Bipartite Random Graphs and Cuckoo Hashing

Reinhard Kutzelnigg

doi:10.46298/dmtcs.3486

Abstract

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.

Highlights

Cuckoo Hashing is a hashing algorithm with the advantage of constant worst case search time, contrary to standard hashing algorithms as Open Addressing or Hashing with Chaining. (See Knuth (1973) for surveys on hashing.) The algorithm was introduced by Pagh and Rodler (2001) and a further analysis was done by Devroye and Morin (2003)
The main idea of Cuckoo Hashing is to split up the available amount of memory
We use a generalization of this result which provides us an asymptotic expansion of [xmym]f (x, y)kg(x, y) for suitable function f and g

Summary

Introduction

Cuckoo Hashing is a hashing algorithm with the advantage of constant worst case search time, contrary to standard hashing algorithms as Open Addressing or Hashing with Chaining. (See Knuth (1973) for surveys on hashing.) The algorithm was introduced by Pagh and Rodler (2001) and a further analysis was done by Devroye and Morin (2003). We allow at most one element to be stored at any position It may still happen, that both possible storage places of a given key x are already occupied. Until we find an empty position or conjecture that we have entered an endless loop In the the latter case, we have to choose new hash functions and rebuild the data structure. Recall that a complex component having k nodes and k + l edges (with l ≥ 1) corresponds to the attempt to put k + l keys in only k locations Because of their close relation, we can obtain results on the expected behaviour of Cuckoo Hashing by an average case analysis of properties of random bipartite graphs. We only consider the case with m(1 − ε) keys, where ε ∈ (0, 1)

Results

Generating Functions

Asymptotic Methods