Document

# Complexity of Linear Operators

## File

LIPIcs.ISAAC.2019.17.pdf
• Filesize: 468 kB
• 12 pages

## Acknowledgements

We thank Paweł Gawrychowski for pointing us out to the paper [Bernard Chazelle and Burton Rosenberg, 1991]. We thank Alexey Talambutsa for fruitful discussions on the theory of semigroups.

## Cite As

Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, and Vladimir Podolskii. Complexity of Linear Operators. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.17

## Abstract

Let A in {0,1}^{n x n} be a matrix with z zeroes and u ones and x be an n-dimensional vector of formal variables over a semigroup (S, o). How many semigroup operations are required to compute the linear operator Ax? As we observe in this paper, this problem contains as a special case the well-known range queries problem and has a rich variety of applications in such areas as graph algorithms, functional programming, circuit complexity, and others. It is easy to compute Ax using O(u) semigroup operations. The main question studied in this paper is: can Ax be computed using O(z) semigroup operations? We prove that in general this is not possible: there exists a matrix A in {0,1}^{n x n} with exactly two zeroes in every row (hence z=2n) whose complexity is Theta(n alpha(n)) where alpha(n) is the inverse Ackermann function. However, for the case when the semigroup is commutative, we give a constructive proof of an O(z) upper bound. This implies that in commutative settings, complements of sparse matrices can be processed as efficiently as sparse matrices (though the corresponding algorithms are more involved). Note that this covers the cases of Boolean and tropical semirings that have numerous applications, e.g., in graph theory. As a simple application of the presented linear-size construction, we show how to multiply two n x n matrices over an arbitrary semiring in O(n^2) time if one of these matrices is a 0/1-matrix with O(n) zeroes (i.e., a complement of a sparse matrix).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Streaming, sublinear and near linear time algorithms
##### Keywords
• algorithms
• linear operators
• commutativity
• range queries
• circuit complexity
• lower bounds
• upper bounds

## Metrics

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

## References

1. Noga Alon and Baruch Schieber. Optimal preprocessing for answering on-line product queries. Technical report, Tel Aviv University, 1987.
2. Peter Butkovič. Max-linear Systems: Theory and Algorithms. Springer, 2010.
3. Bernard Chazelle and Burton Rosenberg. The complexity of computing partial sums off-line. Int. J. Comput. Geometry Appl., 1(1):33-45, 1991. URL: https://doi.org/10.1142/S0218195991000049.
4. 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.
5. J. A. Green and D. Rees. On semi-groups in which x^r = x. Mathematical Proceedings of the Cambridge Philosophical Society, 48(1):35–40, 1952. URL: https://doi.org/10.1017/S0305004100027341.
6. Dima Grigoriev and Vladimir V. Podolskii. Complexity of Tropical and Min-plus Linear Prevarieties. Computational Complexity, 24(1):31-64, 2015. URL: https://doi.org/10.1007/s00037-013-0077-5.
7. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-24508-4.
8. Stasys Jukna. Tropical Complexity, Sidon Sets, and Dynamic Programming. SIAM J. Discrete Math., 30(4):2064-2085, 2016. URL: https://doi.org/10.1137/16M1064738.
9. Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, and Vladimir V. Podolskii. Complexity of Linear Operators. Electronic Colloquium on Computational Complexity (ECCC), 26:2, 2019. URL: https://eccc.weizmann.ac.il/report/2019/002.
10. Andrey Mokhov. Algebraic graphs with class (functional pearl). In Proceedings of the 10th ACM SIGPLAN International Symposium on Haskell, pages 2-13. ACM, 2017.
11. H. Neumann. Varieties of Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. Springer Berlin Heidelberg, 2012. URL: https://books.google.ru/books?id=VaMjCQAAQBAJ.
12. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 664-673, 2014. URL: https://doi.org/10.1145/2591796.2591811.
13. Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 887-898. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214056.
14. Andrew Chi-Chih Yao. Space-Time Tradeoff for Answering Range Queries (Extended Abstract). In Harry R. Lewis, Barbara B. Simons, Walter A. Burkhard, and Lawrence H. Landweber, editors, Proceedings of the 14th Annual ACM Symposium on Theory of Computing, May 5-7, 1982, San Francisco, California, USA, pages 128-136. ACM, 1982. URL: https://doi.org/10.1145/800070.802185.
X

Feedback for Dagstuhl Publishing

### Thanks for your feedback!

Feedback submitted

### Could not send message

Please try again later or send an E-mail