Approximate Point-to-Face Shortest Paths in R^3

We address the point-to-face approximate shortest path problem in R: Given a set of polyhedral obstacles with a total of n vertices, a source point s, an obstacle face f , and a real positive parameter ǫ, compute a path from s to f that avoids the interior of the obstacles and has length at most (1 + ǫ) times the length of the shortest obstacle avoiding path from s to f . We present three approximation algorithms that take O(n(L+log(n/ǫ))/ǫ+n(L+log(n/ǫ))/ǫ) time, O(Tp−p(n)∗ (1/ǫ )) time, and O(nλ(n) log(n/ǫ)/ǫ + n log(nρ) log(n log ρ)) time, respectively, where L is the precision of the integers used, Tp−p(n) is the time complexity of the point-to-point shortest path algorithm used, ρ is the ratio of the length of the longest obstacle edge to the Euclidean distance between s and f , and λ(n) is a very slowly-growing function related to the inverse of the Ackermann’s function.