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Cuckoo Hashing is a hashing scheme invented by pervious study of Pagh and Rodler. It uses d ≥ 2 distinct hash functions to insert n items into the hash table of size m = (1 + ε)n. In their original paper they treated d = 2 and m = (2 + ε)n. It has been an open question for some time as to the expected time for Random Walk Insertion to add items when d > 2. We show that if the number of hash functions d ≥ dε = O(1) then the expected insertion time is O(1) per item. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

The paradigm of many choices has influenced significantly the design of efficient data structures and, most notably, hash tables. Cuckoo hashing is a technique that extends this concept. There, we are given a table with n locations, and we assume that each location can hold one item. Each item to be inserted chooses randomly k ≥ 2 locations and has to be placed in any one of them. How much load can cuckoo hashing handle before collisions prevent the successful assignment of the available items to the chosen locations? Practical evaluations and theoretical analysis of this method have shown that one can allocate a number of elements that is a large proportion of the size of the table, being very close to 1 even for small values of k such as 4 or 5. In this paper we show that there is a critical value for this proportion: with high probability, when the amount of available items is below this value, then these can be allocated successfully, but when it exceeds this value, the allocation becomes impossible. We give explicitly for each k ≥ 3 this critical value. This answers an open question posed by Mitzenmacher ( ESA '09) and underpins theoretically the experimental results. Our proofs are based on the translation of the question into a hypergraph setting, and the study of the related typical properties of random k -uniform hypergraphs.© 2012 Wiley Periodicals, Inc. Random Struct., 2012 [ABSTRACT FROM AUTHOR]