Approximating shortest paths on a convex polytope in three dimensions
Given a convex polytope P with n faces in ℝ3, points s, t ∈ ∂P, and a parameter 0 < ∈ ≤ 1, we present an algorithm that constructs a path on ∂P from s to t whose length is at most (1 + ∈)dP(s, t), where dP(s, t) is the length of the shortest path between s and t on ∂P. The algorithm runs in O(n log 1/∈ + 1/∈3) time, and is relatively simple. The running time is O(n + 1/∈3) if we only want the approximate shortest path distance and not the path itself. We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on ∂P to all vertices of P.
Agarwal, PK; Har-Peled, S; Sharir, M; Varadarajan, KR
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