A hypergraph-partitioning approach for coarse-grain decomposition

Abstract

We propose a new two-phase method for the coarse-grain decomposition of irregular computational domains. This work focuses on the 2D partitioning of sparse matrices for parallel matrix-vector multiplication. However, the proposed model can also be used to decompose computational domains of other parallel reduction problems. This work also introduces the use of multi-constraint hypergraph partitioning, for solving the decomposition problem. The proposed method explicitly models the minimization of communication volume while enforcing the upper bound of p + q ; 2 on the maximum number of messages handled by a single processor, for a parallel system with P = p q processors. Experimental results on a wide range of realistic sparse matrices confirm the validity of the proposed methods, by achieving up to 25 percent better partitions than the standard graph model, in terms of total communication volume, and 59 percent better partitions in terms of number of messages, on the overall average.

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“…1 displays the pseudocode of BSCM applied to the system in Eq. (6). Since the system is composed of two copies of the same matrix with different signs, only the positive one is held during the computations.…”

“…A number of different techniques for checkerboard partitioning of sparse matrices are given in [6,15,25,24]. Among these, only the hypergraph partitioning model of Çatalyürek and Aykanat [6] exploits sparsity to reduce communication cost.…”

“…Among these, only the hypergraph partitioning model of Çatalyürek and Aykanat [6] exploits sparsity to reduce communication cost. Therefore, we use hypergraph partitioning model to obtain an R × C checkerboard partition on the matrix H.…”

“…These algorithms can be implemented using hypercube algorithms to reduce the maximum number of messages handled by a processor to log R + log C ≈ log K . The 2D hypergraph model [6] can be extended to handle the minimization of communication volume overhead in this allto-all communication scheme [2].…”

“…Both one-dimensional (1D), e.g., rowwise, and two-dimensional (2D), e.g., checkerboard, sparse matrix partitioning techniques are investigated (Sections 4 and 5, respectively). The recently proposed hypergraph partitioning models [3,4,6,34] are used for load balancing and communication overhead minimization for both partitioning frameworks. Parallel algorithms for BSCM and PSCM are implemented for a message-passing multicomputer based on the above mentioned partitioning frameworks.…”