Results 1  10
of
32
Grouptheoretic algorithms for matrix multiplication
 In Foundations of Computer Science. 46th Annual IEEE Symposium on 23–25 Oct 2005
, 2005
"... We further develop the grouptheoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication e ..."
Abstract

Cited by 47 (4 self)
 Add to MetaCart
We further develop the grouptheoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2. 1.
Fast Sparse Matrix Multiplication
, 2004
"... Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multi ..."
Abstract

Cited by 38 (2 self)
 Add to MetaCart
Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multiplication algorithm, on the other hand, may need to perform #(mn) operations to accomplish the same task. For , the new algorithm performs an almost optimal number of only n operations. For m the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n ) 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.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we
Breaking the CoppersmithWinograd barrier
, 2011
"... We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727.
Regularity lemmas and combinatorial algorithms
 In Proc. FOCS
"... Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n)) bound for machine models with wordsize w. (For a pointer machine, we can set w = log n.) The algorithms utilize notions from Regularity Lemmas for graphs in a novel way. • We give two randomized combinatorial algorithms for BMM. The first algorithm is essentially a reduction from BMM to the Triangle Removal Lemma. The best known bounds for the Triangle Removal Lemma only imply an O ` (n 3 log β)/(βw log n) ´ time algorithm for BMM where β = (log ∗ n) δ for some δ> 0, but improvements on the Triangle Removal Lemma would yield corresponding runtime improvements. The second algorithm applies the Weak Regularity Lemma of Frieze and Kannan along with “ several information compression ideas, running in O n 3 (log log n) 2 /(log n) 9/4 ”) time with probability exponentially “ close to 1. When w ≥ log n, it can be implemented in O n 3 (log log n) 2 /(w log n) 7/6 ”) time. Our results immediately imply improved combinatorial methods for CFG parsing, detecting trianglefreeness, and transitive closure. Using Weak Regularity, we also give an algorithm for answering queries of the form is S ⊆ V an independent set? in a graph. Improving on prior work, we show how to randomly preprocess a graph in O(n 2+ε) time (for all ε> 0) so that with high probability, all subsequent batches of log n independent “ set queries can be answered deterministically in O n 2 (log log n) 2 /((log n) 5/4 ”) time. When w ≥ log n, w queries can be answered in O n 2 (log log n) 2 /((log n) 7/6 ” time. In addition to its nice applications, this problem is interesting in that it is not known how to do better than O(n 2) using “algebraic ” methods. 1.
Colored Intersection Searching via Sparse Rectangular Matrix Multiplication
, 2006
"... In a Batched Colored Intersection Searching Problem (CI), one is given a set of n geometric objects (of a certain class). Each object is colored by one of c colors, and the goal is to report all pairs of colors (c1, c2) such that there are two objects, one colored c1 and one colored c2, that interse ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
In a Batched Colored Intersection Searching Problem (CI), one is given a set of n geometric objects (of a certain class). Each object is colored by one of c colors, and the goal is to report all pairs of colors (c1, c2) such that there are two objects, one colored c1 and one colored c2, that intersect each other. We also consider the bipartite version of the problem, where we are interested in intersections between objects of one class with objects of another class (e.g., points and halfspaces). In a Sparse
On Sunflowers and Matrix Multiplication
"... We present several variants of the sunflower conjecture of Erdős and Rado [ER60] and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd [CW90] and Cohn et al [CKSU05] regarding possible approaches for ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We present several variants of the sunflower conjecture of Erdős and Rado [ER60] and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd [CW90] and Cohn et al [CKSU05] regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the ErdősRado sunflower conjecture (if true) implies a negative answer to the “no three disjoint equivoluminous subsets ” question of Coppersmith and Winograd [CW90]; we also formulate a “multicolored ” sunflower conjecture in Z n 3 and show that (if true) it implies a negative answer to the “strong USP ” conjecture of [CKSU05] (although it does not seem to impact a second conjecture in [CKSU05] or the viability of the general grouptheoretic approach). A surprising consequence of our results is that the CoppersmithWinograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in Z n 3 is a strengthening of the wellknown (ordinary) sunflower conjecture in Z n 3, and we show via our connection that a construction from [CKSU05] yields a lower bound of (2.51...) n on the size of the largest multicolored 3sunflowerfree set, which beats the current best known lower bound of (2.21...) n [Edel04] on the size of the largest 3sunflowerfree set in Z n 3.