Enumerating Subgraphs of Constant Sizes in External Memory

We present an indivisible I/O-efficient algorithm for subgraph enumeration , where the objective is to list all the subgraphs of a massive graph G := ( V, E ) that are isomorphic to a pattern graph Q having k = O (1) vertices. Our algorithm performs O ( | E | k/ 2 M k/ 2 − 1 B log M/B | E | B + | E | ρ M ρ − 1 B ) I/Os with high probability, where ρ is the fractional edge covering number of Q (it always holds ρ ≥ k/ 2, regardless of Q ), M is the number of words in (internal) memory, and B is the number of words in a disk block. Our solution is optimal in the class of indivisible algorithms for all pattern graphs with ρ > k/ 2. When ρ = k/ 2, our algorithm is still optimal as long as M/B ≥ ( | E | /B ) ϵ for any constant ϵ > 0. 2012 ACM Subject Classification Theory of computation → Graph algorithms analysis; Information systems → Join algorithms