Determining approximate shortest paths on weighted polyhedral surfaces
Lyudmil Aleksandrov, Anil Maheshwari, Jörg-Rüdiger Sack · Journal of the ACM · 2005
In this article, we present an approximation algorithm for solving the single source shortest paths problem on weighted polyhedral surfaces. We consider a polyhedral surface P as consisting of n triangular faces, where each face has an associated positive weight. The cost of travel through a face is the Euclidean distance traveled, multiplied by the face's weight. For a given parameter ε, 0 <ε < 1, the cost of the computed paths is at most 1 + ε times the cost of corresponding shortest paths. Our algorithm is based on a novel way of discretizing polyhedral surfaces and utilizes a generic greedy approach for computing shortest paths in geometric graphs obtained by such discretization. Its running time is O(C(P) n /√ε log n /ε log 1/ε) time, where C(P) captures geometric parameters and the weights of the faces of P .