Tight Thresholds for Cuckoo Hashing via XORSAT (Extended Abstract)

We settle the question of tight thresholds for o ine 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 in nity, 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 o ine 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 nd 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 di ering number of choices, and make a conjecture (based on experimental observations) that extends our result to cuckoo hash tables storing multiple keys in a bucket.