Load Thresholds for Cuckoo Hashing with Overlapping Blocks

We consider a natural variation of cuckoo hashing proposed by Lehman and Panigrahy (2009). Each of cn objects is assigned k = 2 intervals of size ℓ in a linear hash table of size n and both starting points are chosen independently and uniformly at random. Each object must be placed into a table cell within its intervals, but each cell can only hold one object. Experiments suggested that this scheme outperforms the variant with blocks in which intervals are aligned at multiples of ℓ. In particular, the load threshold is higher, i.e., the load c that can be achieved with high probability. For instance, Lehman and Panigrahy (2009) empirically observed the threshold for ℓ = 2 to be around 96.5% as compared to roughly 89.7% using blocks. They pinned down the asymptotics of the thresholds for large ℓ, but the precise values resisted rigorous analysis.

We establish a method to determine these load thresholds for all ℓ ≥ 2, and, in fact, for general k ≥ 2. For instance, for k = ℓ = 2, we get ≈ 96.4995%. We employ a theorem due to Leconte, Lelarge, and Massoulié (2013), which adapts methods from statistical physics to the world of hypergraph orientability. In effect, the orientability thresholds for our graph families are determined by belief propagation equations for certain graph limits. As a side note, we provide experimental evidence suggesting that placements can be constructed in linear time using an adapted version of an algorithm by Khosla (2013).