Parallel Clique Counting and Peeling Algorithms

We present a new parallel algorithm for k-clique counting/listing that has polylogarithmic span (parallel time) and is work-efficient (matches the work of the best sequential algorithm) for sparse graphs. Our algorithm is based on computing low out-degree orientations, which we present new linear-work and polylogarithmic-span algorithms for computing in parallel. We also present new parallel algorithms for producing unbiased estimations of clique counts using graph sparsification. Finally, we design two new parallel work-efficient algorithms for approximating the k-clique densest subgraph, the first of which is a 1/k-approximation and the second of which is a 1/(k(1+ε))-approximation and has polylogarithmic span. Our first algorithm does not have polylogarithmic span, but we prove that it solves a P-complete problem. In addition to the theoretical results, we also implement the algorithms and propose various optimizations to improve their practical performance. On a 30-core machine with two-way hyper-threading, our algorithms achieve 13.23--38.99x and 1.19--13.76x self-relative parallel speedup for k-clique counting and k-clique densest subgraph, respectively. Compared to the state-of-the-art parallel k-clique counting algorithms, we achieve up to 9.88x speedup, and compared to existing implementations of k-clique densest subgraph, we achieve up to 11.83x speedup. We are able to compute the 4-clique counts on the largest publicly-available graph with over two hundred billion edges for the first time.