On Matrix Multiplication Using Programmable Graph Architecture

Recently, Tang et. al. introduced an algorithm called the programmable graph architecture (PGA) algorithm for multiplying matrices in GL(n,mu), the generalized linear group of matrices modulo mu, where mu is an integer, using its Cayley graph representation. The problem of multiplying matrices in GL(n,mu) is mapped to the problem of finding routes on a suitable Cayley graph representation of the group. The main bottleneck in this algorithm is that it is not easily scalable for large values of n and mu because the size of the group GL(n,mu) becomes very large. Since the Cayley graph representation of the group has the same number of vertices as the size of the group, it becomes difficult to implement such a large graph. We present an algorithm that uses the factorization of GL(n,mu) using its principal congruence subgroups to multiply matrices in GL(n.mu). Instead of using one big Cayley graph for GL(n,mu), we use multiple Cayley graphs representing the factor groups of GL(n,mu) with respect to its principal congruence subgroups, which are much smaller in size. This approach provides significant reduction in the space complexity of the PGA algorithm. We also show that Schreier graphs, which are formed using cosets of a group with respect to a subgroup, can be used to multiply matrices using the same approach as used in the Cayley graph-based PGA algorithm only when the chosen subgroup is normal.