A near-linear time exact algorithm for the $L_1$-geodesic Fr'echet distance between two curves on the boundary of a simple polygon

2025-04-18

AI Summary

提出近线性时间算法，用L1-测地线距离计算简单多边形边界上两条曲线的Frechet距离，解决计算效率问题。

Technology Category

Search and Optimization: Distributed SearchConstraint Satisfaction and Optimization: Distributed CSP/OptimizationPlanning, 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 graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let $P$ be a polygon with $k$ vertices. Let $R$ and $B$ be two simple, interior disjoint curves on the boundary of $P$, with $n$ and $m$ vertices. We show how to compute the Fr'echet distance between $R$ and $B$ using the geodesic $L_1$-distance in $P$ in $mathcal{O}(k log nm + (n+m) (log^2 nm log k + log^4 nm))$ time.

Problem

• Computes L1-geodesic Fréchet distance between curves
• Handles curves on simple polygon boundaries
• Achieves near-linear time complexity algorithm

Innovation

• Near-linear time exact algorithm
• L1-geodesic Frechet distance
• Simple polygon boundary curves