Lecture 1: Three Divide and Conquer Algorithms

Background Material

• Chapters 1-6 of Cormen, Lieserson, and Rivest
• Recurrence Relations
• Geometric series
• Summation Notation and Manipulation
• Growth Rates and Asymptotic Analysis

Reading

• Cooley, J. M. and J. W. Tukey, "An algorithm for the machine calculation of complex Fourier series", Math. Comp., 19, 297-301, 1965.
• A. Karatsuba and Y. Ofman, "Multiplication of multidigit numbers on automata," Dokl. Akad. Nauk SSSR 145, 293-294, 1962.
• V. Strassen, "Gaussian elimination is not optimal", Numerische Mathematik, 13, 354-356, 1969.
• S. Huss-Lederman and E.M. Jacobson and A. Tsao and T. Turnbull and J.R. Johnson, Implementation of {S}trassen's algorithm for matrix multiplication, Proceedings of the 1996 ACM/IEEE conference on Supercomputing.

Topics

• Overview of the course.
• Objectives
• Topics
• Grading policy and expectations
• Three Problems/Three Algorithms
  • Matrix Multiplication and Strassen's algorithm
  • Discrete Fourier transform (DFT) and the fast Fourier transform (FFT)
  • Integer multiplication and Karatsuba's algorithm

Lecture Notes

• lec1p0.pdf (Part zero - overview - of lecture notes)
• lec1p1.pdf (Part one - Strassen - of lecture notes)
• lec1p2.pdf (Part two - FFT - of lecture notes)
• lec1p3.pdf (Part three - Karatsuba - of lecture notes)

Resources

• ATLAS
• FFTW
• GNU Multiple Precision Arithmetic Library
• Intel Performance Libraries
• fastmm
• GEMMW - Craig Douglas' implementation of Strassen's algorithm.

Assignments

• Practice Assignment on Strassen's Algorithm (not to be submitted)
• Read and prepare to discuss during the next class: P.D'Alberto and A.Nicolau, Adaptive Winograd's Matrix Multiplications, ACM Transaction on Mathematical Software 2008 [Look at reference number 1 in the given link].