Computing the Discrete Fr\'echet Distance in Subquadratic Time

TL;DR

This paper introduces a subquadratic algorithm for computing the discrete Fréchet distance between two planar point sequences, significantly improving the efficiency over previous quadratic-time methods by leveraging geometric encoding and automata.

Contribution

It presents the first subquadratic algorithm for discrete Fréchet distance in the plane, using geometric insights and automata-based state encoding.

Findings

Algorithm runs in O(mn log log n / log n) time

Uses O(n + m) space

Achieves subquadratic complexity for the problem

Abstract

The Fr\'echet distance is a similarity measure between two curves $A$ and $B$: Informally, it is the minimum length of a leash required to connect a dog, constrained to be on $A$, and its owner, constrained to be on $B$, as they walk without backtracking along their respective curves from one endpoint to the other. The advantage of this measure on other measures such as the Hausdorff distance is that it takes into account the ordering of the points along the curves. The discrete Fr\'echet distance replaces the dog and its owner by a pair of frogs that can only reside on $n$ and $m$ specific pebbles on the curves $A$ and $B$, respectively. These frogs hop from a pebble to the next without backtracking. The discrete Fr\'echet distance can be computed by a rather straightforward quadratic dynamic programming algorithm. However, despite a considerable amount of work on this problem and its variations, there is no subquadratic algorithm known, even for approximation versions of the problem. In this paper we present a subquadratic algorithm for computing the discrete Fr\'echet distance between two sequences of points in the plane, of respective lengths $m\le n$. The algorithm runs in $O(\dfrac{mn\log\log n}{\log n})$ time and uses $O(n+m)$ storage. Our approach uses the geometry of the problem in a subtle way to encode legal positions of the frogs as states of a finite automata.