Geometric kth shortest paths
نویسندگان
چکیده
1 This paper studies algorithmic and combinatorial properties of shortest paths of different homo2 topy types in a polygonal domain with holes. We define the “second shortest path” to be the shortest 3 path that is homotopically different from the (first) shortest path; the kth shortest path for an arbitrary 4 integer k is defined analogously. We introduce the “kth shortest path map”—a structure to answer 5 kth shortest path queries. Given a polygonal domain with n vertices and h holes, we show that the 6 complexity of the kth shortest path map is O(kh + kn), which is tight. Furthermore, we show 7 how to build the kth shortest path map in O((kh + kn) log (kn)) time. We also present a simple 8 visibility-based algorithm to compute the kth shortest path between two points in O(m log n + k) 9 time, where m is the complexity of the visibility graph. This last approach can be extended to com10 pute the kth simple (i.e., without self-intersections) shortest path in O(km(m+ kn) log kn) time. 11 walls of 1-SPM: walls of 2-SPM: walls of 3-SPM: walls of 4-SPM: 12 We invite the reader to play with our applet demonstrating k-SPMs at 13 http://www.cs.helsinki.fi/group/compgeom/kpath_slides/visualize/. 14 ∗B. Speckmann and K. Verbeek were supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.022.707. Sylvester E-B was supported as a Graduate Student Fellow by the National Science Foundation grant no. DGE-1342536. †Courant Institute, NYU [email protected] ‡Mentor Graphics Corporation john [email protected] §Helsinki Institute for IT, CS Dept, University of Helsinki [email protected] ¶Dept. of Mathematics and Computer Science, TU Eindhoven [email protected] ‖Computer Science, University of California Santa Barbara [suri|kverbeek|hakan]@cs.ucsb.edu
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تاریخ انتشار 2013