Approximating the Fr\'echet distance when only one curve is $c$-packed

TL;DR

This paper introduces a nearly-linear time algorithm that approximates the Fréchet distance between a c-packed curve and a general curve in Euclidean space, requiring only one curve to be c-packed, thus relaxing previous assumptions.

Contribution

It presents the first algorithm that approximates the Fréchet distance with only one c-packed curve, improving computational efficiency and applicability.

Findings

Nearly-linear time algorithm for approximation

Works for constant epsilon, dimension, and c

Applicable to real-world curve analysis

Abstract

One approach to studying the Fr\'echet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is $c$-packedness. Existing algorithms for computing the Fr\'echet distance between $c$-packed curves require both curves to be $c$-packed. In this paper, we only require one of the two curves to be $c$-packed. Our result is a nearly-linear time algorithm that $(1+\varepsilon)$-approximates the Fr\'echet distance between a $c$-packed curve and a general curve in $\mathbb R^d$, for constant values of $\varepsilon$, $d$ and $c$.