A faster algorithm for the discrete Fr\'echet distance under translation

TL;DR

This paper presents a more efficient algorithm for computing the minimum discrete Fréchet distance under translation between two point sequences in 2D, improving upon previous computational complexity.

Contribution

The authors develop a faster algorithm for the discrete Fréchet distance under translation in 2D, reducing the time complexity from previous methods.

Findings

Algorithm runs in $O(m^3n^2(1+rac{ ext{log}(n/m)}{ ext{log}(m+n)})$ time

Significantly improves previous $O(m^3n^3 ext{log}(m + n))$ time algorithm

Applicable for sequences where $m \

Abstract

The discrete Fr\'echet distance is a useful similarity measure for comparing two sequences of points $P=(p_1,\ldots, p_m)$ and $Q=(q_1,\ldots,q_n)$. In many applications, the quality of the matching can be improved if we let $Q$ undergo some transformation relative to $P$. In this paper we consider the problem of finding a translation of $Q$ that brings the discrete Fr\'echet distance between $P$ and $Q$ to a minimum. We devise an algorithm that computes the minimum discrete Fr\'echet distance under translation in $\mathbb{R}^2$, and runs in $O(m^3n^2(1+\log(n/m))\log(m+n))$ time, assuming $m\leq n$. This improves a previous algorithm of Jiang et al.~\cite{JXZ08}, which runs in $O(m^3n^3 \log(m + n))$ time.