A polynomial time algorithm for finding an approximate shortest path amid weighted regions

We devise a polynomial-time approximation scheme for the classical geometric problem of finding an ǫ-short path amid weighted regions. In this problem, a triangulated region P comprising of n triangles, a positive weight associated with each triangle, and two points s and t that belong to P are given as the input. The objective is to find a path whose cost is at most (1+ ǫ)OPT where OPT is the cost of an optimal path between s and t. Our algorithm initiates a discretized-Dijkstra wavefront from source s and progresses the wavefront till it strikes t. This result is about a cubic factor (in n) improvement over the Mitchell and Papadimitriou ’91 result [8], which is the only known polynomial time algorithm for this problem to date.