## 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} ṅε ^{-d} )) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1+ε). It requires O(n ^{2}log 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(ṅpolylog(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 ^{2.5}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. (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.

Original language | English |
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Pages (from-to) | 583-600 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 59 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2011 |

Externally published | Yes |

### Bibliographical note

Funding Information:M. Segal was partially supported by REMON (4G) consortium.

## Keywords

- Bounded leg shortest problem
- Distance query
- Geometric spanners