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Parallel Peeling Algorithms

Jiayang Jiang, Michael Mitzenmacher, Justin Thaler
Harvard University, School of Engineering and Applied Sciences. Supported in part by NSF grants CCF-0915922 and IIS-0964473
arXiv:1302.7014 [cs.DS], (27 Feb 2013)

@article{2013arXiv1302.7014J,

   author={Jiang}, J. and {Mitzenmacher}, M. and {Thaler}, J.},

   title={"{Parallel Peeling Algorithms}"},

   journal={ArXiv e-prints},

   archivePrefix={"arXiv"},

   eprint={1302.7014},

   primaryClass={"cs.DS"},

   keywords={Computer Science – Data Structures and Algorithms},

   year={2013},

   month={feb},

   adsurl={http://adsabs.harvard.edu/abs/2013arXiv1302.7014J},

   adsnote={Provided by the SAO/NASA Astrophysics Data System}

}

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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 graphical processing units (GPUs). Our implementation provides insights into how to structure parallel peeling algorithms for efficiency in practice.
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