CLAP: Locality Aware and Parallel Triangle Counting with Content Addressable Memory

Abstract

Triangle counting (TC) is one of the most fundamental graph analysis tools with a wide range of applications. Modern triangle counting algorithms traverse the graph and perform set intersections of neighbor sets to find triangles. However, existing triangle counting approaches suffer from the heavy off-chip memory access and set intersection overhead. Thus, we propose CLAP, the first content addressable memory (CAM) based triangle counting architecture with the software and hardware co-optimizations. To reduce off-chip memory access and the number of set intersections, we propose the first force-based node index reorder method. It simultaneously optimizes both data locality and the computation amount. Compared with random node indices, the reorder method reduces the off-chip memory access and the set intersections by 61% and 64%, respectively, while providing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{2.19}\times$</tex> end-to-end speedup. To improve the set intersection parallelism, we propose the first CAM-based triangle counting architecture under chip area constraints. We enable the high parallel set intersection by translating it into content search on CAM with full parallelism. Thus, the time complexity of the set intersection reduces from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(m+n)$</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n\log m)$</tex> to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n)$</tex> . Extensive experiments on real-world graphs show that CLAP achieves <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{39}\times, \mathbf{27}\times$</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{78}\times$</tex> speedup over state-of-the-art CPU, GPU, and processing-in-memory baselines, respectively. The software code is available at: https://github.com/thu-nics/CLAP-triangle-counting