Counting and finding triangles in graphs is often used in real-world analytics for characterizing the cohesiveness and identifying communities in graphs. In this paper, we present novel sequential and parallel triangle counting algorithms based on identifying horizontal-edges in a breadth-first search (BFS) traversal of the graph. The BFS allows our algorithm to drastically reduce the number of edges examined for set intersections. Our new approach is the first communication-optimal parallel algorithm that asymptotically reduces the 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 algorithms reduces the communication by 21x on a scale 36 and by 176x on a scale 42. Because communication is known to be the dominant cost of parallel triangle counting, our new parallel algorithm, to our knowledge, is now the fastest method for counting triangles in large graphs.
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