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 $d\ge 3$ random 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 $m$ exists with high probability. We show that for any $d\ge 3$ 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 $m$.

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