Constructing Approximate Shortest Path Maps in Three Dimensions

We present a new technique for constructing a data-structure that approximates shortest path maps in IRd . By applying this technique, we get the following two results on approximate shortest path maps in IR3 . (i) Given a polyhedral surface or a convex polytope P with n edges in IR3 , a source point s on P, and a real parameter 0 < ε ≤ 1, we present an algorithm that computes a subdivision of P of size O((n/ε) log(1/ε)) which can be used to answer efficiently approximate shortest path queries. Namely, given any point t on P, one can compute, in O(log (n/ε)) time, a distance ∆P,s(t), such that dP,s(t) ≤ ∆P,s(t) ≤ (1 + ε)dP,s(t), where dP,s(t) is the length of a shortest path between s and t on P. The map can be computed in O(n2 log n + (n/ε) log (1/ε) log (n/ε)) time, for the

• Type: Conference
• Year: 1998
• Where: COMPGEOM
• Authors: Sariel Har-Peled
• Keywords: Approximate Shortest Path, COMPGEOM 1998, Discrete Geometry, Shortest Path, Shortest Path Maps