TY - CHAP

T1 - Fast sparse matrix multiplication

AU - Yuster, Raphael

AU - Zwick, Uri

N1 - Funding Information:
Funded in part by the US National Science Foundation, Grant No. PHY89-22550.

PY - 2004

Y1 - 2004

N2 - Let A and B two n×n matrices over a ring R (e.g., the reals or the integers) each containing at most m non-zero elements. We present a new algorithm that multiplies A and B using O(m0.7n1.2 + n2+o(1)) algebraic operations (i.e., multiplications, additions and subtractions) over R. For m ≤ n1.14, the new algorithm performs an almost optimal number of only n2+o(1) operations. For m ≤ n 1.68, the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n2.38) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices. As the known fast rectangular matrix multiplication algorithms are far from being practical, our result, at least for now, is only of theoretical value.

AB - Let A and B two n×n matrices over a ring R (e.g., the reals or the integers) each containing at most m non-zero elements. We present a new algorithm that multiplies A and B using O(m0.7n1.2 + n2+o(1)) algebraic operations (i.e., multiplications, additions and subtractions) over R. For m ≤ n1.14, the new algorithm performs an almost optimal number of only n2+o(1) operations. For m ≤ n 1.68, the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n2.38) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices. As the known fast rectangular matrix multiplication algorithms are far from being practical, our result, at least for now, is only of theoretical value.

UR - http://www.scopus.com/inward/record.url?scp=35048828796&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-30140-0_54

DO - 10.1007/978-3-540-30140-0_54

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AN - SCOPUS:35048828796

SN - 3540230254

SN - 9783540230250

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 604

EP - 615

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - Albers, Susanne

A2 - Radzik, Tomasz

PB - Springer Verlag

ER -