Transforming Dogs on the Line: On the Fr\'echet Distance Under Translation or Scaling in 1D

TL;DR

This paper introduces efficient algorithms for computing the Fréchet distance under translation and scaling for 1D curves, significantly improving previous bounds and providing a unified framework for these transformations.

Contribution

The paper presents the first sub-quadratic algorithms for continuous Fréchet distance under translation and scaling in 1D, with a novel framework applicable to various scenarios.

Findings

Algorithms run in O(n^{8/3} log^3 n) time.

Match the running times of discrete case, improving over previous bounds.

Framework reduces continuous problem to discrete across scales.

Abstract

The Fr\'echet distance is a computational mainstay for comparing polygonal curves. The Fr\'echet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fr\'echet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fr\'echet distance under some transformations have been studied, with more work concentrating on the discrete Fr\'echet distance, leaving a significant gap between the discrete and continuous versions of the Fr\'echet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fr\'echet distance under translation on 1-dimensional curves of complexity n with a running time of $\mathcal{O}(n^{8/3} log^3 n)$. To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of $\mathcal{O}(n^{8/3} \log^3 n)$ for the Fr\'echet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of $\tilde{\mathcal{O}}(n^4)$. Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.