The input/output complexity of sorting and related problems
Communications of the ACM
Block algorithms for sparse matrix computations on high performance workstations
ICS '96 Proceedings of the 10th international conference on Supercomputing
I/O complexity: The red-blue pebble game
STOC '81 Proceedings of the thirteenth annual ACM symposium on Theory of computing
Optimal sparse matrix dense vector multiplication in the I/O-model
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Challenges and Advances in Parallel Sparse Matrix-Matrix Multiplication
ICPP '08 Proceedings of the 2008 37th International Conference on Parallel Processing
Evaluating non-square sparse bilinear forms on multiple vector pairs in the I/O-model
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Space-round tradeoffs for MapReduce computations
Proceedings of the 26th ACM international conference on Supercomputing
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We consider the multiplication of a sparse N ×N matrix A with a dense N ×N matrix B in the I/O model. We determine the worst-case non-uniform complexity of this task up to a constant factor for all meaningful choices of the parameters N (dimension of the matrices), k (average number of non-zero entries per column or row in A, i.e., there are in total kN non-zero entries), M (main memory size), and B (block size), as long as M≥B2 (tall cache assumption). For large and small k, the structure of the algorithm does not need to depend on the structure of the sparse matrix A, whereas for intermediate densities it is possible and necessary to find submatrices that fit in memory and are slightly denser than on average. The focus of this work is asymptotic worst-case complexity, i.e., the existence of matrices that require a certain number of I/Os and the existence of algorithms (sometimes depending on the shape of the sparse matrix) that use only a constant factor more I/Os.