Approximating shortest paths on a nonconvex polyhedron

We present an approximation algorithm that, given the boundary P of a simple, nonconvex polyhedron in R/sup 3/, and two points s and t on P, constructs a path on P between s and t whose length is at most 7(1+/spl epsi/)d/sub P/(s,t), where d/sub P/(s,t) is the length of the shortest path between s and t on P, and /spl epsi/>0 is an arbitrarily small positive constant. The algorithm runs in O(n/sup 5/3/ log/sup 5/3/ n) time, where n is the number of vertices in P. We also present a slightly faster algorithm that runs in O(n/sup 8/5/ log/sup 8/5/ n) time and returns a path whose length is at most 15(1+/spl epsi/)d/sub P/(s,t).

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