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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fréchet measure defined by a sup min - the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible.
We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs.
We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk. On Flipping the Fréchet Distance. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.51, author = {Filtser, Omrit and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin}, title = {{On Flipping the Fr\'{e}chet Distance}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {51:1--51:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.51}, URN = {urn:nbn:de:0030-drops-175548}, doi = {10.4230/LIPIcs.ITCS.2023.51}, annote = {Keywords: curves, polygons, distancing measure} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the (1+ε,r)-approximate near-neighbor problem for curves (ANNC) under some similarity measure δ, the goal is to construct a data structure for a given set 𝒞 of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ 𝒞 such that δ(Q,C)≤ r, then return a curve C' ∈ 𝒞 with δ(Q,C') ≤ (1+ε)r. There exists an efficient reduction from the (1+ε)-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C ∈ 𝒞 with δ(Q,C) ≤ (1+ε)⋅δ(Q,C^*), where C^* is the curve of 𝒞 most similar to Q.
Given a set 𝒞 of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n⋅ O(1/ε)^{md} storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds.
Further, we also consider the asymmetric version of ANNC, where the length of the query curves is k ≪ m, and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.

Arnold Filtser, Omrit Filtser, and Matthew J. Katz. Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filtser_et_al:LIPIcs.ICALP.2020.48, author = {Filtser, Arnold and Filtser, Omrit and Katz, Matthew J.}, title = {{Approximate Nearest Neighbor for Curves - Simple, Efficient, and Deterministic}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {48:1--48:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.48}, URN = {urn:nbn:de:0030-drops-124555}, doi = {10.4230/LIPIcs.ICALP.2020.48}, annote = {Keywords: polygonal curves, Fr\'{e}chet distance, dynamic time warping, approximation algorithms, (asymmetric) approximate nearest neighbor, range counting} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Let A and B be two sets of points in R^d, where |A|=|B|=n and the distance between them is defined by some bipartite measure dist(A, B). We study several problems in which the goal is to translate the set B, so that dist(A, B) is minimized. The main measures that we consider are (i) the diameter in two and three dimensions, that is diam(A,B) = max {d(a,b) | a in A, b in B}, where d(a,b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is uni(A,B) = diam(A,B) - d(A,B), where d(A,B)=min{d(a,b) | a in A, b in B}, and (iii) the union width in two and three dimensions, that is union_width(A,B) = width(A cup B). For each of these measures we present efficient algorithms for finding a translation of B that minimizes the distance: For diameter we present near-linear-time algorithms in R^2 and R^3, for uniformity we describe a roughly O(n^{9/4})-time algorithm, and for union width we offer a near-linear-time algorithm in R^2 and a quadratic-time one in R^3.

Boris Aronov, Omrit Filtser, Matthew J. Katz, and Khadijeh Sheikhan. Bipartite Diameter and Other Measures Under Translation. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{aronov_et_al:LIPIcs.STACS.2019.8, author = {Aronov, Boris and Filtser, Omrit and Katz, Matthew J. and Sheikhan, Khadijeh}, title = {{Bipartite Diameter and Other Measures Under Translation}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.8}, URN = {urn:nbn:de:0030-drops-102476}, doi = {10.4230/LIPIcs.STACS.2019.8}, annote = {Keywords: Translation-invariant similarity measures, Geometric optimization, Minimum-width annulus} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

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. [Rinat Ben Avraham et al., 2015] presented an O(m^3n^2(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(m^2n^2 log^2(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(m^2n(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. [Martello et al., 1984] 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 the weak DFD under translation in 1D is a special case of it.

Omrit Filtser and Matthew J. Katz. Algorithms for the Discrete Fréchet Distance Under Translation. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filtser_et_al:LIPIcs.SWAT.2018.20, author = {Filtser, Omrit and Katz, Matthew J.}, title = {{Algorithms for the Discrete Fr\'{e}chet Distance Under Translation}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.20}, URN = {urn:nbn:de:0030-drops-88466}, doi = {10.4230/LIPIcs.SWAT.2018.20}, annote = {Keywords: curve similarity, discrete Fr\'{e}chet distance, translation, algorithms, BOP} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Frechet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS).
In this paper we consider for the first time the complexity of GCPS under both the discrete Frechet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fr'echet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an widetilde-O((n+m)^6 min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n+m)^4) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.

Chenglin Fan, Omrit Filtser, Matthew J. Katz, and Binhai Zhu. On the General Chain Pair Simplification Problem. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fan_et_al:LIPIcs.MFCS.2016.37, author = {Fan, Chenglin and Filtser, Omrit and Katz, Matthew J. and Zhu, Binhai}, title = {{On the General Chain Pair Simplification Problem}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {37:1--37:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.37}, URN = {urn:nbn:de:0030-drops-64510}, doi = {10.4230/LIPIcs.MFCS.2016.37}, annote = {Keywords: chain simplification, discrete Frechet distance, dynamic programming, geometric arrangements, protein structural resemblance} }

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