Algorithms for the discrete Fréchet distance under translation

Omrit Filtser, Matthew J. Katz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

Abstract

The (discrete) Fréchet distance (DFD) is a popular similarity measure for curves. Often the input curves are not aligned, so one of them must undergo some transformation for the distance computation to be meaningful. Ben Avraham et al. [5] presented an O(m3n2(1+log(n/m)) log(m+ n))-time algorithm for DFD between two sequences of points of sizes m and n in the plane under translation. In this paper we consider two variants of DFD, both under translation. For DFD with shortcuts in the plane, we present an O(m2n2 log2(m + n))-time algorithm, by presenting a dynamic data structure for reachability queries in the underlying directed graph. In 1D, we show how to avoid the use of parametric search and remove a logarithmic factor from the running time of (the 1D versions of) these algorithms and of an algorithm for the weak discrete Fréchet distance; the resulting running times are thus O(m2n(1 + log(n/m))), for the discrete Fréchet distance, and O(mn log(m + n)), for its two variants. Our 1D algorithms follow a general scheme introduced by Martello et al. [21] for the Balanced Optimization Problem (BOP), which is especially useful when an e cient dynamic version of the feasibility decider is available. We present an alternative scheme for BOP, whose advantage is that it yields e cient algorithms quite easily, without having to devise a specially tailored dynamic version of the feasibility decider. We demonstrate our scheme on the most uniform path problem (significantly improving the known bound), and observe that the weak DFD under translation in 1D is a special case of it.

Original languageEnglish
Title of host publication16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018
EditorsDavid Eppstein
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages201-2014
Number of pages1814
ISBN (Electronic)9783959770682
DOIs
StatePublished - 1 Jun 2018
Event16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 - Malmo, Sweden
Duration: 18 Jun 201820 Jun 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume101
ISSN (Print)1868-8969

Conference

Conference16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018
Country/TerritorySweden
CityMalmo
Period18/06/1820/06/18

Keywords

  • Algorithms
  • BOP
  • Discrete Fréchet distance
  • Phrases curve similarity
  • Translation

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