Maximum Matchings in Random Bipartite Graphs and the Space Utilization of Cuckoo Hashtables

TL;DR

This paper analyzes the size of maximum matchings in random bipartite graphs and determines conditions for optimal space utilization in Cuckoo Hashing, providing exact thresholds for certain parameters.

Contribution

It establishes precise thresholds for maximum matchings in random bipartite graphs and their implications for Cuckoo Hashing efficiency.

Findings

Maximum matching size equals n with high probability when d ≥ 4.

Exact threshold identified for Phase 1 of Karp-Sipser algorithm to find maximum matchings.

Provides insights into space utilization in Cuckoo Hash tables.

Abstract

We study the the following question in Random Graphs. We are given two disjoint sets $L,R$ with $|L|=n=\alpha m$ and $|R|=m$. We construct a random graph $G$ by allowing each $x\in L$ to choose $d$ random neighbours in $R$. The question discussed is as to the size $\mu(G)$ of the largest matching in $G$. When considered in the context of Cuckoo Hashing, one key question is as to when is $\mu(G)=n$ whp? We answer this question exactly when $d$ is at least four. We also establish a precise threshold for when Phase 1 of the Karp-Sipser Greedy matching algorithm suffices to compute a maximum matching whp.