Document Open Access Logo

Hamming Distance Oracles

Authors Itai Boneh , Dvir Fried , Shay Golan , Matan Kraus , Ely Porat

PDF

File

Thumbnail PDF
PDF
LIPIcs.CPM.2026.1.pdf
  • Filesize: 0.85 MB
  • 12 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Hamming distance
  • Fine-grained complexity
  • Data structure
  • Oracle

Metrics

Abstract

In this paper, we present and study the Hamming distance oracle problem. In this problem, the task is to preprocess two strings S and T of lengths n and m, respectively, to obtain a data structure that is able to return the Hamming distance between a substring of S and a substring of T.
For strings over a constant-size alphabet, we show that for every x ≤ min{n,m} there is a data structure with Õ(nm/x) preprocessing time and O(x) query time. We also provide a conditional lower bound, showing that for every ε > 0 there is no combinatorial data structure with query time O(x) and preprocessing time O((nm/x)^{1-ε}) unless combinatorial fast matrix multiplication is possible.
For strings over a general alphabet, we present a data structure with Õ(nm/√x) pre-processing time and O(x) query time for every x ≤ min {n,m}. Moreover, for every ε > 0 we provide a data structure with a preprocessing time of Õ((n+m)/ε³) that returns with high probability a (1±ε) approximation of the Hamming distance of two input substrings. The query time of the approximation data structure is Õ(1/ε²).

Cite As Get BibTex

Itai Boneh, Dvir Fried, Shay Golan, Matan Kraus, and Ely Porat. Hamming Distance Oracles. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 1:1-1:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.1

Author Details

Itai Boneh
  • University of Wrocław, Poland
Dvir Fried
  • Bar-Ilan University, Ramat Gan, Israel
Shay Golan
  • Ariel University, Israel
Matan Kraus
  • Bar-Ilan University, Ramat Gan, Israel
Ely Porat
  • Bar-Ilan University, Ramat Gan, Israel

Funding

  • Boneh, Itai: partially supported by Israel Science Foundation grant 810/21 and by the Polish National Science Centre grant number 2023/51/B/ST6/01505.
  • Fried, Dvir: supported by ISF grant no. 1926/19, by a BSF grant 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).
  • Golan, Shay: partially supported by Israel Science Foundation grant 810/21.
  • Kraus, Matan: supported by the ISF grant no. 1926/19, by the BSF grant 2018364, and by the ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).
  • Porat, Ely: supported by the ISF grant no. 1926/19, by the BSF grant 2018364, and by the ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).

Acknowledgements

We would like to warmly thank Panagiotis Charalampopoulos for suggesting the approximation version of the problem addressed in this paper.

References

  1. G. M. Adelson-Velsky and E. M. Landis. An algorithm for the organization of information. Doklady Akademii Nauk SSSR, 146:263-266, 1962. Google Scholar
  2. Amihood Amir, Panagiotis Charalampopoulos, Solon P Pissis, and Jakub Radoszewski. Dynamic and internal longest common substring. Algorithmica, 82(12):3707-3743, 2020. URL: https://doi.org/10.1007/S00453-020-00744-0.
  3. Amihood Amir, Moshe Lewenstein, and Sharma V. Thankachan. Range LCP queries revisited. In Costas S. Iliopoulos, Simon J. Puglisi, and Emine Yilmaz, editors, String Processing and Information Retrieval - 22nd International Symposium, SPIRE 2015, London, UK, September 1-4, 2015, Proceedings, volume 9309 of Lecture Notes in Computer Science, pages 350-361. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-23826-5_33.
  4. Timothy M. Chan, Shay Golan, Tomasz Kociumaka, Tsvi Kopelowitz, and Ely Porat. Approximating text-to-pattern hamming distances. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 643-656. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384266.
  5. Timothy M. Chan, Ce Jin, Virginia Vassilevska Williams, and Yinzhan Xu. Faster algorithms for text-to-pattern hamming distances. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 2188-2203. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00136.
  6. Panagiotis Charalampopoulos, Pawel Gawrychowski, Shay Mozes, and Oren Weimann. An almost optimal edit distance oracle. 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 48:1-48:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.48.
  7. Panagiotis Charalampopoulos, Tomasz Kociumaka, and Philip Wellnitz. Faster approximate pattern matching: A unified approach. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 978-989. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00095.
  8. Raphaël Clifford. Matrix multiplication and pattern matching under Hamming norm. URL: https://web.archive.org/web/20160818144748/, http://www.cs.bris.ac.uk/Research/Algorithms/events/BAD09/BAD09/Talks/BAD09-Hammingnotes.pdf, 2009.
  9. Martin Farach. Optimal suffix tree construction with large alphabets. In 38th Annual Symposium on Foundations of Computer Science, FOCS 1997, 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. Michael J Fischer and Michael S Paterson. String-matching and other products. Technical report, Massachusetts Institute of Technology, 1974. Google Scholar
  11. Zvi Galil and Raffaele Giancarlo. Data structures and algorithms for approximate string matching. Journal of Complexity, 4(1):33-72, 1988. URL: https://doi.org/10.1016/0885-064X(88)90008-8.
  12. Pawel Gawrychowski and Przemyslaw Uznanski. Towards unified approximate pattern matching for hamming and l_1 distance. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 62:1-62:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.62.
  13. L. J. Guibas and R. Sedgewick. A dichromatic framework for balanced trees. Information and Control, 15(1):1-29, 1978. Google Scholar
  14. Richard W Hamming. Error detecting and error correcting codes. The Bell system technical journal, 29(2):147-160, 1950. Google Scholar
  15. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338-355, 1984. URL: https://doi.org/10.1137/0213024.
  16. Ce Jin and Yinzhan Xu. Shaving logs via large sieve inequality: Faster algorithms for sparse convolution and more. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 1573-1584. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649605.
  17. Orgad Keller, Tsvi Kopelowitz, Shir Landau Feibish, and Moshe Lewenstein. Generalized substring compression. Theoretical Computer Science, 525:42-54, 2014. URL: https://doi.org/10.1016/J.TCS.2013.10.010.
  18. Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. Efficient data structures for the factor periodicity problem. In String Processing and Information Retrieval: 19th International Symposium, SPIRE 2012, Cartagena de Indias, Colombia, October 21-25, 2012. Proceedings 19, pages 284-294. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-34109-0_30.
  19. Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, and Tomasz Walen. Internal pattern matching queries in a text and applications. SIAM J. Comput., 53(5):1524-1577, 2024. URL: https://doi.org/10.1137/23M1567618.
  20. Gad M. Landau and Uzi Vishkin. Fast string matching with k differences. Journal of Computer and System Sciences, 37(1):63-78, 1988. URL: https://doi.org/10.1016/0022-0000(88)90045-1.
  21. Alexandre Tiskin. Semi-local string comparison: Algorithmic techniques and applications. Math. Comput. Sci., 1(4):571-603, 2008. URL: https://doi.org/10.1007/S11786-007-0033-3.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact