Bipartite Random Graphs and Cuckoo Hashing

Reinhard Kutzelnigg (1)

1 TU Wien - Institute of Discrete Mathematics and Geometry [Vienne]

The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically $1-c(\varepsilon)/m+O(1/m^2)$ for some explicit $c(\varepsilon)$, where $m$ denotes the size of each of the two tables, $n=m(1- \varepsilon)$ is the number of keys and $\varepsilon \in (0,1)$. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.

Mots clés: hashing; random bipartite graphs; generating functions; double saddle point method

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Reinhard Kutzelnigg. Bipartite Random Graphs and Cuckoo Hashing. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.403-406, ⟨10.46298/dmtcs.3486⟩. ⟨hal-01184689⟩