The performance of a dynamic dictionary is measured mainly by its update time, lookup time, and space consumption. In terms of update time and lookup time there are known constructions that guarantee constant-time operations in the worst case with high probability, and in terms of space consumption there are known constructions that use essentially optimal space. In this paper we settle two fundamental open problems: • We construct the first dynamic dictionary that enjoys the best of both worlds: we present a two-level variant of cuckoo hashing that stores n elements using (1+ϵ)n memory words, and guarantees constant-time operations in the worst case with high probability. Specifically, for any ϵ = Ω((log log n / log n) 1/2) and for any sequence of polynomially many operations, with high probability over the randomness of the initialization phase, all operations are performed in constant time which is independent of ϵ. The construction is based on augmenting cuckoo hashing with a “backyard ” that handles a large fraction of the elements, together with a de-amortized perfect hashing scheme for eliminating the dependency on ϵ.

Let G(n, m) be an undirected random graph with n vertices and m multiedges that may include loops, where each edge is realized by choosing its two vertices independently and uniformly at random with replacement from the set of all n vertices. The random graph G(n, m) is said to be k-orientable, where k ≥ 2 is an integer, if there exists an orientation of the edges such that the maximum out-degree is at most k. Let ck = sup {c: G(n, cn) is k-orientable w.h.p.}. We prove that for k large enough, 1 − 2 k exp −k + 1 + e −k/4) < ck/k < 1 − exp

Abstract. The purpose of this brief note is to describe recent work in the area of cuckoo hashing, including a clear description of several open problems, with the hope of spurring further research. 1