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, (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.
Regarding the parallel computations, Jiang et al. REF introduced parallel algorithms to find the number of iterations needed to find the empty k-core in random hypergraphs.
Parallel Peeling Algorithms
TOPC
Mathematics Computer Science