Approximate Distance Queries For Weighted Polyhedral Surfaces

Let P be a planar polyhedral surface consisting of n triangular faces, each assigned with a positive weight. The weight of a path p on P is defined as the weighted sum of the Euclidean lengths of the portions of p in each face multiplied by the corresponding face weights. We show that, for every epsilon is an element of (0, 1), there exists a data structure, termed distance oracle, computable in time O(n epsilon(-2) log(3) (n/epsilon) log(2) (1/epsilon)) and of size O(n epsilon(-3/2) log(2) (n/epsilon) log(1/epsilon)), such that (1+epsilon)-approximate distance queries in P can be answered in time O(epsilon(-1) log(1/epsilon) + log log n). As in previous work (Aleksandrov, Maheshwari, and Sack (J. ACM 2005) and others), the big-O notation hides constants depending logarithmically on the ratio of the largest and smallest face weights and reciprocally on the sine of the smallest angle of P. The tradeoff between space and query time of our distance oracle is a significant improvement in terms of n over the previous best tradeoff obtained by a distance oracle of Aleksandrov, Djidjev, Guo, Maheshwari, Nussbaum, and Sack (Discrete Comput. Geom. 2010), which requires space roughly quadratic in n for a comparable query time.