Parallel Peeling Algorithms

TL;DR

This paper analyzes the efficiency of parallel peeling algorithms on hypergraphs, revealing thresholds for the number of rounds needed to find the k-core, with practical GPU implementation insights.

Contribution

It provides a theoretical analysis of parallel peeling rounds relative to hypergraph thresholds and demonstrates GPU-based implementation for practical efficiency.

Findings

Below the threshold, only (log log n)/log(k-1)(r-1) + O(1) rounds needed

Above the threshold, Omega(log n) rounds are required

GPU implementation confirms theoretical predictions

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.