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Tight Approximations of Degeneracy in Large Graphs

  • Martín Farach-Colton
  • Meng-Tsung TsaiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

Abstract

Given an n-node m-edge graph G, the degeneracy of graph G and the associated node ordering can be computed in linear time in the RAM model by a greedy algorithm that iteratively removes the node of min-degree [28]. In the semi-streaming model for large graphs, where memory is limited to \(\mathcal {O}(n \,\mathrm{polylog}\,n)\) and edges can only be accessed in sequential passes, the greedy algorithm requires too many passes, so another approach is needed.

In the semi-streaming model, there is a deterministic log-pass algorithm for generating an ordering whose degeneracy approximates the minimum possible to within a factor of \((2+\varepsilon )\) for any constant \(\varepsilon > 0\) [12]. In this paper, we propose a randomized algorithm that improves the approximation factor to \((1+\varepsilon )\) with high probability and needs only a single pass. Our algorithm can be generalized to the model that allows edge deletions, but then it requires more computation and space usage.

The generated node ordering not only yields a \((1+\varepsilon )\)-approximation for the degeneracy but gives constant-factor approximations for arboricity and thickness.

Keywords

Degeneracy Arboricity Thickness Semi-streaming algorithm Space lower bound 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Rutgers UniversityNew BrunswickUSA

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