Fast output-sensitive matrix multiplication

TY - GEN

T1 - Fast output-sensitive matrix multiplication

AU - Jacob, Riko

AU - Stöckel, Morten

N1 - Conference code: 23

PY - 2015

Y1 - 2015

N2 - We consider the problem of multiplying two U×U matrices A and C of elements from a field F. We present a new randomized algorithm that can use the known fast square matrix multiplication algorithms to perform fewer arithmetic operations than the current state of the art for output matrices that are sparse. In particular, let ω be the best known constant such that two dense U × U matrices can be multiplied with O(Uω) arithmetic operations. Further denote by N the number of nonzero entries in the input matrices while Z is the number of nonzero entries of matrix product AC. We present a new Monte Carlo algorithm that uses (Formula presented) arithmetic operations and outputs the nonzero entries of AC with high probability. For dense input, i.e., N = U2, if Z is asymptotically larger than U, this improves over state of the art methods, and it is always at most O(Uω). For general input density we improve upon state of the art when N is asymptotically larger than U4−ωZω−5/2. The algorithm is based on dividing the input into “balanced” subproblems which are then compressed and computed. The new subroutine that computes a matrix product with balanced rows and columns in its output uses time Õ (UZ(ω−1)/2 + N) which is better than the current state of the art for balanced matrices when N is asymptotically larger than UZω/2−1, which always holds when N = U2. In the I/O model — where M is the memory size and B is the block size — our algorithm is the first nontrivial result that exploits cancellations and sparsity of the output. The I/O complexity of our algorithm is Õ (U2(Z/U)ω−2/(Mω/2−1B) + Z/B + N/B), which is asymptotically faster than the state of the art unless M is large.

AB - We consider the problem of multiplying two U×U matrices A and C of elements from a field F. We present a new randomized algorithm that can use the known fast square matrix multiplication algorithms to perform fewer arithmetic operations than the current state of the art for output matrices that are sparse. In particular, let ω be the best known constant such that two dense U × U matrices can be multiplied with O(Uω) arithmetic operations. Further denote by N the number of nonzero entries in the input matrices while Z is the number of nonzero entries of matrix product AC. We present a new Monte Carlo algorithm that uses (Formula presented) arithmetic operations and outputs the nonzero entries of AC with high probability. For dense input, i.e., N = U2, if Z is asymptotically larger than U, this improves over state of the art methods, and it is always at most O(Uω). For general input density we improve upon state of the art when N is asymptotically larger than U4−ωZω−5/2. The algorithm is based on dividing the input into “balanced” subproblems which are then compressed and computed. The new subroutine that computes a matrix product with balanced rows and columns in its output uses time Õ (UZ(ω−1)/2 + N) which is better than the current state of the art for balanced matrices when N is asymptotically larger than UZω/2−1, which always holds when N = U2. In the I/O model — where M is the memory size and B is the block size — our algorithm is the first nontrivial result that exploits cancellations and sparsity of the output. The I/O complexity of our algorithm is Õ (U2(Z/U)ω−2/(Mω/2−1B) + Z/B + N/B), which is asymptotically faster than the state of the art unless M is large.

U2 - 10.1007/978-3-662-48350-3_64

DO - 10.1007/978-3-662-48350-3_64

M3 - Article in proceedings

SN - 978-3-662-48350-3

T3 - Lecture notes in computer science

SP - 766

EP - 778

BT - Algorithms - ESA 2015

A2 - Bansal, Nikhil

A2 - Finocchi, Irene

PB - Springer

T2 - Annual European Symposium 2015

Y2 - 14 September 2015 through 16 September 2015

ER -