A 5/2n 2-Lower Bound for the Multiplicative Complexity of n × n-Matrix Multiplication

2001 ◽  
pp. 99-109 ◽  
Author(s):  
Markus Bläser
Keyword(s):  
Lower Bound ◽  
Matrix Multiplication ◽  
Multiplicative Complexity

scholarly journals Latency Analysis in the 2-Dimensional Systolic Arrays for Matrix Multiplication

2021 ◽  
Vol 15 ◽  
pp. 1-7
Author(s):  
Halil Snopce ◽  
Azir Aliu
Keyword(s):  
Lower Bound ◽  
Systolic Array ◽  
Matrix Multiplication ◽  
Systolic Arrays ◽  
Latency Analysis

This paper deals with the latency analysis in a twodimensional systolic array for matrix multiplication. The latency for all possible connection schemes is discussed. In this way there is obtained the lower bound of the latency that can be achieved using such arrays.

scholarly journals Lower Bounds for Dynamic Algebraic Problems

1998 ◽  
Vol 5 (11) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Johan P. Hansen ◽  
Peter Bro Miltersen
Keyword(s):  
Fourier Transform ◽  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Symmetric Functions ◽  
Matrix Multiplication ◽  
Worst Case ◽  
Dynamic Matrix ◽  
Wide Range ◽  
On Line

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form "change input xi to value v" or "what is the value of output yi?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent<br />algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal  Omega(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an <br />Omega(sqrt(n)) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and<br />Tate's O(sqrt(n log n)) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Omega(sqrt(n)) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting<br />of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an ((log n)2= log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.

A Lower Bound for Matrix Multiplication

1989 ◽  
Vol 18 (4) ◽  
pp. 759-765 ◽  
Author(s):  
Nader H. Bshouty
Keyword(s):  
Lower Bound ◽  
Matrix Multiplication

LOW CROSSTALK ADDRESS ENCODINGS FOR OPTICAL MESSAGE SWITCHING SYSTEMS

1996 ◽  
Vol 06 (01) ◽  
pp. 87-100
Author(s):  
YOSI BEN-ASHER ◽  
ASSAF SCHUSTER
Keyword(s):  
Lower Bound ◽  
Matrix Multiplication ◽  
Maximum Intensity ◽  
Massively Parallel ◽  
Switching Systems ◽  
Switching System ◽  
Clock Skew ◽  
Optical Elements ◽  
Linear Elements ◽  
Vector Matrix

An optical message switching system delivers messages from N sources to N destinations using beams of light. The redirection of the beams involves vector-matrix multiplication and a threshold operation. The input vectors are set by the sources and may be viewed as the addresses of the desired destinations. In a massively parallel system, it is highly desirable to reduce the number of threshold (non-linear) elements, which require extra wiring and increase clock skew. Moreover, the threshold devices have a sensitivity parameter (implied by the technology) defined as the gap in which the outcome of the device is not determined. This gap is largely effected by the crosstalk which is the maximum number of joint set bits in any pair of addresses, implying a lower bound on the maximum intensity for which the outcome of the threshold operation is determined. In this work we consider the design of addresses which are both short (so that the number of threshold devices is reduced) and have low crosstalk (so that the sensitivity gap may grow). We show that addresses of O( log N) bits exist, for which the crosstalk is a constant fraction of the number of set bits in each address, hence allowing for a Θ( log N) sized sensitivity gap. More generally, we show the precise coefficient which depends on the desired gap. It is established that when using O( log N) bit addresses, the crosstalk cannot be further reduced. An exact construction of O( log 2 N) bit addresses is given, where the involved constant depends on the desired crosstalk. Finally we describe briefly the basic optical elements that can be used in order to construct a message switching system which use these address schemes.

scholarly journals A Tight I/O Lower Bound for Matrix Multiplication

2020 ◽  
Author(s):  
Tyler M Smith ◽  
Robert A van de Geijn
Keyword(s):  
Lower Bound ◽  
Matrix Multiplication
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