## 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 Avrahametal.[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))), forthediscreteFréchetdistance, O(mnlog(m+n)), fortheshortcuts variant, and O(mnlog(m + n)(loglog(m + n))3) for the weak variant. Our 1D algorithms follow a general scheme introduced by Martello et al. [22] for the Balanced Optimization Problem (BOP), which is especially useful when an eﬃcient dynamic version of the feasibility decider is available. We present an alternative scheme for BOP, whose advantage is that it yields eﬃcient 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 weak DFD under translation in 1D is a special case of it.

Original language | English |
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Pages (from-to) | 156-175 |

Number of pages | 20 |

Journal | Journal of Computational Geometry |

Volume | 11 |

Issue number | 1 |

State | Published - 2020 |

Externally published | Yes |

### Bibliographical note

Funding Information:‡Ben-Gurion University of the Negev, matya@cs.bgu.ac.il. Work on this paper by Matthew Katz was supported by grant 1884/16 from the Israel Science Foundation, and by grant 2014/170 from the US-Israel Binational Science Foundation.

Funding Information:

†Stony Brook University, omrit.filtser@gmail.com. Work on this paper by Omrit Filtser was supported by the Eric and Wendy Schmidt Fund for Strategic Innovation, by the Council for Higher Education of Israel, and by Ben-Gurion University of the Negev.

Publisher Copyright:

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