The Fréchet distance revisited and extended

Abstract

Given two simplicial complexes, and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the Frechet distance between these curves is minimized. As a polygonal curve is a complex, this generalizes the regular notion of Frechet distance between curves. We also generalize the algorithm to handle an input of k simplicial complexes.

Using this new algorithm we can solve a slew of new problems, from computing a median curve for a given collection of curves, to various motion planning problems. Additionally, we show that for the median curve problem, when the k input curves are c-packed, one can (1+epsilon)-approximate the median curve in near linear time, for fixed k and epsilon.