TY - GEN

T1 - Algorithms for the discrete Fréchet distance under translation

AU - Filtser, Omrit

AU - Katz, Matthew J.

N1 - Funding Information:
O. Filtser was supported by the Ministry of Science, Technology & Space, Israel, and by the Lynn and William Frankel Center. 2 M. Katz was supported by grant 1884/16 from the Israel Science Foundation and grant 2014/170 from the US-Israel Binational Science Foundation.
Publisher Copyright:
© Omrit Filtser and Matthew J. Katz.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Algorithms

KW - BOP

KW - Discrete Fréchet distance

KW - Phrases curve similarity

KW - Translation

UR - http://www.scopus.com/inward/record.url?scp=85049018349&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SWAT.2018.20

DO - 10.4230/LIPIcs.SWAT.2018.20

M3 - Conference contribution

AN - SCOPUS:85049018349

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 201

EP - 2014

BT - 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018

A2 - Eppstein, David

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018

Y2 - 18 June 2018 through 20 June 2018

ER -