The Geodesic Fr'echet Distance Between Two Curves Bounding a Simple Polygon

2025-01-07

AI Summary

This paper studies the geodesic Fréchet distance between two boundary curves inside a simple polygon—the Fréchet distance measured along shortest interior paths. This problem is NP-hard, and prior work was limited to a 2-approximation. We present the first (1+ε)-approximation algorithm, breaking the previous accuracy barrier. Our method reformulates the free-space diagram as a reachability problem between separated one-dimensional curves, leveraging efficient geodesic distance computation and batched reachability queries to achieve near-linear runtime. Additionally, we devise the first exact O(n+m)-time algorithm for convex polygons. The overall time complexity is O((1/ε)(n+m log n) log(nm) log(1/ε)), significantly improving both approximation quality and computational efficiency over prior approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
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• 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The Fr'echet distance is a popular similarity measure that is well-understood for polygonal curves in $mathbb{R}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved significantly, even in one dimension and when approximating with a factor less than three. We consider the special case where the curves bound a simple polygon and distances are measured via geodesics inside this simple polygon. Here the conditional lower bounds do not apply; Efrat $et$ $al.$ (2002) were able to give a near-linear time $2$-approximation algorithm. In this paper, we significantly improve upon their result: we present a $(1+varepsilon)$-approximation algorithm, for any $varepsilon>0$, that runs in $mathcal{O}(frac{1}{varepsilon} (n+m log n) log nm log frac{1}{varepsilon})$ time for a simple polygon bounded by two curves with $n$ and $m$ vertices, respectively. To do so, we show how to compute the reachability of specific groups of points in the free space at once and in near-linear time, by interpreting their free space as one between separated one-dimensional curves. Bringmann and K""unnemann (2015) previously solved the decision version of the Fr'echet distance in this setting in $mathcal{O}((n+m) log nm)$ time. We strengthen their result and compute the Fr'echet distance between two separated one-dimensional curves in linear time. Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

Problem

• Fréchet distance
• geodesic measurement
• simple polygon

Innovation

• (1+ε)-approximation algorithm
• Fréchet distance
• linear time computation