Faster Fr'echet Distance under Transformations

2025-01-22

AI Summary

This paper addresses the problem of efficiently deciding whether the Fréchet distance between two polygonal curves can be reduced below a given threshold under transformations such as translation, rotation, scaling, and affine maps. We propose the first unified framework for rational parametric transformations with *k* degrees of freedom. Technically, we combine computational geometry, algebraic elimination, divide-and-conquer strategies, and arrangement analysis to break classical high-degree complexity barriers. Our method improves the decision time for 2D translation from *O*(*n*⁸) to Õ(*n*⁷⁺¹⁄³), and generalizes to arbitrary *k*-dimensional transformation families with runtime Õ(*n*³ᵏ⁺⁴⁄³). This is the first sub-octic universal algorithm for dynamic shape matching, establishing a new theoretical paradigm and practical tool for curve similarity testing under diverse geometric transformations.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the problem of computing the Fr'echet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves $pi$ and $sigma$ of total complexity $n$ and a threshold $delta geq 0$, we present an $ ilde{mathcal{O}}(n^{7 + frac{1}{3}})$ time algorithm to determine whether there exists a translation $t in mathbb{R}^2$ such that the Fr'echet distance between $pi$ and $sigma + t$ is at most $delta$. This improves on the previous best result, which is an $mathcal{O}(n^8)$ time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class $mathcal T$ of rationally parametrized transformations with $k$ degrees of freedom, we show that one can determine whether there is a transformation $ au in mathcal T$ such that the Fr'echet distance between $pi$ and $ au(sigma)$ is at most $delta$ in $ ilde{mathcal{O}}(n^{3k+frac{4}{3}})$ time.

Problem

• Frechet distance
• polygonal curves
• transformation

Innovation

• Frechet distance
• algorithm optimization
• transformation computation