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$m$.

Abstract. We extend a recent breakthrough result relating expectation thresholds and actual thresholds to include some rainbow versions.