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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

The fine-grained complexity of computing the {Fréchet distance } has been a topic of much recent work, starting with the quadratic SETH-based conditional lower bound by Bringmann from 2014. Subsequent work established largely the same complexity lower bounds for the {Fréchet distance } in 1D. However, the imbalanced case, which was shown by Bringmann to be tight in dimensions d ≥ 2, was still left open. Filling in this gap, we show that a faster algorithm for the {Fréchet distance } in the imbalanced case is possible: Given two 1-dimensional curves of complexity n and n^{α} for some α ∈ (0,1), we can compute their {Fréchet distance } in O(n^{2α} log² n + n log n) time. This rules out a conditional lower bound of the form O((nm)^{1-ε}) that Bringmann showed for d ≥ 2 and any ε > 0 in turn showing a strict separation with the setting d = 1. At the heart of our approach lies a data structure that stores a 1-dimensional curve P of complexity n, and supports queries with a curve Q of complexity m for the continuous {Fréchet distance } between P and Q. The data structure has size in 𝒪(nlog n) and uses query time in 𝒪(m² log² n). Our proof uses a key lemma that is based on the concept of visiting orders and may be of independent interest. We demonstrate this by substantially simplifying the correctness proof of a clustering algorithm by Driemel, Krivošija and Sohler from 2015.

Lotte Blank and Anne Driemel. A Faster Algorithm for the Fréchet Distance in 1D for the Imbalanced Case. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blank_et_al:LIPIcs.ESA.2024.28, author = {Blank, Lotte and Driemel, Anne}, title = {{A Faster Algorithm for the Fr\'{e}chet Distance in 1D for the Imbalanced Case}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.28}, URN = {urn:nbn:de:0030-drops-210999}, doi = {10.4230/LIPIcs.ESA.2024.28}, annote = {Keywords: \{Fr\'{e}chet distance\}, distance oracle, data structures, time series} }

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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

We study the Fréchet queries problem. It is a data structure problem for range reporting, where we are given a set S of n polygonal curves and a distance threshold ρ. The data structure should support queries with a polygonal curve q for the elements of S, for which the continuous Fréchet distance to q is at most ρ. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves of constant complexity and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point q should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with size in 𝒪(n log^{t-1} n) and query time in 𝒪(log^{t-1} n+k), where k denotes the output size and t can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. Note that we omit factors depending on the complexity of the curves that do not depend on n. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter d = ⌊t/2⌋ can be applied to our problem via reduction.

Lotte Blank and Anne Driemel. Range Reporting for Time Series via Rectangle Stabbing. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blank_et_al:LIPIcs.SWAT.2024.15, author = {Blank, Lotte and Driemel, Anne}, title = {{Range Reporting for Time Series via Rectangle Stabbing}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.15}, URN = {urn:nbn:de:0030-drops-200559}, doi = {10.4230/LIPIcs.SWAT.2024.15}, annote = {Keywords: Data Structures, Fr\'{e}chet distance, Rectangle Stabbing, Orthogonal Range Searching} }

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