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Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions

Authors Noam Horowicz , Tsvi Kopelowitz

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • data structures
  • fast matrix multiplication
  • fine-grained complexity
  • pattern matching
  • distance oracles

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Abstract

In the snippets problem, the goal is to preprocess a text T so that given two pattern queries, P₁ and P₂, one can quickly locate the occurrences of the two patterns in T that are closest to each other, or report the distance between these occurrences. Kopelowitz and Krauthgamer [CPM2016] showed upper bound tradeoffs and conditional lower bounds tradeoffs for the snippets problem, by utilizing connections between the snippets problem and the problem of constructing a color distance oracle (CDO), which is a data structure that preprocess a set of points with associated colors so that given two colors c and c' one can quickly find the (distance between the) closest pair of points where one has color c and the other has color c'. However, the existing upper bound and lower bound curves are not tight. 
Inspired by recent advances by Kopelowitz and Vassilevska-Williams [ICALP2020] regarding tradeoff curves for Set-disjointness data structures, in this paper we introduce new conditionally optimal algorithms for a (1+ε) approximation version of the snippets problem and a (1+ε) approximation version of the CDO problem, by applying fast matrix multiplication. For example, for CDO on n points in an array, if the preprocessing time is Õ(n^a) and the query time is Õ(n^b) then, assuming that ω = 2 (where ω is the exponent of n in the runtime of the fastest matrix multiplication algorithm on two squared matrices of size n× n), we show that approximate CDO can be solved with the following tradeoff
a + 2b = 2 (if 0 ≤ b ≤ 1/3)
2a + b = 3 (if 1/3 ≤ b ≤ 1).
Moreover, we prove that for exact CDO on points in an array, the algorithm of Kopelowitz and Krauthgamer [CPM2016], which obtains a tradeoff of a+b = 2, is essentially optimal assuming that the strong all-pairs shortest paths hypothesis holds for randomized algorithms. Thus, we demonstrate that the exact version of CDO is strictly harder than the approximate version. Moreover, this separation carries over to the snippets problem.

Cite As Get BibTex

Noam Horowicz and Tsvi Kopelowitz. Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.72

Author Details

Noam Horowicz
  • Bar-Ilan University, Ramat Gan, Israel
Tsvi Kopelowitz
  • Bar-Ilan University, Ramat Gan, Israel

Funding

Supported by a BSF grant 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).

Acknowledgements

The authors thank an anonymous reviewer who suggested some ideas for tightening the lower bounds proved in Theorems 8 and 9.

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