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

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