## 30 September 2022

# Parallel Peeling Algorithms

## Ryuto Jonathan Kitagawa

Abstract:
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 paper, we analyze parallel peeling processes, where 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 \((\log \log n)/\log(k-1)(r-1) + O(1)\) rounds of peeling are needed to obtain the empty k-core for r-uniform hypergraphs. Interestingly, we show that above this threshold, Omega(log n) rounds of peeling are required to find the non-empty 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.

Authors: Jiayang Jiang, Michael Mitzenmacher, Justin Thaler