We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have n keys to be hashed into m buckets each capable of holding a single key. Each key has k ≥ 3 (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds ck such that, as n goes to infinity, if n/m ≤ c for some c < ck then a hash table can be constructed successfully with high probability, and if n/m ≥ c for some c> ck a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of k> 2 by showing that they are in fact the same as the previously known thresholds for the random k-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and provide evidence in the form of an algorithm for a conjecture extending this result to cuckoo hash tables that store multiple keys in a bucket.

Cuckoo hashing with a stash is a robust high-performance hashing scheme that can be used in many real-life applications. It complements cuckoo hashing by adding a small stash storing the elements that cannot fit into the main hash table due to collisions. However, the exact required size of the stash and the tradeoff between its size and the memory over-provisioning of the hash table are still unknown. We settle this question by investigating the equivalent maximum matching size of a random bipartite graph, with a constant left-side vertex degree d = 2. Specifically, we provide an exact expression for the expected maximum matching size and show that its actual size is close to its mean, with high probability. This result relies on decomposing the bipartite graph into connected components, and then separately evaluating the distribution of the matching size in each of these components. In particular, we provide an exact expression for any finite bipartite graph size and also deduce asymptotic results as the number of vertices goes to infinity. We also extend our analysis to cases where only part of the left-side vertices have a degree of 2; as well as to the case where the set of right-size vertices is partitioned into two subsets, and each