Library
- J. O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987. Google ScholarDigital Library
- J. Pach and G. Tardos. Conflict-free colourings of graphs and hypergraphs. Combinatorics, Probability and Computing, 18(05):819--834, Sep 2009. Google ScholarDigital Library
- J. Pach and G. Tardos. Coloring axis-parallel rectangles. J. Combin. Theory Ser. A, 117(6):776--782, Aug 2010. Google ScholarDigital Library
- T. C. Shermer. Recent results in art galleries (geometry). Proceedings of the IEEE, 80(9):1384--1399, Sep 1992.Google ScholarCross Ref
- S. Smorodinsky. Combinatorial Problems in Computational Geometry. PhD thesis, Tel-Aviv University, School of Computer Science, Jun 2003.Google Scholar
- S. Smorodinsky. On the chromatic number of geometric hypergraphs. SIAM J. Discrete Math., 21(3):676--687, Sep 2007. Google ScholarDigital Library
- S. Smorodinsky. Conflict-Free Coloring and its Applications. In I. Barany, K. Boroczky, G. Toth, and J. Pach, editors, Geometry---Intuitive, Discrete, and Convex, volume 24 of Bolyai Society Mathematical Studies. Springer, Berlin, 2014.Google Scholar
- S. Suri. A linear time algorithm for minimum link paths inside a simple polygon. Comput. Vision Graph., 35(1):99--110, Jul 1986. Google ScholarDigital Library
- S. Suri. On some link distance problems in a simple polygon. IEEE Transactions on Robotics and Automation, 6(1):108--113, Feb 1990.Google ScholarCross Ref
- J. Urrutia. Art gallery and illumination problems. In J. Sack and J. Urrutia, editors, Handbook on Computational Geometry, pages 973--1026. Elsevier Science Publishers, Amsterdam, 2000.Google ScholarCross Ref
- F. A. Valentine. Minimal sets of visibility. Proceedings of the American Mathematical Society, 4(6):917--921, Dec 1953.Google ScholarCross Ref
Index Terms
- Improved bounds for the conflict-free chromatic art gallery problem
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