Parallel distance-k coloring algorithms for numerical optimization

Abstract

Matrix partitioning problems that arise in the efficient estimation of sparse Jacobians andHessians can be modeledusing variants of graph coloring problems. In a previous work [6], we argue that distance-2 and distance-(formula presented) graph coloring are robust andflexible formulations of the respective matrix estimation problems. The problem size in large-scale optimization contexts makes the matrix estimation phase an expensive part of the entire computation both in terms of execution time andmemory space. Hence, there is a needfor both sharedand distributed-memory parallel algorithms for the stated graph coloring problems. In the current work, we present the first practical shared address space parallel algorithms for these problems. The main idea in our algorithms is to randomly partition the vertex set equally among the available processors, let each processor speculatively color its vertices using information about already colored vertices, detect eventual conflicts in parallel, andfinally re-color conflicting vertices sequentially. Randomization is also usedin the coloring phases to further reduce conflicts. Our PRAM-analysis shows that the algorithms shouldgiv e almost linear speedup for sparse graphs that are large relative to the number of processors. Experimental results from our OpenMP implementations on a Cray Origin2000 using various large graphs show that the algorithms indeed yield reasonable speedup for modest numbers of processors.