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Approximating shortest paths on a convex polytope in three dimensions
Journal article   Open access

Approximating shortest paths on a convex polytope in three dimensions

Pankaj K. Agarwal, Sariel Har-Peled, Micha Sharir and Kasturi R. Varadarajan
Journal of the ACM, Vol.44(4), pp.567-584
07/1997
DOI: 10.1145/263867.263869
url
https://doi.org/10.1145/263867.263869View
Published (Version of record) Open Access

Abstract

Given a convex polytope P with n faces in ℝ 3 , points ∈ ∂P, and a parameter 0 < ϵ ≤ 1, we present an algorithm that constructs a path on ∂P from s to t whose length is at most (1 + ϵ) d p (s, t) , where d p (s, t) is the length of the shortest path between s and t on ∂P. The algorithm runs in O(n log 1/ϵ + 1/ϵ 3 ) time, and is relatively simple. The running time is O(n + 1/ϵ 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 .

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