Path planning above a polyhedral terrain

We consider the problem of path planning above a polyhedral terrain and present a new algorithm that for any p >= 1, computes a (c + epsilon)-approximation to the L-p-shortest path above a polyhedral terrain in O(n/epsilon log n log log n) time and O(n log n) space, where n. is the number of vertices of the terrain, and c = 2((p-1)/p). This leads to an E-approximation algorithm for the problem in L-1 metric, and a (root 2 + epsilon)-factor approximation algorithm in Euclidean space.