O(1) Insertion for Random Walk d-ary Cuckoo Hashing up to the Load Threshold

The random walk d-ary cuckoo hashing algorithm was defined by Fotakis, Pagh, Sanders, and Spirakis to generalize and improve upon the standard cuckoo hashing algorithm of Pagh and Rodler. Random walk d-ary cuckoo hashing has low space overhead, guaranteed fast access, and fast in practice insertion time. In this paper, we give a theoretical insertion time bound for this algorithm. More precisely, for every $d\geq 3$ hashes, let $c_{d}^{*}$ be the sharp threshold for the load factor at which a valid assignment of $cm$ objects to a hash table of size $m$ likely exists. We show that for any $d\geq 4$ hashes and load factor $c < c_{d}^{*}$, the expectation of the random walk insertion time is $O(1)$, that is, a constant depending only on $d$ and $c$ but not $m$.