On the row merge tree for sparse LU factorization with partial pivoting

We consider the problem of structure prediction for sparse LU factorization with partial pivoting. In this context, it is well known that the column elimination tree plays an important role for matrices satisfying an irreducibility condition, called the strong Hall property. Our primary goal in this paper is to address the structure prediction problem for matrices satisfying a weaker assumption, which is the Hall property. For this we consider the row merge matrix, an upper bound that contains the nonzeros in L and U for all possible row permutations that can be performed during the numerical factorization with partial pivoting. We discuss the row merge tree, a structure that represents information obtained from the row merge matrix; that is, information on the dependencies among the columns in Gaussian elimination with partial pivoting and on structural upper bounds of the factors L and U. We present new theoretical results that show that the nonzero structure of the row merge matrix can be described in terms of branches and subtrees of the row merge tree. These results lead to an efficient algorithm for the computation of the row merge tree, that uses as input the structure of A, and has a time complexity almost linear in the number of nonzeros in A. We also investigate experimentally the usage of the row merge tree for structure prediction purposes on a set of matrices that satisfy only the Hall property. We analyze in particular the size of upper bounds of the structure of L and U, the reordering of the matrix based on a postorder traversal and its impact on the factorization runtime. We show experimentally that for some matrices, the row merge tree is a preferred alternative to the column elimination tree.