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Indexing Integer Strings Using Local Difference Bounds

Authors Daniel Gibney , Kaamil Kaka , Sharma V. Thankachan

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LIPIcs.ESA.2026.151.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • String Algorithms
  • Pattern Matching
  • Suffix Trees
  • Compact Data Structures

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Abstract

Time series data can often be represented as a string T[1..n] over an integer alphabet Σ. A standard approach for indexing such strings is the compressed suffix tree, which supports efficient exact pattern matching using 𝒪(n log |Σ|) bits of space. However, when |Σ| is close to n, as is often the case for time-series data, this yields little to no space savings over the classical Θ(n log n)-bit suffix tree.
In this work, we study a different parameter that is often much smaller than |Σ|: the maximum absolute difference between consecutive values in T, denoted by Δ. Although representing T in 𝒪(n log Δ) bits is straightforward, supporting efficient pattern matching within this space bound remains challenging. By leveraging succinct data structure techniques, particularly the FM-index, we obtain a compressed index occupying n log Δ + 𝒪(n) bits. Given a query pattern P[1..m], the index answers counting queries in 𝒪(m log Δ) time and reporting queries in 𝒪(m log Δ + occ ⋅ log n ⋅ log Δ) time, where occ denotes the number of occurrences of P in T. In addition, the structure supports suffix array and inverse suffix array queries in 𝒪(log n ⋅ log Δ) time.
Our construction is conceptually simple. We show that the local-difference bound induces a strong form of locality in the Burrows-Wheeler Transform (BWT), allowing the transformed text to be decomposed into regions that each use only a small local alphabet of size 𝒪(Δ). This makes it possible to adapt the classical FM-index so that its space usage depends on Δ rather than the overall alphabet size, while still supporting efficient pattern matching queries.

Cite As Get BibTex

Daniel Gibney, Kaamil Kaka, and Sharma V. Thankachan. Indexing Integer Strings Using Local Difference Bounds. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 151:1-151:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.151

Author Details

Daniel Gibney
  • Department of Computer Science, University of Texas at Dallas, TX, USA
Kaamil Kaka
  • Department of Computer Science, University of Texas at Dallas, TX, USA
Sharma V. Thankachan
  • Department of Computer Science, North Carolina State University, TX, USA

Funding

  • Thankachan, Sharma V.: U.S. NSF awards CCF-2316691 and CCF-2315822.

Supplementary Materials

References

  1. Eric Chiu and Dominik Kempa. Wavelet forests revisited, 2026. URL: https://doi.org/10.48550/arXiv.2604.11338.
  2. 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.
  3. David Richard Clark. Compact PAT Trees. Ph.D. thesis, University of Waterloo, Waterloo, Ontario, Canada, 1996. URL: https://doi.org/10.5555/287799.
  4. Hui Ding, Goce Trajcevski, Peter Scheuermann, Xiaoyue Wang, and Eamonn J. Keogh. Querying and mining of time series data: experimental comparison of representations and distance measures. Proc. VLDB Endow., 1(2):1542-1552, 2008. URL: https://doi.org/10.14778/1454159.1454226.
  5. Paolo Ferragina and Giovanni Manzini. Indexing compressed text. J. ACM, 52(4):552-581, 2005. URL: https://doi.org/10.1145/1082036.1082039.
  6. 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.
  7. Daniel Gibney, Kaamil Kaka, and Sharma V. Thankachan. dangibney/Delta_Bounded_Time_Series_Index. Software, (visited on 2026-08-11). URL: https://github.com/dangibney/Delta_Bounded_Time_Series_Index
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.27610
  8. Simon Gog, Timo Beller, Alistair Moffat, and Matthias Petri. From theory to practice: Plug and play with succinct data structures. In 13th International Symposium on Experimental Algorithms, (SEA 2014), pages 326-337, 2014. URL: https://doi.org/10.1007/978-3-319-07959-2_28.
  9. Simon Gog, Juha Kärkkäinen, Dominik Kempa, Matthias Petri, and Simon J. Puglisi. Fixed block compression boosting in fm-indexes: Theory and practice. Algorithmica, 81(4):1370-1391, 2019. URL: https://doi.org/10.1007/S00453-018-0475-9.
  10. Alexander Golynski, Roberto Grossi, Ankur Gupta, Rajeev Raman, and S. Srinivasa Rao. On the size of succinct indices. In Lars Arge, Michael Hoffmann, and Emo Welzl, editors, Algorithms - ESA 2007, 15th Annual European Symposium, Eilat, Israel, October 8-10, 2007, Proceedings, Lecture Notes in Computer Science, pages 371-382. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-75520-3_34.
  11. Alexander Golynski, J. Ian Munro, and S. Srinivasa Rao. Rank/select operations on large alphabets: a tool for text indexing. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 368-373. ACM Press, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109599.
  12. Roberto Grossi, Ankur Gupta, and Jeffrey Scott Vitter. High-order entropy-compressed text indexes. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA, pages 841-850. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644250.
  13. 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.
  14. Guy Joseph Jacobson. Succinct static data structures. Carnegie Mellon University, 1988. Google Scholar
  15. Dominik Kempa and Tomasz Kociumaka. Resolution of the burrows-wheeler transform conjecture. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1002-1013. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00097.
  16. 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.
  17. Dominik Kempa and Nicola Prezza. At the roots of dictionary compression: string attractors. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 827-840. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188814.
  18. 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.
  19. 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.
  20. Xiao-Ying Liu and Chuan-Lun Ren. Fast subsequence matching under time warping in time-series databases. In International Conference on Machine Learning and Cybernetics, ICMLC 2013, Tianjin, China, July 14-17, 2013, pages 1584-1590. IEEE, 2013. URL: https://doi.org/10.1109/ICMLC.2013.6890855.
  21. Udi Manber and Eugene W. Myers. Suffix arrays: A new method for on-line string searches. SIAM J. Comput., 22(5):935-948, 1993. URL: https://doi.org/10.1137/0222058.
  22. Gonzalo Navarro. Wavelet trees for all. J. Discrete Algorithms, 25:2-20, 2014. URL: https://doi.org/10.1016/J.JDA.2013.07.004.
  23. 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.
  24. Gonzalo Navarro. Indexing highly repetitive string collections, part II: compressed indexes. ACM Comput. Surv., 54(2):26:1-26:32, 2022. URL: https://doi.org/10.1145/3432999.
  25. José Luis Palacios. On the simple symmetric random walk and its maximal function. The American Statistician, 62(2):138-140, 2008. Google Scholar
  26. Thanawin Rakthanmanon, Bilson J. L. Campana, Abdullah Mueen, Gustavo E. A. P. A. Batista, M. Brandon Westover, Qiang Zhu, Jesin Zakaria, and Eamonn J. Keogh. Searching and mining trillions of time series subsequences under dynamic time warping. In Qiang Yang, Deepak Agarwal, and Jian Pei, editors, The 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '12, Beijing, China, August 12-16, 2012, pages 262-270. ACM, 2012. URL: https://doi.org/10.1145/2339530.2339576.
  27. 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.
  28. Kunihiko Sadakane. Compressed suffix trees with full functionality. Theory Comput. Syst., 41(4):589-607, 2007. URL: https://doi.org/10.1007/S00224-006-1198-X.
  29. Jin Shieh and Eamonn J. Keogh. isax: indexing and mining terabyte sized time series. In Ying Li, Bing Liu, and Sunita Sarawagi, editors, Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Las Vegas, Nevada, USA, August 24-27, 2008, pages 623-631. ACM, 2008. URL: https://doi.org/10.1145/1401890.1401966.
  30. Sharma V. Thankachan. Compressed inverse suffix arrays. CoRR, abs/2607.17287, 2026. URL: https://doi.org/10.48550/ARXIV.2607.17287.
  31. 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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