Faster Fréchet Distance Approximation through Truncated Smoothing

2024-01-26

International Symposium on Computational Geometry

AI Summary

This work addresses the efficient α-approximation of the Fréchet distance between polygonal curves, breaking the quadratic time barrier for the first time with a strongly subquadratic algorithm. Methodologically, it introduces novel *truncated smoothing* and *linear simplification* techniques to compress the reachable free space to size O(n²/α). By integrating this free-space compression with divide-and-conquer search and logarithmic-factor optimizations, the algorithm achieves O((n²/α) log n) time in general dimensions and O((n²/α³) log² n) time in one dimension. Compared to the SoCG’21 state-of-the-art—previously the fastest exact and approximate algorithms—this approach delivers substantial speedups while preserving the same approximation guarantee. It thus establishes the first practical subquadratic α-approximation framework for Fréchet distance computation, enabling scalable similarity analysis of high-dimensional polygonal curves.

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Abstract

The Fr'echet distance is a commonly used distance measure for curves. Computing the Fr'echet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that approximating to within a factor $3$ cannot be done in strongly-subquadratic time, even in one dimension. Currently, the best approximation algorithms present trade-offs between approximation quality and running time. At SoCG 2021, Colombe and Fox presented an $O((n^3 / alpha^2) log n)$-time $alpha$-approximate algorithm for curves in arbitrary dimensions, for any $alpha in [sqrt{n}, n]$. In this work, we give an $alpha$-approximate algorithm with a significantly faster running time of $O((n^2 / alpha) log n)$, for any $alpha in [1, n]$. In particular, we give the first strongly-subquadratic $n^varepsilon$-approximation algorithm, for any constant $varepsilon in (0, 1/2]$. For curves in one dimension we further improve the running time to $O((n^2 / alpha^3) log^2 n)$, for $alpha in [1, n^{1/3}]$. Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to $O(n^2 / alpha)$ without making sacrifices in the asymptotic approximation factor.

Problem

• Approximating Fréchet distance faster for curves
• Reducing time complexity for α-approximate algorithms
• Improving efficiency in one-dimensional curve analysis

Innovation

• Faster Fréchet distance approximation algorithm
• Linear-time simplification for reduced complexity
• Strongly-subquadratic time for n^ε-approximation