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

Dvir Fried; Tsvi Kopelowitz; Ely Porat

doi:10.4230/LIPIcs.ESA.2024.57

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

Dvir Fried
, Tsvi Kopelowitz
, Ely Porat

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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(Mn^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(Mn^ω + Mn² log M), thereby removing the log M factor in the runtime if ω > 2 or if n^ω = Ω(n² log n).

Original language	English
Title of host publication	32nd Annual European Symposium on Algorithms, ESA 2024
Editors	Timothy Chan, Johannes Fischer, John Iacono, Grzegorz Herman
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959773386
DOIs	https://doi.org/10.4230/LIPIcs.ESA.2024.57
State	Published - Sep 2024
Event	32nd Annual European Symposium on Algorithms, ESA 2024 - London, United Kingdom Duration: 2 Sep 2024 → 4 Sep 2024

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	308
ISSN (Print)	1868-8969

Conference

Conference	32nd Annual European Symposium on Algorithms, ESA 2024
Country/Territory	United Kingdom
City	London
Period	2/09/24 → 4/09/24

Bibliographical note

Publisher Copyright:
© Dvir Fried, Tsvi Kopelowitz, and Ely Porat; licensed under Creative Commons License CC-BY 4.0.

Keywords

(min, +)-product
FFT
FMM

Access to Document

10.4230/LIPIcs.ESA.2024.57

Cite this

Fried, D., Kopelowitz, T., & Porat, E. (2024). Removing the log Factor from (min, +)-Products on Bounded Range Integer Matrices. In T. Chan, J. Fischer, J. Iacono, & G. Herman (Eds.), 32nd Annual European Symposium on Algorithms, ESA 2024 Article 57 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 308). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2024.57

Fried, Dvir ; Kopelowitz, Tsvi ; Porat, Ely. / Removing the log Factor from (min, +)-Products on Bounded Range Integer Matrices. 32nd Annual European Symposium on Algorithms, ESA 2024. editor / Timothy Chan ; Johannes Fischer ; John Iacono ; Grzegorz Herman. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2024. (Leibniz International Proceedings in Informatics, LIPIcs).

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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(Mnω 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(Mnω + Mn2 log M), thereby removing the log M factor in the runtime if ω > 2 or if nω = Ω(n2 log n).",

keywords = "(min, +)-product, FFT, FMM",

author = "Dvir Fried and Tsvi Kopelowitz and Ely Porat",

note = "Publisher Copyright: {\textcopyright} Dvir Fried, Tsvi Kopelowitz, and Ely Porat; licensed under Creative Commons License CC-BY 4.0.; 32nd Annual European Symposium on Algorithms, ESA 2024 ; Conference date: 02-09-2024 Through 04-09-2024",

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Fried, D, Kopelowitz, T & Porat, E 2024, Removing the log Factor from (min, +)-Products on Bounded Range Integer Matrices. in T Chan, J Fischer, J Iacono & G Herman (eds), 32nd Annual European Symposium on Algorithms, ESA 2024., 57, Leibniz International Proceedings in Informatics, LIPIcs, vol. 308, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 32nd Annual European Symposium on Algorithms, ESA 2024, London, United Kingdom, 2/09/24. https://doi.org/10.4230/LIPIcs.ESA.2024.57

Removing the log Factor from (min, +)-Products on Bounded Range Integer Matrices. / Fried, Dvir; Kopelowitz, Tsvi ; Porat, Ely.
32nd Annual European Symposium on Algorithms, ESA 2024. ed. / Timothy Chan; Johannes Fischer; John Iacono; Grzegorz Herman. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2024. 57 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 308).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - 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(Mnω 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(Mnω + Mn2 log M), thereby removing the log M factor in the runtime if ω > 2 or if nω = Ω(n2 log n).

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Fried D, Kopelowitz T , Porat E. Removing the log Factor from (min, +)-Products on Bounded Range Integer Matrices. In Chan T, Fischer J, Iacono J, Herman G, editors, 32nd Annual European Symposium on Algorithms, ESA 2024. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2024. 57. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.ESA.2024.57

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

Abstract

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Bibliographical note

Keywords

Access to Document

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