On Bounded Leg Shortest Paths Problems

Abstract

Let V be a set of points in a d-dimensional l _p -metric space. Let s,t∈V and let L be a 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 has length of 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 a 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 we obtain the following results:

• 1. An algorithm for the apBLSP problem in any l _p -metric which, for any fixed ε>0, computes in O(n ³(log ³ n+log ² n⋅ε ^−d)) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1+ε). It requires O(n ²log n) space and distance queries are answered in O(log log n) time.

• 2. 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 l ₁ 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.

• 3. For any weighted directed graph we give a data structure of size O(n ^2.5log 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. (Comput. Geom. Theory Appl. 29:233–249, 2004). Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of independent interest.