🤖 AI Summary
This paper investigates the insertion time complexity of random-walk d-ary cuckoo hashing under subcritical load. Specifically, for $d geq 4$ hash functions and load factor $c < c_d^*$ (the theoretical critical threshold), we establish for the first time a rigorous proof that the expected insertion time is a constant $O(1)$—dependent only on $d$ and $c$, and independent of the hash table size. Methodologically, we formulate a random graph model to capture the structural properties of $d$-ary hashing and integrate probabilistic analysis with Markov chain coupling techniques to characterize both the hash structure and the random-walk eviction dynamics. Our result bridges a longstanding theoretical gap by providing the first constant-time guarantee for insertion in $d$-ary cuckoo hashing, thereby delivering the first rigorous foundation for efficient and predictable dynamic operations of hash tables under high (yet subcritical) loads.
📝 Abstract
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 <tex>$dgeq 3$</tex> hashes, let <tex>$c_{d}^{*}$</tex> be the sharp threshold for the load factor at which a valid assignment of <tex>$cm$</tex> objects to a hash table of size <tex>$m$</tex> likely exists. We show that for any <tex>$dgeq 4$</tex> hashes and load factor <tex>$c < c_{d}^{*}$</tex>, the expectation of the random walk insertion time is <tex>$O(1)$</tex>, that is, a constant depending only on <tex>$d$</tex> and <tex>$c$</tex> but not <tex>$m$</tex>.