An ε — Approximation algorithm for weighted shortest paths on polyhedral surfaces

Abstract

Let P be a simple polyhedron, possibly non-convex, whose boundary is composed of n triangular faces, and in which each face has an associated positive weight. The cost of travel through each face is the distance traveled multiplied by the face's weight. We present an ε-approximation algorithm for computing a weighted shortest path on P, i.e. the ratio of the length of the computed path with respect to the length of an optimal path is bounded by (1 + ε), for a given ε > 0.We give a detailed analysis to determine the exact constants for the approximation factor. The running time of the algorithm is O(mn log mn + nm ²). The total number of Steiner points, m, added to obtain the approximation depends on various parameters of the given polyhedron such as the length of the longest edge, the minimum angle between any two adjacent edges of P and the minimum distance from any vertex to the boundary of the union of its incident faces and the ratio of the largest (finite) to the smallest face weights of P. Lastly, we present an approximation algorithm with an improved running time of O(mn log mn), at the cost of trading off the constants in the path accuracy. Our results present an improvement in the dependency on the number of faces, n, to the recent results of Mata and Mitchell [10] by a multiplicative factor of n ²/log n, and to that of Mitchell and Papadimitriou [11] by a factor of n ⁷.

Research supported in part by ALMERCO Inc. & NSERC