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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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References

  1. Philip Bille, Inge Li Gørtz, Mathias Bæk Tejs Knudsen, Moshe Lewenstein, and Hjalte Wedel Vildhøj. Longest common extensions in sublinear space. In Ferdinando Cicalese, Ely Porat, and Ugo Vaccaro, editors, Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Ischia Island, Italy, June 29 - July 1, 2015, Proceedings, volume 9133 of Lecture Notes in Computer Science, pages 65-76. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-19929-0_6.
  2. Philip Bille, Inge Li Gørtz, Benjamin Sach, and Hjalte Wedel Vildhøj. Time-space trade-offs for longest common extensions. J. Discrete Algorithms, 25:42-50, 2014. URL: https://doi.org/10.1016/J.JDA.2013.06.003.
  3. Nieves R. Brisaboa, Travis Gagie, Adrián Gómez-Brandón, and Gonzalo Navarro. Two-dimensional block trees. Comput. J., 67(1):391-406, 2024. URL: https://doi.org/10.1093/COMJNL/BXAC182.
  4. Stefan Burkhardt and Juha Kärkkäinen. Fast lightweight suffix array construction and checking. In Ricardo A. Baeza-Yates, Edgar Chávez, and Maxime Crochemore, editors, Combinatorial Pattern Matching, 14th Annual Symposium, CPM 2003, Morelia, Michocán, Mexico, June 25-27, 2003, Proceedings, volume 2676 of Lecture Notes in Computer Science, pages 55-69. Springer, 2003. URL: https://doi.org/10.1007/3-540-44888-8_5.
  5. Lorenzo Carfagna and Giovanni Manzini. Compressibility measures for two-dimensional data. In Franco Maria Nardini, Nadia Pisanti, and Rossano Venturini, editors, String Processing and Information Retrieval - 30th International Symposium, SPIRE 2023, Pisa, Italy, September 26-28, 2023, Proceedings, volume 14240 of Lecture Notes in Computer Science, pages 102-113. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-43980-3_9.
  6. Lorenzo Carfagna and Giovanni Manzini. The landscape of compressibility measures for two-dimensional data. IEEE Access, 12:87268-87283, 2024. URL: https://doi.org/10.1109/ACCESS.2024.3417621.
  7. Anders Roy Christiansen, Mikko Berggren Ettienne, Tomasz Kociumaka, Gonzalo Navarro, and Nicola Prezza. Optimal-time dictionary-compressed indexes. ACM Trans. Algorithms, 17(1):8:1-8:39, 2021. URL: https://doi.org/10.1145/3426473.
  8. Charles J. Colbourn and Alan C. H. Ling. Quorums from difference covers. Inf. Process. Lett., 75(1-2):9-12, 2000. URL: https://doi.org/10.1016/S0020-0190(00)00080-6.
  9. Martin Farach. Optimal suffix tree construction with large alphabets. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 137-143. IEEE Computer Society, 1997. URL: https://doi.org/10.1109/SFCS.1997.646102.
  10. Paolo Ferragina and Giovanni Manzini. Opportunistic data structures with applications. In 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, 12-14 November 2000, Redondo Beach, California, USA, pages 390-398. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/SFCS.2000.892127.
  11. Travis Gagie, Gonzalo Navarro, and Nicola Prezza. Fully functional suffix trees and optimal text searching in BWT-runs bounded space. J. ACM, 67(1):2:1-2:54, 2020. URL: https://doi.org/10.1145/3375890.
  12. Arnab Ganguly, Dhrumil Patel, Rahul Shah, and Sharma V. Thankachan. LF successor: Compact space indexing for order-isomorphic pattern matching. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 71:1-71:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.71.
  13. Arnab Ganguly, Rahul Shah, and Sharma V. Thankachan. pbwt: Achieving succinct data structures for parameterized pattern matching and related problems. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 397-407. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.25.
  14. Arnab Ganguly, Rahul Shah, and Sharma V. Thankachan. Fully functional parameterized suffix trees in compact space. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 65:1-65:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ICALP.2022.65.
  15. Pawel Gawrychowski, Tomasz Kociumaka, Wojciech Rytter, and Tomasz Walen. Faster longest common extension queries in strings over general alphabets. In Roberto Grossi and Moshe Lewenstein, editors, 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016, June 27-29, 2016, Tel Aviv, Israel, volume 54 of LIPIcs, pages 5:1-5:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CPM.2016.5.
  16. Raffaele Giancarlo. A generalization of the suffix tree to square matrices, with applications. SIAM J. Comput., 24(3):520-562, 1995. URL: https://doi.org/10.1137/S0097539792231982.
  17. Gaston H Gonnet. Efficient searching of text and pictures. UW Centre for the New Oxford English Dictionary, 1990. Google Scholar
  18. Roberto Grossi and Jeffrey Scott Vitter. Compressed suffix arrays and suffix trees with applications to text indexing and string matching. SIAM J. Comput., 35(2):378-407, 2005. URL: https://doi.org/10.1137/S0097539702402354.
  19. Dominik Kempa and Tomasz Kociumaka. String synchronizing sets: sublinear-time BWT construction and optimal LCE data structure. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 756-767. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316368.
  20. Dominik Kempa and Tomasz Kociumaka. Resolution of the Burrows-Wheeler transform conjecture. Commun. ACM, 65(6):91-98, 2022. URL: https://doi.org/10.1145/3531445.
  21. Dominik Kempa and Tomasz Kociumaka. Collapsing the hierarchy of compressed data structures: Suffix arrays in optimal compressed space. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1877-1886. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00114.
  22. Dong Kyue Kim, Yoo Ah Kim, and Kunsoo Park. Generalizations of suffix arrays to multi-dimensional matrices. Theor. Comput. Sci., 302(1-3):223-238, 2003. URL: https://doi.org/10.1016/S0304-3975(02)00828-9.
  23. Dong Kyue Kim, Joong Chae Na, Jeong Seop Sim, and Kunsoo Park. Linear-time construction of two-dimensional suffix trees. Algorithmica, 59(2):269-297, 2011. URL: https://doi.org/10.1007/S00453-009-9350-Z.
  24. Tomasz Kociumaka, Gonzalo Navarro, and Francisco Olivares. Near-optimal search time in δ-optimal space, and vice versa. Algorithmica, 86(4):1031-1056, 2024. URL: https://doi.org/10.1007/S00453-023-01186-0.
  25. Tomasz Kociumaka, Gonzalo Navarro, and Nicola Prezza. Toward a definitive compressibility measure for repetitive sequences. IEEE Trans. Inf. Theory, 69(4):2074-2092, 2023. URL: https://doi.org/10.1109/TIT.2022.3224382.
  26. Gonzalo Navarro. Indexing highly repetitive string collections, part I: repetitiveness measures. ACM Comput. Surv., 54(2):29:1-29:31, 2022. URL: https://doi.org/10.1145/3434399.
  27. Takaaki Nishimoto, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Fully dynamic data structure for LCE queries in compressed space. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, volume 58 of LIPIcs, pages 72:1-72:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.MFCS.2016.72.
  28. Dhrumil Patel and Rahul Shah. Inverse suffix array queries for 2-dimensional pattern matching in near-compact space. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 60:1-60:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ISAAC.2021.60.
  29. Rajeev Raman, Venkatesh Raman, and Srinivasa Rao Satti. Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans. Algorithms, 3(4):43, 2007. URL: https://doi.org/10.1145/1290672.1290680.
  30. Sofya Raskhodnikova, Dana Ron, Ronitt Rubinfeld, and Adam D. Smith. Sublinear algorithms for approximating string compressibility. Algorithmica, 65(3):685-709, 2013. URL: https://doi.org/10.1007/S00453-012-9618-6.
  31. Giuseppe Romana, Marinella Sciortino, and Cristian Urbina. Exploring repetitiveness measures for two-dimensional strings, 2024. URL: https://doi.org/10.48550/arXiv.2404.07030.
  32. Kunihiko Sadakane. Succinct representations of LCP information and improvements in the compressed suffix arrays. In David Eppstein, editor, Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 6-8, 2002, San Francisco, CA, USA, pages 225-232. ACM/SIAM, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545410.
  33. Sharma V. Thankachan. Compact text indexing for advanced pattern matching problems: Parameterized, order-isomorphic, 2d, etc. (invited talk). In Hideo Bannai and Jan Holub, editors, 33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022, June 27-29, 2022, Prague, Czech Republic, volume 223 of LIPIcs, pages 3:1-3:3. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.CPM.2022.3.
  34. Peter Weiner. Linear pattern matching algorithms. In 14th Annual Symposium on Switching and Automata Theory, Iowa City, Iowa, USA, October 15-17, 1973, pages 1-11. IEEE Computer Society, 1973. URL: https://doi.org/10.1109/SWAT.1973.13.
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