# On bounded leg shortest paths problems

Liam Roditty, Michael Segal

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

## Abstract

Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics. Let V be a set of points in a d-dimensional lp-metric space. Let s; t ∈ V and let L be any real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set is at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s-t Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and any real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper present first an algorithm for the apBLSP problem in any lp-metric which, for any fixed ϵ > 0, computes in O(n3(log3 n + log2 n · ϵ-d)) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1 + ϵ). It requires O(n2 log n) space and distance queries are answered in O(log log n) time. This improves on an algorithm with running time of O(n5) given by Bose et al. in [8]. We present also an algorithm for the stBLSP problem that, given s; t ∈ V and a real number L, computes in O(n · polylog(n)) the exact L-bounded shortest path from s to t. This algorithm works in l1 and l∞ metrics. In the Euclidean metric we also obtain an exact algorithm but with a running time of O(n4/3+ϵ), for any ϵ > 0. We end by showing that for any weighted directed graph there is a data structure of size O(n2.5 log n) which is capable of answering path queries with a multiplicative error of (1 + ϵ) in O(log log n + ℓ) time, where ℓ is the length of the reported path. Our results improve upon the results given by Bose et al. [8]. Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of an independent interest.
Original language English Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms 07-09-January-2007 Published - 1 Jan 2007

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