Querying approximate shortest paths in anisotropic regions

  • Authors:

  • Siu-Wing Cheng;Hyeon-Suk Na;Antoine Vigneron;Yajun Wang

  • Affiliations:

  • HKUST, Hong Kong, Hong Kong;Soongsil University, Seoul, South Korea;INRA, Jouy-en-Josas, France;HKUST, Hong Kong, Hong Kong

  • Venue:
  • SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
  • Year:
  • 2007

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Abstract

We present a data structure for answering approximate shortest path queries ina planar subdivision from a fixed source. Let ρ ≥ 1 be a real number.Distances in each face of this subdivision are measured by a possiblyasymmetric convex distance function whose unit disk is contained in aconcentric unit Euclidean disk, and contains a concentric Euclidean disk withradius 1/ρ. Different convex distance functions may be used for differentfaces, and obstacles are allowed. Let ε be any number strictly between 0and 1. Our data structure returns a (1+ε)approximation of the shortest path cost from the fixed source to a querydestination in O(logρn/ε) time. Afterwards, a(1+ε)-approximate shortest path can be reported in time linear in itscomplexity. The data structure uses O(ρ2 n4/ε2 log ρn/ε) space and can be built in O((ρ2 n4)/(ε2)(log ρn/ε)2) time. Our time and space bounds do not depend onany geometric parameter of the subdivision such as the minimum angle.