Four Soviets Walk the Dog-Improved Bounds for Computing the Fr\'echet Distance

TL;DR

This paper presents improved algorithms for computing the Fréchet distance between polygonal curves, breaking the quadratic time barrier and providing the first subquadratic algorithm for the weak Fréchet distance, advancing understanding of this classic problem.

Contribution

It introduces a randomized algorithm with subquadratic time complexity and establishes an algebraic decision tree of depth below quadratic, significantly improving upon the classical algorithm.

Findings

New randomized algorithm with $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ time on pointer machine.

Existence of an algebraic decision tree of depth $O(n^{2-\varepsilon})$ for the decision problem.

First subquadratic algorithm for the weak Fréchet distance on a word RAM.

Abstract

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fr\'echet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original $O(n^2 \log n)$ algorithm by Alt and Godau for computing the Fr\'echet distance remains the state of the art (here, $n$ denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fr\'echet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fr\'echet distance between two polygonal curves in time $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ on a pointer machine and in time $O(n^2(\log\log n)^2)$ on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth $O(n^{2-\varepsilon})$, for some $\varepsilon > 0$. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fr\'echet distance on a word RAM.