A General Graph Model For Representing Exact Communication Volume in Parallel Sparse Matrix–Vector Multiplication Aleksandar Trifunovi´ c and William Knottenbelt {at701,wjk}@doc.ic.ac.uk Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, UK Abstract. In this paper, we present a new graph model of sparse ma- trix decomposition for parallel sparse matrix–vector multiplication. Our model differs from previous graph-based approaches in two main re- spects. Firstly, our model is based on edge colouring rather than ver- tex partitioning. Secondly, our model is able to correctly quantify and minimise the total communication volume of the parallel sparse matrix– vector multiplication while maintaining the computational load balance across the processors. We show that our graph edge colouring model is equivalent to the fine-grained hypergraph partitioning-based sparse ma- trix decomposition model. We conjecture that the existence of such a graph model should lead to faster serial and parallel sparse matrix de- composition heuristics and associated tools. 1 Introduction Parallel sparse matrix–vector multiplication is the core operation in iterative solvers for large-scale linear systems and eigensystems. Major application areas include Markov modelling, linear programming and PageRank computation. Efficient parallel sparse matrix–vector multiplication requires intelligent a priori partitioning of the sparse matrix non-zeros across the processors to en- sure that interprocessor communication is minimised subject to a load balancing constraint. The problem of sparse matrix decomposition can be reformulated in terms of a graph or hypergraph partitioning problem. These partitioning prob- lems are NP-hard [10], so (sub-optimal) heuristic algorithms are used in practice. The resulting graph or hypergraph partition is then used to direct the distribu- tion of matrix elements across processors. The limits of the existing graph partitioning approaches are outlined in [11, 8,4]. For example, in the case of one-dimensional row-wise or column-wise par- titioning of a sparse matrix for parallel sparse matrix–vector multiplication, ex- isting graph models cannot optimise the exact communication volume; instead, they operate indirectly by optimising an upper bound on the communication volume. On the other hand, hypergraph models that correctly represent the total com- munication volume have been proposed and are thus preferred to graph models