Parallel Peeling Algorithms

The analysis of several algorithms and data structures can be framed as a peeling process on a random hypergraph: vertices with degree less than k are removed until there are no vertices of degree less than k left. The remaining hypergraph is known as the k-core. In this article, we analyze parallel peeling processes, in which in each round, all vertices of degree less than k are removed. It is known that, below a specific edge-density threshold, the k-core is empty with high probability. We show that, with high probability, below this threshold, only 1/log((k-1)(r-1)) log log n + O(1) rounds of peeling are needed to obtain the empty k-core for r-uniform hypergraphs. This bound is tight up to an additive constant. Interestingly, we show that, above this threshold, Ω(log n) rounds of peeling are required to find the nonempty k-core. Since most algorithms and data structures aim to peel to an empty k-core, this asymmetry appears fortunate. We verify the theoretical results both with simulation and with a parallel implementation using graphics processing units (GPUs). Our implementation provides insights into how to structure parallel peeling algorithms for efficiency in practice.