A column approximate minimum degree ordering algorithm
ACM Transactions on Mathematical Software (TOMS)
Algorithm 849: A concise sparse Cholesky factorization package
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Experiences of sparse direct symmetric solvers
ACM Transactions on Mathematical Software (TOMS)
Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate
ACM Transactions on Mathematical Software (TOMS)
Dynamic Supernodes in Sparse Cholesky Update/Downdate and Triangular Solves
ACM Transactions on Mathematical Software (TOMS)
Experiments on union-find algorithms for the disjoint-set data structure
SEA'10 Proceedings of the 9th international conference on Experimental Algorithms
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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.