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

Let an undirected graph G be given, along with a specified depth-first spanning tree T . We give almost-linear-time algorithms 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. Our 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. Our 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...