Halfspace range search: an algorithmic application of k-sets
Discrete & Computational Geometry
Solving query-retrieval problems by compacting Voronoi diagrams
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
A sparse graph almost as good as the complete graph on points in K dimensions
Discrete & Computational Geometry
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
New Techniques for Exact and Approximate Dynamic Closest-point
SIAM Journal on Computing
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Undirected single-source shortest paths with positive integer weights in linear time
Journal of the ACM (JACM)
Dynamic algorithms for geometric spanners of small diameter: randomized solutions
Computational Geometry: Theory and Applications
Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications
SIAM Journal on Computing
Dynamic planar convex hull operations in near-logarithmic amortized time
Journal of the ACM (JACM)
(1 + &egr;&Bgr;)-spanner constructions for general graphs
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Fly cheaply: on the minimum fuel consumption problem
Journal of Algorithms
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
A new approach to all-pairs shortest paths on real-weighted graphs
Theoretical Computer Science - Special issue on automata, languages and programming
Approximating geometric bottleneck shortest paths
Computational Geometry: Theory and Applications
A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A simple linear time algorithm for computing a (2k - 1)-spanner of o(n1+1/k) size in weighted graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Bounded-leg distance and reachability oracles
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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Let V be a set of points in a d-dimensional lp-metric space. Let s, t ε 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 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(n2log 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(n2.5log 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. [8]. Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of an independent interest.