Fr\'echet Distance Revisited and Extended

TL;DR

This paper introduces new algorithms for computing and approximating the Fréchet distance between curves and complexes, extending existing notions and enabling solutions to complex motion planning and mean curve problems.

Contribution

It generalizes the Fréchet distance computation to simplicial complexes and k complexes, and provides efficient algorithms for mean curve approximation and strong Fréchet distance calculation.

Findings

Near linear time (1+epsilon)-approximation for mean curves with c-packed curves

Simpler algorithm for strong Fréchet distance avoiding parametric search

Extension of Fréchet distance to complexes and multiple complexes

Abstract

Given two simplicial complexes in R^d, and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the Fr\'echet distance between these curves is minimized. As a polygonal curve is a complex, this generalizes the regular notion of weak Fr\'echet 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 mean curve for a given collection of curves, to various motion planning problems. Additionally, we show that for the mean curve problem, when the k input curves are c-packed, one can (1+epsilon)-approximate the mean curve in near linear time, for fixed k and epsilon. Additionally, we present an algorithm for computing the strong Fr\'echet distance between two curves, which is simpler than previous algorithms, and avoids using parametric search.