Triangulation refinement and approximate shortest paths in weighted regions

Abstract

Let T be a planar subdivision with n vertices. Each face of T has a weight from [1, ρ] ∪ {∞}. A path inside a face has cost equal to the product of its length and the face weight. In general, the cost of a path is the sum of the subpath costs in the faces intersected by the path. For any ε ∈ (0, 1), we present a fully polynomial-time approximation scheme that finds a (1 + ε)-approximate shortest path between two given points in T in O ([EQUATION]) time, where k is the smallest integer such that the sum of the k smallest angles in T is at least π. Therefore, our running time can be as small as O ([EQUATION]) if there are O(1) small angles and it is O ([EQUATION]) in the worst case. Our algorithm relies on a new triangulation refinement method, which produces a triangulation of size O(n + k2) such that no triangle has two angles less than min{π/(2k), π/12}.