Document Open Access Logo

Searching 2D-Strings for Matching Frames

Authors Itai Boneh , Dvir Fried , Shay Golan , Matan Kraus , Adrian Miclăuş , Arseny Shur

PDF
Thumbnail PDF

File

LIPIcs.CPM.2024.10.pdf
  • Filesize: 1.09 MB
  • 19 pages

Document Identifiers

Author Details

Itai Boneh
  • Reichman University, Herzliya, Israel
  • University of Haifa, Israel
Dvir Fried
  • Bar Ilan University, Ramat Gan, Israel
Shay Golan
  • Reichman University, Herzliya, Israel
  • University of Haifa, Israel
Matan Kraus
  • Bar Ilan University, Ramat Gan, Israel
Adrian Miclăuş
  • Faculty of Mathematics and Computer Science, University of Bucharest, Romania
Arseny Shur
  • Bar Ilan University, Ramat Gan, Israel

Funding

  • Boneh, Itai: supported by Israel Science Foundation grant 810/21.
  • 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: 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).
  • Miclăuş, Adrian: supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2021-0253, within PNCDI III.
  • Shur, Arseny: supported by the ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064) and by the State of Israel through the Center for Absorption in Science of the Ministry of Aliyah and Immigration.

Cite As Get BibTex

Itai Boneh, Dvir Fried, Shay Golan, Matan Kraus, Adrian Miclăuş, and Arseny Shur. Searching 2D-Strings for Matching Frames. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CPM.2024.10

Abstract

We study a natural type of repetitions in 2-dimensional strings. Such a repetition, called a matching frame, is a rectangular substring of size at least 2× 2 with equal marginal rows and equal marginal columns. Matching frames first appeared in literature in the context of Wang tiles.
We present two algorithms finding a matching frame with the maximum perimeter in a given n× m input string. The first algorithm solves the problem exactly in Õ(n^{2.5}) time (assuming n ≥ m). The second algorithm finds a (1-ε)-approximate solution in Õ((nm)/ε⁴) time, which is near linear in the size of the input for constant ε. In particular, by setting ε = O(1) the second algorithm decides the existence of a matching frame in a given string in Õ(nm) time. Some technical elements and structural properties used in these algorithms can be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • 2D string
  • matching frame
  • LCP
  • multidimensional range query

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Georgii Maksimovich Adelson-Velskii and Evgenii Mikhailovich Landis. An algorithm for organization of information. Dokl. Akad. Nauk SSSR, 146:263-266, 1962. Google Scholar
  2. Amihood Amir and Gary Benson. Two-dimensional periodicity and its applications. In Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms, pages 440-452, 1992. Google Scholar
  3. Amihood Amir and Gary Benson. Two-dimensional periodicity in rectangular arrays. SIAM Journal on Computing, 27(1):90-106, 1998. Google Scholar
  4. Amihood Amir, Itai Boneh, Panagiotis Charalampopoulos, and Eitan Kondratovsky. Repetition detection in a dynamic string. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA, volume 144 of LIPIcs, pages 5:1-5:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.5.
  5. Amihood Amir, Gad M Landau, Shoshana Marcus, and Dina Sokol. Two-dimensional maximal repetitions. Theoretical Computer Science, 812:49-61, 2020. Google Scholar
  6. Hideo Bannai, Tomohiro I, Shunsuke Inenaga, Yuto Nakashima, Masayuki Takeda, and Kazuya Tsuruta. The "runs" theorem. SIAM J. Comput., 46(5):1501-1514, 2017. URL: https://doi.org/10.1137/15M1011032.
  7. Djamal Belazzougui and Simon J. Puglisi. Range predecessor and lempel-ziv parsing. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 2053-2071. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch143.
  8. Robert Berger. The undecidability of the domino problem. Amer. Math. Soc., 1966. Google Scholar
  9. Itai Boneh, Dvir Fried, Shay Golan, Matan Kraus, Adrian Miclaus, and Arseny Shur. Searching 2d-strings for matching frames. arXiv preprint arXiv:2310.02670, 2023. Google Scholar
  10. Kirill Borozdin, Dmitry Kosolobov, Mikhail Rubinchik, and Arseny M. Shur. Palindromic length in linear time. In Juha Kärkkäinen, Jakub Radoszewski, and Wojciech Rytter, editors, 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017, volume 78 of LIPIcs, pages 23:1-23:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CPM.2017.23.
  11. Sarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, and Andrew Winslow. Two hands are better than one (up to constant factors): Self-assembly in the 2HAM vs. aTAM. In Natacha Portier and Thomas Wilke, editors, 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, volume 20 of LIPIcs, pages 172-184. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2013. URL: https://doi.org/10.4230/LIPICS.STACS.2013.172.
  12. Panagiotis Charalampopoulos, Tomasz Kociumaka, Jakub Radoszewski, Solon P. Pissis, Wojciech Rytter, Tomasz Walen, and Wiktor Zuba. Approximate circular pattern matching. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, volume 244 of LIPIcs, pages 35:1-35:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.35.
  13. Panagiotis Charalampopoulos, Jakub Radoszewski, Wojciech Rytter, Tomasz Walen, and Wiktor Zuba. The number of repetitions in 2d-strings. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA, volume 173 of LIPIcs, pages 32:1-32:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.32.
  14. Bernard Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17(3):427-462, 1988. URL: https://doi.org/10.1137/0217026.
  15. Peter Clifford and Raphaël Clifford. Simple deterministic wildcard matching. Inf. Process. Lett., 101(2):53-54, 2007. URL: https://doi.org/10.1016/j.ipl.2006.08.002.
  16. Jonas Ellert and Johannes Fischer. Linear time runs over general ordered alphabets. 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 63:1-63:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.63.
  17. Jonas Ellert, Pawel Gawrychowski, and Garance Gourdel. Optimal square detection over general alphabets. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 5220-5242. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch189.
  18. Nathan J Fine and Herbert S Wilf. Uniqueness theorems for periodic functions. Proceedings of the American Mathematical Society, 16(1):109-114, 1965. Google Scholar
  19. Guilhem Gamard, Gwénaël Richomme, Jeffrey O. Shallit, and Taylor J. Smith. Periodicity in rectangular arrays. Inf. Process. Lett., 118:58-63, 2017. URL: https://doi.org/10.1016/J.IPL.2016.09.011.
  20. Younan Gao, Meng He, and Yakov Nekrich. Fast preprocessing for optimal orthogonal range reporting and range successor with applications to text indexing. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 54:1-54:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.54.
  21. Pawel Gawrychowski, Samah Ghazawi, and Gad M. Landau. Lower bounds for the number of repetitions in 2d strings. In Thierry Lecroq and Hélène Touzet, editors, String Processing and Information Retrieval - 28th International Symposium, SPIRE, volume 12944 of Lecture Notes in Computer Science, pages 179-192. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86692-1_15.
  22. Pawel Gawrychowski, Oleg Merkurev, Arseny M. Shur, and Przemyslaw Uznanski. Tight tradeoffs for real-time approximation of longest palindromes in streams. Algorithmica, 81(9):3630-3654, 2019. URL: https://doi.org/10.1007/S00453-019-00591-8.
  23. 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.
  24. Emmanuel Jeandel and Michaël Rao. An aperiodic set of 11 Wang tiles. CoRR, abs/1506.06492, 2015. URL: https://arxiv.org/abs/1506.06492.
  25. Juha Kärkkäinen and Peter Sanders. Simple linear work suffix array construction. In Jos C. M. Baeten, Jan Karel Lenstra, Joachim Parrow, and Gerhard J. Woeginger, editors, Automata, Languages and Programming, 30th International Colloquium, ICALP 2003, 2003. Proceedings, volume 2719 of Lecture Notes in Computer Science, pages 943-955. Springer, 2003. URL: https://doi.org/10.1007/3-540-45061-0_73.
  26. Dong Kyue Kim, Jeong Seop Sim, Heejin Park, and Kunsoo Park. Linear-time construction of suffix arrays. In Ricardo A. Baeza-Yates, Edgar Chávez, and Maxime Crochemore, editors, Combinatorial Pattern Matching, 14th Annual Symposium, CPM 2003, volume 2676 of Lecture Notes in Computer Science, pages 186-199. Springer, 2003. URL: https://doi.org/10.1007/3-540-44888-8_14.
  27. Donald E. Knuth, James H. Morris Jr., and Vaughan R. Pratt. Fast pattern matching in strings. SIAM J. Comput., 6(2):323-350, 1977. URL: https://doi.org/10.1137/0206024.
  28. Pang Ko and Srinivas Aluru. Space efficient linear time construction of suffix arrays. J. Discrete Algorithms, 3(2-4):143-156, 2005. URL: https://doi.org/10.1016/j.jda.2004.08.002.
  29. Roman M. Kolpakov and Gregory Kucherov. Finding maximal repetitions in a word in linear time. In 40th Annual Symposium on Foundations of Computer Science, FOCS '99, 17-18 October, 1999, New York, NY, USA, pages 596-604. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814634.
  30. Manasi S Kulkarni and Kalpana Mahalingam. Two-dimensional palindromes and their properties. In International Conference on Language and Automata Theory and Applications, pages 155-167. Springer, 2017. Google Scholar
  31. Glenn K. Manacher. A new linear-time "on-line" algorithm for finding the smallest initial palindrome of a string. J. ACM, 22(3):346-351, 1975. URL: https://doi.org/10.1145/321892.321896.
  32. Udi Manber and Gene Myers. Suffix arrays: A new method for on-line string searches. In David S. Johnson, editor, Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1990, pages 319-327. SIAM, 1990. URL: http://dl.acm.org/citation.cfm?id=320176.320218.
  33. Oleg Merkurev and Arseny M. Shur. Searching runs in streams. In Nieves R. Brisaboa and Simon J. Puglisi, editors, String Processing and Information Retrieval - 26th International Symposium, SPIRE 2019, Segovia, Spain, October 7-9, 2019, Proceedings, volume 11811 of Lecture Notes in Computer Science, pages 203-220. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-32686-9_15.
  34. Ge Nong, Sen Zhang, and Wai Hong Chan. Linear suffix array construction by almost pure induced-sorting. In James A. Storer and Michael W. Marcellin, editors, 2009 Data Compression Conference (DCC 2009), pages 193-202. IEEE Computer Society, 2009. Google Scholar
  35. Ge Nong, Sen Zhang, and Wai Hong Chan. Linear time suffix array construction using d-critical substrings. In Gregory Kucherov and Esko Ukkonen, editors, Combinatorial Pattern Matching, 20th Annual Symposium, CPM 2009, Proceedings, volume 5577 of Lecture Notes in Computer Science, pages 54-67. Springer, 2009. Google Scholar
  36. Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 459-468. ACM, 2000. URL: https://doi.org/10.1145/335305.335358.
  37. Mikhail Rubinchik and Arseny M. Shur. Palindromic k-factorization in pure linear time. In Javier Esparza and Daniel Král', editors, 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 81:1-81:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.MFCS.2020.81.
  38. Taylor Smith. Properties of two-dimensional words. Master’s thesis, University of Waterloo, 2017. Google Scholar
  39. Esko Ukkonen. On-line construction of suffix trees. Algorithmica, 14(3):249-260, 1995. URL: https://doi.org/10.1007/BF01206331.
  40. Hao Wang. Proving theorems by pattern recognition II. Bell System Tech. J., 40:1-41, 1961. Google Scholar
  41. Hao Wang. Games, logic and computers. Scientific American, 213(5):107, 1965. Google Scholar
  42. Peter Weiner. Linear pattern matching algorithms. In 14th Annual Symposium on Switching and Automata Theory, 1973, pages 1-11. IEEE Computer Society, 1973. URL: https://doi.org/10.1109/SWAT.1973.13.
  43. Dan E. Willard. New data structures for orthogonal range queries. SIAM J. Comput., 14(1):232-253, 1985. URL: https://doi.org/10.1137/0214019.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact