Optimal Dag Partitioning for Partially Inverting Triangular Systems

An approach for solving sparse triangular systems of equations on highly parallel computers employs a partitioned representation of the inverse of the triangular matrix so that the solution can be obtained by a series of matrix-vector multiplications. This approach requires a number of global communication steps that is proportional to the number of factors in the partitioning. The problem of nding the minimal number of factors subject to the requirement that these factors do not need more storage space than the original triangular factor has been studied by several authors. We formulate a new related graph problem and give an algorithm to solve this problem. We prove that the partitioning resulting from this algorithm requires less factors than existing partitioning algorithms.