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 .

We consider the classical geometric problem of determining a shortest path through a weighted domain.We present approximation algorithms that compute e-short paths, i.e., paths whose costs are within a factor of 1 + e of the shortest path costs, for an arbitrary constant e > O, for the following geometric configurations: O n 1 1 runs in (~ log ; (~ +log n)) time.The run time improves to O(;~-log ~logn)) when all weights are equal.This can be used to solve the shortest path problem amidst obstacles in 3-dimensional Euclidean space (ESP-3D).

This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive $\varepsilon $ we show that any n-vertex graph G of genus g can be divided in $O(n + g)$ time into components whose sizes do not exceed $\varepsilon n$ by removing a set C of $O(\sqrt {(g + 1/\varepsilon )} n)$ vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut off from G a piece of no more than $(1 - \varepsilon )n$ vertices by removing a set of $O(\sqrt {n\varepsilon (g\varepsilon + 1)} )$ vertices. Both results are optimal up to a constant factor.