Description
Overview:
Strassen’s divide and conquer matrix multiplication algorithm for n by n matrices is asymptotically
faster than the conventional O(n
3
) 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 may be faster. Indeed, the textbook Algorithms in C (1990 edition) suggests that n would have
to be in the thousands before offering an improvement to standard multiplication, and “Thus the algorithm
is a thoeretical, not practical, contribution.” Here we test this armchair analysis.
Here is a key point, though (for any recursive algorithm!). 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 all the way down to a “base case” of a 1 by 1
matrix, but to switch earlier and use conventional matrix multiplication.
That is, there’s no reason to do
a “base case” of a 1 by 1 matrix; it might be faster to use a larger-sized base case, as conventional matrix
multiplication might be faster up to some reasonable size. 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 dividing
two 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, somehow,
numerically.) 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 to
find 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 size 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. We will test on integer matrices, possibly of this form. (You may assume 32-bit integer
inputs.) You need not try all of these, but do test your algorithm adequately.
This assignment is partially autograded. The only languages which have been configured which work
with autograder are C11, C++ 17, Java 8, and Python 3.6. Correspondingly, your code should
be contained within a single file that is called one of: strassen.c, strassen.cpp, strassen.java,
1
or strassen.py.
3. Triangle in random graphs: Recall that you can represent the adjacency matrix of a graph by a
matrix A. Consider an undirected graph. It turns out that A3
can be used to determine the number
of triangles in a graph. To see this, you can see that (ij)th entry in the matrix A2
counts the paths
from i to j of lenth two, and the (ij)th entry in the matrix A3
counts the path from i to j of length
3. To count the number of triangles in in graph, we can simply add the entries in the diagonal, and
divide by 6. This is because the jth diagonal entry counts the number of paths of length 3 from j to
j. Each such path is a triangle, and each triangle is counted 6 times (for each of the vertices in the
triangle, it is counted once in each direction).
Create a random graph on 1024 vertices where each edge is included with probability p for each of the
following values of p: p = 0.01, 0.02, 0.03, 0.04, and 0.05. Use your (Strassen’s) matrix multiplication
code to count the number of triangles in each of these graphs, and compare it to the expected number
of trianles, which is
1024
3
p
3
. Create a chart showing your results compared to the expectation.
Code setup:
Your code will be compiled, run, and tested by an autograding script. The way you will run the code
will depend on the language, but it should take in three command line arguments: a flag, a dimension,
and an input file. For example, if you did the assignment in C, we will run:
$ ./strassen 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
an ASCII file with 2d
2 CS124 Programming Assignment 2
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 a0,0, followed by a0,1, a0,2, . . . , a0,d−1; next
comes a1,0, a1,1, and so on, for the first d
2 numbers. The next d
2 numbers are similar for matrix B.
Your program should put on standard output (printf, cout, System.out, etc) a list of the values of
the diagonal entries c0,0, c1,1, . . . , cd−1,d−1, one per line, including a trailing newline. The output will be
checked by a script – add no clutter. (You should not output the whole matrix, although of course all
entries should be computed.)
The inputs we present will be small (signed) integers, but you should make sure your matrix multiplication can deal with results that are up to 32 bits.
When you submit your code to Gradescope, some tests will run to make sure that your code compiles
and to make sure that it is correct. These tests will make up a significant percantage of your grade. Note
that depending on your choice of language and the server load on Gradescope at the time of submission,
these tests may take several minutes to complete, so be sure to submit well before the deadline or to
resubmit often.
What to hand in:
• A PDF file containing your project report
• A single code file that is one of: strassen.c, strassen.cpp, strassen.py, or strassen.java.
Submissions which rely on external files or libraries may fail to compile.
As before, you may work in pairs, or by yourself. 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 excessive memory allocation and deallocation. This requires some thinking.
• 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.
3 CS124 Programming Assignment 2
