Geometric k Shortest Paths
نویسندگان
چکیده
We consider the problem of computing k shortest paths in a two-dimensional environment with polygonal obstacles, where the jth path, for 1 j k, is the shortest path in the free space that is also homotopically distinct from each of the first j 1 paths. In fact, we consider a more general problem: given a source point s, construct a partition of the free space, called the kth shortest path map (k-SPM), in which the homotopy of the kth shortest path in a region has the same structure. Our main combinatorial result establishes a tight bound of ⇥(k2h + kn) on the worst-case complexity of this map. We also describe an O((k3h + k2n) log (kn)) time algorithm for constructing the map. In fact, the algorithm constructs the jth map for every j k. Finally, we present a simple visibility-based algorithm for computing the k shortest paths between two fixed points. This algorithm runs in O(m log n + k) time and uses O(m + k) space, where m is the size of the visibility graph. This latter algorithm can be extended to compute k shortest simple (non-self-intersecting) paths, taking O(k2m(m+ kn) log(kn)) time. We invite the reader to play with our applet demonstrating k-SPMs [10]. B. Speckmann and K. Verbeek were partially supported by the Netherlands’ Organisation for Scientific Research (NWO) under project nos. 639.022.707 and 639.023.208. S. Eriksson-Bique was supported as a Graduate Student Fellow by the National Science Foundation grant no. DGE-1342536. S. Eriksson-Bique, V. Polishchuk and T. Talvitie were supported by the Academy of Finland grant 1138520 and University of Helsinki Research Funds. The research of Subhash Suri, Kevin Verbeek and Hakan Yildiz was partially supported by the NSF grant CCF-1161495. Courant Institute, NYU. [email protected] Mentor Graphics Corporation. john [email protected] Communications and Transport Systems, ITN, Linköping University. [email protected] Dept. of Mathematics and Computer Science, TU Eindhoven.
منابع مشابه
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