Two space-saving algorithms for computing the permuted transpose of a sparse matrix

Abstract

Given an m× n sparse matrix A having n a nonzeros and a permutation matrix P, we consider the problem of finding ( PA) T in a space-saving manner. Two algorithms, TRANSPERM1 and TRANSPERM2, are presented. Both algorithms make use of the in-place inversion of permutation vectors. To save even more space TRANSPERM1 forms a piecewise linear model of the nonzero distribution. The computational complexity of TRANSPERM2 is O( n a , n, m). For a particularly treachrous nonzero distribution we derive an upper bound for the complexity of TRANSPERM1 of O( n a , τn a n/ n l , m, n, n l ), where n l is the number of intervals in the piecewise model and τ≤ 1 2 . These two algorithms were compared with Gustavson's efficient HALFPERM algorithm. 3 We expected our algorithms to be slower than HALFPERM since they were written with the intention of saving space. In tests on 21 matrices, HALFPERM was on average 2·5 times faster than TRANSPERM1 and 1·6 times faster than TRANSPERM2. However, for these 21 matrices, TRANSPERM1's storage requirements were on average only 51% of HALFPERM's, and TRANSPERM2's requirements were only 67% of HALFPERM's.