Approximating Weighted Shortest Paths on Polyhedral Surfaces yz

Consider a simple polyhedron P, possibly non-convex, composed of n triangular regions (faces), in which each region has an associated positive weight. The cost of travel through each region is the distance traveled times its weight. We present and experimentally study several algorithms to compute an approximate weighted geodesic shortest path, 0 (s; t), between two points s and t on the surface of P. Our algorithms are simple, practical, less prone to numerical problems, adaptable to a wide spectrum of weight functions, and use only elementary data structures. An additional feature of our algorithms is that execution time and space utilization can be traded oo for accuracy; likewise, a sequence of approximate shortest paths for a given pair of points can be computed with increasing accuracy (and execution time) if desired. Dynamic changes to the polyhe-dron (removal, insertions of vertices or faces) are easily handled. The key step in these algorithms is the construction of a graph by introducing Steiner points on the edges of the given polyhedron and compute a shortest path in the resulting graph using Dijkstra's algorithm. Diierent strategies for Steiner point placement are examined. Our experimental results obtained on Triangular Irregular Networks (TINs) modeling terrains in Geographical Information Systems (GIS) show that a constant number of Steiner points per edge suuce to obtain high-quality approximate shortest paths. The time complexity of these algorithms for TINs (obtained using real data and randomly generated data) which we Research supported in part by ALMERCO Inc. & NSERC. y For the full version of this paper see 9]. z See 10] for an accompanying video. experimentally investigated is O(n log n). Our analysis bounds the approximate shortest path cost, 0 (s; t), by (s; t) + w max jl e j, where (s; t) denotes the geodesic shortest path between s and t on the boundary of P, l e is the longest edge and w max is the maximum weight of the faces of P, respectively. The worst case time complexity is bounded by O(n 5). We present an alternate algorithm, using graph spanners, that runs in O(n 3 logn) worst case time and reports an approximate path such that 0 (s; t) ((s; t)+w max jl e j), where > 1 is a constant. Already, for planar subdivisions , the best known algorithm for computing exact geodesic weighted shortest path runs in O(n 8 logn) time and …