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Orientability of Random Hypergraphs and the Power of Multiple Choices

  • Nikolaos Fountoulakis
  • Konstantinos Panagiotou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

A hypergraph H = (V, E) is called s-orientable, if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than s edges. Let H n,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the 1-orientability of H n,m,k for all k ≥ 3, i.e., we determine a critical quantity \(c_k^*\) such that with probability 1 − o(1) the graph H n,cn,k has a 1-orientation if \(c < c_k^*\), but fails doing so if \(c > c_k^*\).

We present two applications of this result that involve the paradigm of multiple choices. First, we show how it implies sharp load thresholds for cuckoo hash tables, where each element chooses k out of n locations. Particularly, for each k ≥ 3 we prove that with probability 1 − o(1) the maximum number of elements that can be hashed is \((1 - o(1))c_k^* n\), and more items prevent the successful allocation. Second, we study random graph processes, where in each step we have the choice among any edge connecting k random vertices. Here we show the existence of a phase transition for avoiding a giant connected component, and quantify precisely the dependence on k.

Keywords

Random Graph Multiple Choice Degree Sequence Giant Component Poisson Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nikolaos Fountoulakis
    • 1
  • Konstantinos Panagiotou
    • 1
  1. 1.Max-Planck-Institute for InformaticsSaarbrückenGermany

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