2026-07-19T16:42:31Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20008 2024-06-06T06:21:16Z ddc:004 open_access

Faster Fréchet Distance Approximation Through Truncated Smoothing van der Horst, Thijs Ophelders, Tim Frécht distance approximation algorithms simplification 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Thijs van der Horst and Tim Ophelders 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.63 urn:nbn:de:0030-drops-200083 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.63 eng https://creativecommons.org/licenses/by/4.0/legalcode