On bounded leg shortest paths problems

Let V be a set of points in a d-dimensional l_p-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 l_p-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(n^4/3+ϵ), for any ϵ > 0. We end by showing that for any weighted directed graph there is a data structure of size O(n^2.5 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.