An Efficient Algorithm to Compute Row and Column Counts for Sparse Cholesky Factorization

Abstract

Let an undirected graph $G$ be given, along with a specified depth-first spanning tree $T$. Almost-linear-time algorithms are given to solve the following two problems. First, for every vertex $v$, compute the number of descendants $w$ of $v$ for which some descendant of $w$ is adjacent (in $G$) to $v$. Second, for every vertex $v$, compute the number of ancestors of $v$ that are adjacent (in $G$) to at least one descendant of $v$. These problems arise in Cholesky and $QR$ factorizations of sparse matrices. The authors' algorithms can be used to determine the number of nonzero entries in each row and column of the triangular factor of a matrix from the zero/nonzero structure of the matrix. Such a prediction makes storage allocation for sparse matrix factorizations more efficient. The authors' algorithms run in time linear in the size of the input times a slowly growing inverse of Ackermann's function. The best previously known algorithms for these problems ran in time linear in the sum of the nonzero counts, which is usually much larger. Experimental results are given demonstrating the practical efficiency of the new algorithms.