Revisiting Hypergraph Models for Sparse Matrix Partitioning

Abstract

Abstract.We provide an exposition of hypergraph models for parallelizing sparse matrix-vector multiplies. Our aim is to emphasize the expressive power of hypergraph models. First, we set forth an elementary hypergraph model for the parallel matrix-vector multiply based on one-dimensional (1D) matrix partitioning. In the elementary model, the vertices represent the data of a matrix-vector multiply, and the nets encode dependencies among the data. We then apply a recently proposed hypergraph transformation operation to devise models for 1D sparse matrix partitioning. The resulting 1D partitioning models are equivalent to the previously proposed computational hypergraph models and are not meant to be replacements for them. Nevertheless, the new models give us insights into the previous ones and help us explain a subtle requirement, known as the consistency condition, of hypergraph partitioning models. Later, we demonstrate the flexibility of the elementary model on a few 1D partitioning problems that are hard to solve using the previously proposed models. We also discuss extensions of the proposed elementary model to two-dimensional matrix partitioning.

A Matrix Partitioning Interface to PaToH in MATLAB

“…Finding a partition on the vectors x and y is referred to as the vector partitioning operation, and it can be performed in three different ways: by decoding the partition given on A [2]; in a post-processing step using the partition on the matrix [7,8,13]; or explicitly partitioning the vectors during partitioning the matrix [9]. In any of these cases, the vector partitioning for matrix-vector operations is called symmetric if x and y have the same partition, and non-symmetric otherwise.…”

“…During the last decade, several successful hypergraph-based models and methods were proposed for sparse matrix partitioning [1][2][3][4][5][6][7][8][9][10]. These models and methods have gained wide acceptance in the literature for efficient parallelization of sparse matrix-vector multiply operations.…”

Multi-level direct K-way hypergraph partitioning with multiple constraints and fixed vertices

“…In the literature, combinatorial models based on hypergraph partitioning are proposed for various complex and irregular problems arising in parallel scientific computing [4,10,17,26,50,53], VLSI design [2,42], software engineering [6], and database design [22,23,41,43,46]. These models formulate an original problem as a hypergraph partitioning problem, trying to optimize a certain objective function (e.g., minimizing the total volume of communication in parallel volume rendering, optimizing the placement of circuitry on a dice area, minimizing the access to disk pages in processing GIS queries) while maintaining a constraint (e.g., balancing the computational load in a parallel system, using disk page capacities as an upper bound in data allocation) imposed by the problem.…”

Replicated partitioning for undirected hypergraphs

“…Models and methods based on hypergraph partitioning (HP) have been successfully used for different objectives in a wide range of areas such as parallel scientific computing [4,11,15,44], very large scale integration (VLSI) circuit layout design [1,32], parallel information retrieval (IR) [8], parallel volume rendering [9], and database systems [12,13,40].…”

“…We should note here that almost all of the state-of-the-art HP tools [2,14,26,43,45] are designed to partition undirected hypergraphs. Hence, some special techniques such as consistency condition [11] and the elementary hypergraph model [44] are utilized to model some types of directed relations correctly via undirectional HP models.…”