Overview:
Strassen’s divide and conquer matrix multiplication algorithm for n by n matrices is asymptotically faster
than the conventional O(n3) algorithm. This means that for sufficiently large values of n, Strassen’s algorithm
will run faster than the conventional algorithm. For small values of n, however, the conventional algorithm is
faster.
Since Strassen’s algorithm is a recursive algorithm, at some point in the recursion, once the matrices are small
enough, we may want to switch from recursively calling Strassen’s algorithm and just do a conventional
matrix multiplication. That is, the proper way to do Strassen’s algorithm is to not recurse to a “base case” of
a 1 by 1 matrix, but to switch earlier and use conventional matrix multiplication. Let us define the cross-over
point between the two algorithms to be the value of n for which we want to stop using Strassen’s algorithm
and switch to conventional matrix multiplication. The goal of this assignment is to implement the
conventional algorithm and Strassen’s algorithm and to determine their cross-over point, both analytically
and experimentally. One important factor our simple analysis will not take into account is memory
management, which may significantly affect the speed of your implementation.
Tasks:
1. Assume that the cost of any single arithmetic operation (adding, subtracting, multiplying, or
dividingtwo real numbers) is 1, and that all other operations are free. Consider the following variant of
Strassen’s algorithm: to multiply two n by n matrices, start using Strassen’s algorithm, but stop the
recursion at some size n0, and use the conventional algorithm below that point. You have to find a
suitable value for n0 – the cross-over point. Analytically determine the value of n0 that optimizes the
running time of this algorithm in this model. (That is, solve the appropriate equations.) This gives a
crude estimate for the cross-over point between Strassen’s algorithm and the standard matrix
multiplication algorithm.
2. Implement your variant of Strassen’s algorithm and the standard matrix multiplication algorithm
tofind the cross-over point experimentally. Experimentally optimize for n0 and compare the
experimental results with your estimate from above. Make both implementations as efficient as
possible. The actual cross-over point, which you would like to make as small as possible, will depend
on how efficiently you implement Strassen’s algorithm. Your implementation should work for any
matrices, not just those whose dimensions are a power of 2.
To test your algorithm, you might try matrices where each entry is randomly selected to be 0 or 1;
similarly, you might try matrices where each entry is randomly selected to be 0, 1 or 2, or instead 0, 1,
or −1. You might also try matrices where each entry is a randomly selected real number in the range
[0,1]. You need not try all of these, but do test your algorithm adequately.
Code setup:
So that we may test your code ourselves as necessary, please make sure your code runs as follows:
$ python strassen.py 0 dimension inputfile
The flag 0 is meant to provide you some flexibility; you may use other values for your own testing, debugging,
or extensions. The dimension, which we refer to henceforth as d, is the dimension of the matrix you are
multiplying, so that 32 means you are multiplying two 32 by 32 matrices together. The inputfile is
1
an ASCII file with 2d2 integer numbers, one per line, representing two matrices A and B; you are to find the
product AB = C. The first integer number is matrix entry a1,1, followed by a1,2,a1,3,...,a1,d, where ai,j denotes the
entry of A in the ith row and jth column; next comes a2,1,a2,2, and so on, for the first d2 numbers. The next d2
numbers are similar for matrix B.
Your program should print to standard output a list of the values of the diagonal entries c1,1,c2,2,...,cd,d, one per
line, including a trailing newline. We reserve the right to check the output by a script, so add no clutter. (You
should not output the whole matrix, although of course all entries should be computed.)
What to hand in:
As before, you may work in pairs, or by yourself. Refer to the Collaboration and Academic Honesty Policies
on the course website for rules regarding resources and discussion other than your partner. You should not
use any code outside of the Python Standard Library.
Hand in a project report (on paper) describing your analytical and experimental work (for example, carefully
describe optimizations you made in your implementations). Be sure to discuss the results you obtain, and try
to give explanations for what you observe. How low was your cross-over point? What difficulties arose? What
types of matrices did you multiply, and does this choice matter?
Your grade will be based primarily on the correctness of your program, the crossover point you find, your
interpretation of the data, and your discussion of the experiment.
Hints:
It is hard to make the conventional algorithm inefficient; however, you may get better caching performance
by looping through the variables in the right order (really, try it!). For Strassen’s algorithm:
• Avoid copying large blocks of data unnecessarily. This requires some thinking.
• Your implementation of Strassen’s algorithm should work even when n is odd! This requires some
additional work, and thinking. (One option is to pad with 0’s; how can this be done most effectively?)
However, you may want to first get it to work when n is a power of 2 – this will get you most of the
credit – and then refine it to work for more general values of n.