Parallel Distance-k Coloring Algorithms for Numerical Optimization

Abstract

Matrix partitioning problems that arise in the efficient estimation of sparse Jacobians and Hessians can be modeled using variants of graph coloring problems. In a previous work [7], we argue that distance-2 and distance-3 2 graph coloring are robust and flexible 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 and memory space. Hence, there is a need for both shared-and 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, and finally re-color conflicting vertices sequentially. Randomization is also used in the coloring phases to further reduce conflicts. Our PRAM-analysis shows that the algorithms should give 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.

Speeding up Parallel Graph Coloring

“…Figure 1 outlines this scheme as presented in [8]. This algorithm was later extended to various coloring problems in numerical optimization including distance-2 (d2) coloring, a case in which a vertex v is required to receive a color distinct from the colors of vertices within distance 2 edges from v [7]. This shared memory approach is the only algorithm that we know of that has been shown through parallel implementations to actually give lower running time as more processors are applied.…”

“…In [7] it was shown that for dense graphs the number of conflicts could become sufficiently high that the final sequential conflict resolution step starts to dominate the overall runtime. Also for dense graphs the effect of graph partitioning is less pronounced.…”

“…Gebremedhin et al [7] suggested that one use a geometrically distributed random variable to determine which of the available legal colors to use. This was implemented by processing the available legal colors in increasing order, each time choosing a color with probability q.…”

“…In order to compare the TH algorithm with the GMP-algorithm [7], we ran the GMPalgorithm on the d2-random graph using various values for the parameter q. Due to space limitations we do not show these numbers here but we observed that the optimal value of q in terms of speed seems to decrease as the number of processors increases.…”

“…Several papers have appeared in the literature over the last couple of years dealing with this issue [5,7,8,11]. We consider the algorithm in [8] and show how its performance can be enhanced both by paying attention to its sequential memory access pattern, and by employing graph partitioning to assign vertices to processors.…”