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(min, + ) Matrix and Vector Products for Inputs Decomposable into Few Monotone Subsequences

Lingas, Andrzej LU and Persson, Mia LU (2024) 29th International Computing and Combinatorics Conference, COCOON 2023 In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 14423 LNCS. p.55-68
Abstract

We study the time complexity of computing the (min, + ) matrix product of two n× n integer matrices in terms of n and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most m1 monotone subsequences and each column of the second matrix can be decomposed into at most m2 monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the (min, + ) product of the matrices can be computed in O(m1m2n2.569) time. On the other hand, we observe that if all the rows of the first matrix... (More)

We study the time complexity of computing the (min, + ) matrix product of two n× n integer matrices in terms of n and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most m1 monotone subsequences and each column of the second matrix can be decomposed into at most m2 monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the (min, + ) product of the matrices can be computed in O(m1m2n2.569) time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or vice versa then this case is as hard as the general one. Similarly, we also study the time complexity of computing the (min, + ) convolution of two n-dimensional integer vectors in terms of n and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most m1 monotone subsequences and the second vector can be decomposed into at most m2 subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or vice versa then their (min, + ) convolution can be computed in O~ (m1m2n1.5) time. On the other, the case when both vectors are non-decreasing or both of them are non-increasing is as hard as the general case.

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author
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organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
host publication
Computing and Combinatorics - 29th International Conference, COCOON 2023, Proceedings
series title
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
editor
Wu, Weili and Tong, Guangmo
volume
14423 LNCS
pages
14 pages
publisher
Springer Science and Business Media B.V.
conference name
29th International Computing and Combinatorics Conference, COCOON 2023
conference location
Hawaii, United States
conference dates
2023-12-15 - 2023-12-17
external identifiers
  • scopus:85180531292
ISSN
0302-9743
1611-3349
ISBN
9783031491924
DOI
10.1007/978-3-031-49193-1_5
language
English
LU publication?
yes
id
3db16d83-c174-435c-a50f-6b0649f1a89b
date added to LUP
2025-01-15 12:15:03
date last changed
2025-07-31 04:57:56
@inproceedings{3db16d83-c174-435c-a50f-6b0649f1a89b,
  abstract     = {{<p>We study the time complexity of computing the (min, + ) matrix product of two n× n integer matrices in terms of n and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most m<sub>1</sub> monotone subsequences and each column of the second matrix can be decomposed into at most m<sub>2</sub> monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the (min, + ) product of the matrices can be computed in O(m<sub>1</sub>m<sub>2</sub>n<sup>2.569</sup>) time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or vice versa then this case is as hard as the general one. Similarly, we also study the time complexity of computing the (min, + ) convolution of two n-dimensional integer vectors in terms of n and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most m<sub>1</sub> monotone subsequences and the second vector can be decomposed into at most m<sub>2</sub> subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or vice versa then their (min, + ) convolution can be computed in O~ (m<sub>1</sub>m<sub>2</sub>n<sup>1.5</sup>) time. On the other, the case when both vectors are non-decreasing or both of them are non-increasing is as hard as the general case.</p>}},
  author       = {{Lingas, Andrzej and Persson, Mia}},
  booktitle    = {{Computing and Combinatorics - 29th International Conference, COCOON 2023, Proceedings}},
  editor       = {{Wu, Weili and Tong, Guangmo}},
  isbn         = {{9783031491924}},
  issn         = {{0302-9743}},
  language     = {{eng}},
  pages        = {{55--68}},
  publisher    = {{Springer Science and Business Media B.V.}},
  series       = {{Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)}},
  title        = {{(min, + ) Matrix and Vector Products for Inputs Decomposable into Few Monotone Subsequences}},
  url          = {{http://dx.doi.org/10.1007/978-3-031-49193-1_5}},
  doi          = {{10.1007/978-3-031-49193-1_5}},
  volume       = {{14423 LNCS}},
  year         = {{2024}},
}