Abstract
We study the the following question in Random Graphs. We are given two disjoint sets L,R with |L| = n 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. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012
Highlights
For a graph G we let μ(G) denote the size of the maximum matching in G
In essence this paper provides an analysis of μ(G) in the following model of a random bipartite graph
Under the assumption that the hash functions are completely random we see that G has the same distribution as the random graph defined in the previous paragraph
Summary
For a graph G we let μ(G) denote the size of the maximum matching in G. In essence this paper provides an analysis of μ(G) in the following model of a random bipartite graph. Under the assumption that the hash functions are completely random we see that G has the same distribution as the random graph defined in the previous paragraph. We assume each location can hold only one item. When an item x is inserted into the table, it can be placed immediately if one of its d locations is currently empty. One of the items in its d locations must be displaced and moved to another of its d choices to make room for x. Having inserted k items, we have constructed a matching M of size k in G. All n items will be insertable in this way iff G contains a matching of size n. We will revert to the abstract question posed in first paragraph of the paper
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