Triangulation Refinement and Approximate Shortest Paths in Weighted Regions

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 ( kn+k ε log 2 ρn ε ) 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 ( n ε log 2 ρn ε ) if there are O(1) small angles and it is O ( n ε log 2 ρn ε ) in the worst case. ∗Research supported by Research Grants Council, Hong Kong, China (project no. 611812) †Hong Kong University of Science and Technology. Email: [email protected] ‡Google Inc. Email: [email protected] §King Abdullah University of Science and Technology. Email: [email protected]