Hierarchical Core Decomposition in Parallel: From Construction to Subgraph Search

Theoretically and practically efficient parallel nucleus decomposition

“…For (1, 2) nucleus (𝑘-core) decomposition, we compare to the parallel phcd [11] and the sequential nh [49]. Our fastest implementation for 𝑘-core is anh-te, and we see that anh-te is up to 2.57x slower than phcd overall, but 1.87x faster than phcd on dblp.…”

“…We also note that their space-usage can in theory be as large as 𝑂 (𝑛𝛼 𝑠 −2 ), i.e., proportional to the number of 𝑠-cliques in the graph. Chu et al [11] present a parallel algorithm for generating the 𝑘-core decomposition hierarchy, which is a special case of nucleus decomposition for 𝑟 = 1 and 𝑠 = 2, but their algorithm does not trivially generalize to higher 𝑟 and 𝑠. Their serial and parallel algorithms both use union-find and run in 𝑂 (𝑚𝛼 (𝑛)) work (where 𝛼 (𝑛) is the inverse Ackermann function), which is not work-efficient, and their parallel algorithm has depth that depends on the peeling-complexity of the input.…”

“…nh, like anh-bl and anh-el, constructs the hierarchy while computing the (𝑟, 𝑠)-clique-core numbers of the graph. For the special case of 𝑘-core, we also compare to phcd, the state-of-the-art parallel 𝑘-core hierarchy implementation by Chu et al [11].…”

“…We first compare approx-arb-nucleus to arb-nucleus [55] and see a speedup of up to 16.16x for 𝛿 = 0.1, up to 8.35x for 𝛿 = 0.5, and up to 10.88x for 𝛿 = 1. Figure 9: Multiplicative slowdowns, comparing anh-te, anh-el, the parallel phcd [11], and the sequential nd [49], for (1, 2), (2, 3), and (3, 4) nucleus decomposition. We give the multiplicative slowdown over the fastest implementation for each graph and each (𝑟, 𝑠 ), where the fastest running time is labeled in parentheses below each graph.…”