eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
63:1
63:15
10.4230/LIPIcs.SoCG.2024.63
article
Faster Fréchet Distance Approximation Through Truncated Smoothing
van der Horst, Thijs
1
2
Ophelders, Tim
1
2
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
The Fréchet distance is a popular distance measure for curves. Computing the Fréchet distance between two polygonal curves of n vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor 3 cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation quality and running time. Recently, van der Horst et al. (SODA, 2023) presented an O((n²/α) log³ n) time α-approximate algorithm for curves in arbitrary dimensions, for any α ∈ [1, n]. Our main contribution is an approximation algorithm for curves in one dimension, with a significantly faster running time of O(n log³ n + (n²/α³) log²n log log n). Additionally, we give an algorithm for curves in arbitrary dimensions that improves upon the state-of-the-art running time by a logarithmic factor, to O((n²/α) log² n). 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(n²/α) without making sacrifices in the asymptotic approximation factor.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.63/LIPIcs.SoCG.2024.63.pdf
Frécht distance
approximation algorithms
simplification