Counting and finding triangles in graphs is often used in real-world analytics to characterize cohesiveness and identify communities in graphs. In this paper, we propose the novel concept of a cover-edge set that can be used to find triangles more efficiently. We use a breadth-first search (BFS) to quickly generate a compact cover-edge set. Novel sequential and parallel triangle counting algorithms are presented that employ cover-edge sets. The sequential algorithm avoids unnecessary triangle-checking operations, and the parallel algorithm is communication-efficient. The parallel algorithm can asymptotically reduce communication on massive graphs such as from real social networks and synthetic graphs from the Graph500 Benchmark. In our estimate from massive-scale Graph500 graphs, our new parallel algorithm can reduce the communication on a scale 36 graph by 1156x and on a scale 42 graph by 2368x.

翻译：在现实世界分析中，对图中的三角形进行计数和查找通常用于表征图的凝聚性和识别社群。本文提出覆盖边集这一新颖概念，可更高效地发现三角形。我们采用广度优先搜索（BFS）快速生成紧凑的覆盖边集。随后提出了基于覆盖边集的新型串行与并行三角计数算法。串行算法避免了不必要的三角形检验操作，而并行算法具有通信高效的特性。该并行算法在大规模图（如真实社交网络图和Graph500基准测试中的合成图）上可渐进式降低通信开销。根据我们对大规模Graph500图的估算，该并行算法在规模为36的图上可将通信量降低1156倍，在规模为42的图上降低2368倍。