Optimal guarding of polygons and monotone chains
نویسندگان
چکیده
In this paper we study several problems concerning the guarding of a polygon or a x-monotone polygonal chain P with n vertices from a set of points lying on it. Our results are: (1) An O(n logn) time sequential algorithm for computing the shortest guarding boundary chain of a polygon P. (2) An O(n logn) time sequential algorithm for computing the smallest set of consecutive edges guarding a polygon P. (3) Parallel algorithms for each of the two previous problems that run in O(logn) time using O(n) processors in the CREW-PRAM computational model. (4) A linear sequential algorithm for computing the smallest left-guarding set of vertices of an x-monotone polygonal chain P. (5) An optimal (n log n) sequential algorithm for computing the smallest guarding set of relays of an x-monotone polygonal chain P. (6) Finally, we consider the problem of nding the problem of placing on a x-monotone polygonal chain P one or several vertex guards which collectively cover the entire surface and show that this problem is NP-complete. The previously best known sequential algorithms for problems (1) and (2) take O(n 2 log n) time.
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