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Two-Dimensional Longest Common Extension Queries in Compact Space

Authors Arnab Ganguly , Daniel Gibney , Rahul Shah , Sharma V. Thankachan

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Author Details

Arnab Ganguly
  • University of Wisconsin, Whitewater, WI, USA
Daniel Gibney
  • University of Texas at Dallas, TX, USA
Rahul Shah
  • Louisiana State University, Baton Rouge, LA, USA
Sharma V. Thankachan
  • North Carolina State University, Raleigh, NC, USA

Funding

Supported by the US National Science Foundation (NSF) under Grant Numbers 2315822 (S Thankachan) and 2137057 (R Shah).

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Arnab Ganguly, Daniel Gibney, Rahul Shah, and Sharma V. Thankachan. Two-Dimensional Longest Common Extension Queries in Compact Space. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.38

Abstract

For a length n text over an alphabet of size σ, we can encode the suffix tree data structure in 𝒪(nlog σ) bits of space. It supports suffix array (SA), inverse suffix array (ISA), and longest common extension (LCE) queries in 𝒪(log^ε_σ n) time, which enables efficient pattern matching; here ε > 0 is an arbitrarily small constant. Further improvements are possible for LCE queries, where 𝒪(1) time queries can be achieved using an index of space 𝒪(nlog σ) bits. However, compactly indexing a two-dimensional text (i.e., an n× n matrix) has been a major open problem. We show progress in this direction by first presenting an 𝒪(n²log σ)-bit structure supporting LCE queries in near 𝒪((log_σ n)^{2/3}) time. We then present an 𝒪(n²log σ + n²log log n)-bit structure supporting ISA queries in near 𝒪(log n ⋅ (log_σ n)^{2/3}) time. Within a similar space, achieving SA queries in poly-logarithmic (even strongly sub-linear) time is a significant challenge. However, our 𝒪(n²log σ + n²log log n)-bit structure can support SA queries in 𝒪(n²/(σ log n)^c) time, where c is an arbitrarily large constant, which enables pattern matching in time faster than what is possible without preprocessing. 
We then design a repetition-aware data structure. The δ_2D compressibility measure for two-dimensional texts was recently introduced by Carfagna and Manzini [SPIRE 2023]. The measure ranges from 1 to n², with smaller δ_2D indicating a highly compressible two-dimensional text. The current data structure utilizing δ_2D allows only element access. We obtain the first structure based on δ_2D for LCE queries. It takes 𝒪^{~}(n^{5/3} + n^{8/5}δ_2D^{1/5}) space and answers queries in 𝒪(log n) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • String matching
  • text indexing
  • two-dimensional text

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