Constant Approximation of Fr'echet Distance in Strongly Subquadratic Time

2025-03-17

AI Summary

提出随机算法，在强次二次时间内以7+ε的常数因子近似计算多边形曲线间的弗雷歇距离，解决现有方法近似比高的问题。

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Let $ au$ and $sigma$ be two polygonal curves in $mathbb{R}^d$ for any fixed $d$. Suppose that $ au$ and $sigma$ have $n$ and $m$ vertices, respectively, and $mle n$. While conditional lower bounds prevent approximating the Fr'echet distance between $ au$ and $sigma$ within a factor of 3 in strongly subquadratic time, the current best approximation algorithm attains a ratio of $n^c$ in strongly subquadratic time, for some constant $cin(0,1)$. We present a randomized algorithm with running time $O(nm^{0.99}log(n/varepsilon))$ that approximates the Fr'echet distance within a factor of $7+varepsilon$, with a success probability at least $1-1/n^6$. We also adapt our techniques to develop a randomized algorithm that approximates the emph{discrete} Fr'echet distance within a factor of $7+varepsilon$ in strongly subquadratic time. They are the first algorithms to approximate the Fr'echet distance and the discrete Fr'echet distance within constant factors in strongly subquadratic time.

Problem

• Approximating Fréchet distance for curves in constant factor
• Achieving strongly subquadratic time complexity for approximation
• Overcoming conditional lower bounds with randomized algorithm techniques

Innovation

• Randomized algorithm with O(nm^0.99 log(n/ε)) time
• Approximates Fréchet distance within 7+ε factor
• Works for both continuous and discrete Fréchet variants