Geometric k shortest paths

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2015-01-01
Eriksson-Bique, Sylvester
Hershberger, John
Polishchuk, Valentin
Speckii, Bettina
Suri, Subhash
Talvitie, Topi
Verbeek, Kevin
Yıldız, Hakan
Copyright © 2015 by the Society for Industrial and Applied Mathmatics.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 θ (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 to 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 (fcn)) time. We invite the reader to play with our applet demonstrating fc-SPMs [10].

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Citation Formats
S. Eriksson-Bique et al., “Geometric k shortest paths,” California, Amerika Birleşik Devletleri, 2015, vol. 2015-January, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/93695.