We describe a O(log n)-approximation algorithm for computing the

homotopic Frechet distance between two polygonal curves that lie on the boundary of a surface.

Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles.

A key technical ingredient in our analysis is an O(log n)-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic Frechet distance, or the minimum height of a homotopy, is in NP.

Joint work with Sariel Har-Peled, Mohammad Salavatipour and Anastasios Sidiropoulos