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A Faster Algorithm for the Fréchet Distance in 1D for the Imbalanced Case

Authors Lotte Blank , Anne Driemel

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LIPIcs.ESA.2024.28.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • {Fréchet distance}
  • distance oracle
  • data structures
  • time series

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.ESA.2024.28

Author Details

Lotte Blank
  • University of Bonn, Germany
Anne Driemel
  • University of Bonn, Germany

Funding

This work was funded by 390685813 (Germany’s Excellence Strategy – EXC-2047/1: Hausdorff Center for Mathematics); 416767905; and the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) – 459420781 (FOR AlgoForGe).

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