Flying over a polyhedral terrain

Abstract

We consider the problem of computing shortest paths in three-dimensions in the presence of a single-obstacle polyhedral terrain, and present a new algorithm that for any p ⩾ 1 , computes a ( c + ε )-approximation to the L p -shortest path above a polyhedral terrain in O ( n ε 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 a FPTAS for the problem in L 1 metric, a ( 2 + ε ) -factor approximation algorithm in Euclidean space, and a 2-approximation algorithm in the general L p metric.