Document Open Access Logo

Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices

Authors Dvir Fried , Tsvi Kopelowitz , Ely Porat

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.57.pdf
  • Filesize: 0.62 MB
  • 6 pages

Document Identifiers

Author Details

Dvir Fried
  • Bar-Ilan University, Ramat-Gan, Israel
Tsvi Kopelowitz
  • Bar-Ilan University, Ramat-Gan, Israel
Ely Porat
  • Bar-Ilan University, Ramat-Gan, Israel

Funding

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).

Cite AsGet BibTex

Dvir Fried, Tsvi Kopelowitz, and Ely Porat. Removing the log Factor from (min,+)-Products on Bounded Range Integer Matrices. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 57:1-57:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.57

Abstract

We revisit the problem of multiplying two square matrices over the (min, +) semi-ring, where all entries are integers from a bounded range [-M : M] ∪ {∞}. The current state of the art for this problem is a simple O(M n^{ω} log M) time algorithm by Alon, Galil and Margalit [JCSS'97], where ω is the exponent in the runtime of the fastest matrix multiplication (FMM) algorithm. We design a new simple algorithm whose runtime is O(M n^ω + M n² log M), thereby removing the logM factor in the runtime if ω > 2 or if n^ω = Ω (n²log n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • FMM
  • (min
  • +)-product
  • FFT

Metrics

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

References

  1. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Dániel Marx, editor, SODA 2021, pages 522-539. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.32.
  2. Noga Alon, Zvi Galil, and Oded Margalit. On the exponent of the all pairs shortest path problem. J. Comput. Syst. Sci., 54(2):255-262, 1997. URL: https://doi.org/10.1006/jcss.1997.1388.
  3. Noga Alon, Zvi Galil, Oded Margalit, and Moni Naor. Witnesses for boolean matrix multiplication and for shortest paths. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 417-426. IEEE Computer Society, 1992. URL: https://doi.org/10.1109/SFCS.1992.267748.
  4. Karl Bringmann, Fabrizio Grandoni, Barna Saha, and Virginia Vassilevska Williams. Truly subcubic algorithms for language edit distance and RNA folding via fast bounded-difference min-plus product. SIAM J. Comput., 48(2):481-512, 2019. URL: https://doi.org/10.1137/17M112720X.
  5. Shucheng Chi, Ran Duan, and Tianle Xie. Faster algorithms for bounded-difference min-plus product. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1435-1447. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.60.
  6. Shucheng Chi, Ran Duan, Tianle Xie, and Tianyi Zhang. Faster min-plus product for monotone instances. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, 2022, pages 1529-1542. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520057.
  7. James W Cooley and John W Tukey. An algorithm for the machine calculation of complex fourier series. Mathematics of computation, 19(90):297-301, 1965. Google Scholar
  8. Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9(3):251-280, 1990. URL: https://doi.org/10.1016/S0747-7171(08)80013-2.
  9. Ran Duan and Seth Pettie. Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In Claire Mathieu, editor, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 384-391. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.43.
  10. Ran Duan, Hongxun Wu, and Renfei Zhou. Faster matrix multiplication via asymmetric hashing. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 2129-2138. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00130.
  11. Pavlos Eirinakis, Matthew D. Williamson, and K. Subramani. On the shoshan-zwick algorithm for the all-pairs shortest path problem. J. Graph Algorithms Appl., 21(2):177-181, 2017. URL: https://doi.org/10.7155/JGAA.00410.
  12. Dvir Fried, Shay Golan, Tomasz Kociumaka, Tsvi Kopelowitz, Ely Porat, and Tatiana Starikovskaya. An improved algorithm for the k-dyck edit distance problem. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 3650-3669. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.144.
  13. Zvi Galil and Oded Margalit. All pairs shortest distances for graphs with small integer length edges. Inf. Comput., 134(2):103-139, 1997. URL: https://doi.org/10.1006/inco.1997.2620.
  14. Zvi Galil and Oded Margalit. All pairs shortest paths for graphs with small integer length edges. J. Comput. Syst. Sci., 54(2):243-254, 1997. URL: https://doi.org/10.1006/jcss.1997.1385.
  15. François Le Gall. Powers of tensors and fast matrix multiplication. In Katsusuke Nabeshima, Kosaku Nagasaka, Franz Winkler, and Ágnes Szántó, editors, International Symposium on Symbolic and Algebraic Computation, ISSAC '14, Kobe, Japan, July 23-25, 2014, pages 296-303. ACM, 2014. URL: https://doi.org/10.1145/2608628.2608664.
  16. Francois Le Gall and Florent Urrutia. Improved rectangular matrix multiplication using powers of the Coppersmith-Winograd tensor. In SODA 2018, pages 1029-1046, 2018. URL: https://doi.org/10.1137/1.9781611975031.67.
  17. Jirí Matousek. Computing dominances in e^n. Inf. Process. Lett., 38(5):277-278, 1991. URL: https://doi.org/10.1016/0020-0190(91)90071-O.
  18. Ran Raz. On the complexity of matrix product. SIAM J. Comput., 32(5):1356-1369, 2003. URL: https://doi.org/10.1137/S0097539702402147.
  19. Asaf Shapira, Raphael Yuster, and Uri Zwick. All-pairs bottleneck paths in vertex weighted graphs. Algorithmica, 59(4):621-633, 2011. URL: https://doi.org/10.1007/s00453-009-9328-x.
  20. Avi Shoshan and Uri Zwick. All pairs shortest paths in undirected graphs with integer weights. In 40th Annual Symposium on Foundations of Computer Science, FOCS '99, 17-18 October, 1999, New York, NY, USA, pages 605-615. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814635.
  21. Volker Strassen. Gaussian elimination is not optimal. Matematika, 13(5):354-356, 1969. Google Scholar
  22. Leslie G. Valiant. General context-free recognition in less than cubic time. J. Comput. Syst. Sci., 10(2):308-315, 1975. URL: https://doi.org/10.1016/S0022-0000(75)80046-8.
  23. Virginia Vassilevska, Ryan Williams, and Raphael Yuster. All pairs bottleneck paths and max-min matrix products in truly subcubic time. Theory Comput., 5(1):173-189, 2009. URL: https://doi.org/10.4086/toc.2009.v005a009.
  24. R. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. SIAM J. Comput., 47(5):1965-1985, 2018. URL: https://doi.org/10.1137/15M1024524.
  25. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. URL: https://doi.org/10.1145/3186893.
  26. Virginia Vassilevska Williams and Yinzhan Xu. Truly subcubic min-plus product for less structured matrices, with applications. In SODA 2020, pages 12-29. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.2.
  27. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds for matrix multiplication: from alpha to omega, 2023. URL: https://arxiv.org/abs/2307.07970.
  28. Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. CoRR, cs.DS/0008011, 2000. URL: https://arxiv.org/abs/cs/0008011.
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 Imprint Privacy Contact