A global, distributed ordering library

Computing a fill-reducing ordering of a sparse matrix is a critical step for sparse direct solvers. In the early days of the field, “bottom-up” or “greedy” methods were popular. These include the Minimum Degree algorithm [7] and its variants, and profile-reducing methods such as the Cuthill-McKee algorithm [2]. More recently, “top-down” or “divide-and-conquer” methods such as Nested Dissection [3] have been favored, in particular for problems arising from the discretization of PDEs on 3D domains. The idea is, given the adjacency graph G = (V, E) of the matrix, where V are vertices and E are edges, to partition the vertices V = BtStW where S separates B from W; S is called a vertex separator. Then, recursively, B and W are partitioned similarly. Most practical implementations, especially in distributed memory environments, rely on the multilevel framework, in which the graph is coarsened in order to find separators. ParMETIS [6] and PT-Scotch [1] are widely used graph partitioning packages that rely on the multilevel framework. The approach we propose here is a top-down algorithm, but it is not multilevel. Instead, we perform all the operations on the whole, uncoarsened, graph. We refer to this as a “global” method. In the next sections we describe briefly our algorithm and our distributedmemory implementation.