Related papers: Fine-Grained Computation in 3-Space: Matrix Multip…
The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication…
We describe two algorithms for multiplying n x n matrices using time and energy n^2 polylog(n) under basic models of classical physics. The first algorithm is for multiplying integer-valued matrices, and the second, quite different…
Efficient multiple precision linear numerical computation libraries such as MPLAPACK are critical in dealing with ill-conditioned problems. Specifically, there are optimization methods for matrix multiplication, such as the Strassen…
Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and $o(\log n)$ rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on…
The Strassen algorithm and Winograd's variant accelerate matrix multiplication by using fewer arithmetic operations than standard matrix multiplication. Although many papers have been published to accelerate single- as well as…
A parallel algorithm has perfect strong scaling if its running time on P processors is linear in 1/P, including all communication costs. Distributed-memory parallel algorithms for matrix multiplication with perfect strong scaling have only…
A tight $\Omega((n/\sqrt{M})^{\log_2 7}M)$ lower bound is derived on the \io complexity of Strassen's algorithm to multiply two $n \times n$ matrices, in a two-level storage hierarchy with $M$ words of fast memory. A proof technique is…
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…
This paper initiates the study of I/O algorithms (minimizing cache misses) from the perspective of fine-grained complexity (conditional polynomial lower bounds). Specifically, we aim to answer why sparse graph problems are so hard, and why…
Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic…
Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for…
In science and engineering, intelligent processing of complex signals such as images, sound or language is often performed by a parameterized hierarchy of nonlinear processing layers, sometimes biologically inspired. Hierarchical systems…
In this work the algorithms of fast multiplication of matrices are considered. To any algorithm there associated a certain group of automorphisms. These automorphism groups are found for some well-known algorithms, including algorithms of…
Recently, reinforcement algorithms discovered new algorithms that really jump-started a wave of excitements and a flourishing of publications. However, there is little on implementations, applications, and, especially, no absolute…
Designing problems using matrices is very important in Computer Science. Fields like graph computer, graphs theory, and machine learning use matrices very often to solve their own problems. The most often matrix operation is the…
We consider the famous Strassen algorithm for fast multiplication of matrices. We show that this algorithm has a nontrivial finite group of automorphisms of order 36 (namely the direct product of two copies of the symmetric group on 3…
This paper gives simple distributed algorithms for the fundamental problem of computing graph distances in the Congested Clique model. One of the main components of our algorithms is fast matrix multiplication, for which we show an…
Fast matrix-by-matrix multiplication (hereafter MM) is a highly recognized research subject. The record upper bound 3 of 1968 on the exponent of the complexity MM decreased below 2.38 by 1987, applies to celebrated problems in many areas of…