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 $d\ge 3$ 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 $d\ge 4$ 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 $m$.

翻译：随机游走d元布谷鸟哈希算法由Fotakis、Pagh、Sanders和Spirakis提出，旨在推广并改进Pagh与Rodler的标准布谷鸟哈希算法。该算法具有低空间开销、有保障的快速访问特性以及实际应用中较快的插入速度。本文从理论上给出了该算法的插入时间上界。具体而言，对于任意$d\ge 3$个哈希函数，令$c_d^*$表示将$cm$个对象分配到大小为$m$的哈希表中可能存在有效分配方案的负载因子的尖锐阈值。我们证明：对于任意$d\ge 4$个哈希函数及负载因子$c<c_d^*$，随机游走插入时间的期望值为$O(1)$，即该常数仅取决于$d$和$c$，而与$m$无关。