To Tile or not to Tile, That is the Question

A wide variety of graph algorithms expressed as linear algebra operations, i.e., triangle counting, k-truss analysis, breath first search, betweenness centrality, depend on the masked sparse matrix time sparse matrix multiplication (masked-SpGEMM) kernel. SuiteSparse:GraphBLAS, the de-facto sparse linear algebra library for graph analytics, offers support for this specific computation. Under a simple API, the library offers multiple masked-SpGEMM implementations. While attempting to provide a flexible solution that adapts to the input graphs/data, the system uses heuristics to choose between implementations. It hides the mechanisms behind layers of complex code, making it hard for users to reason about performance. In this work, we provide an in-depth analysis of the design choices that affect the performance of the masked-SpGEMM, using triangle counting as the benchmark. We look at 1) techniques for load balancing the sparse computation across multiple threads, 2) the iteration space for traversing the matrix multiplication and masking operation, and 3) the implementation of the accumulator used to store the intermediate results. We discuss the trade-offs and show a detailed performance analysis of the implementations on shared memory systems for a wide variety of input graphs, comparing SuiteSparse:GraphBLAS and a highly optimized masked-SpGEMM implementation, and discuss future research directions given our observations.