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).

我们考虑由 Lehman 和 Panigrahy (2009) 提出的布谷鸟哈希的自然变体。在大小为n的线性哈希表中，为cn个对象中的每一个分配了k = 2 个大小为 ℓ 的区间，并且两个起点都是独立且均匀地随机选择的。每个对象必须在其间隔内放入一个表格单元格中，但每个单元格只能容纳一个对象。实验表明，该方案优于其中间隔以 ℓ 的倍数对齐的块的变体。特别是负载阈值较高，即负载c这很有可能实现。例如，Lehman 和 Panigrahy (2009) 根据经验观察到 ℓ = 2 的阈值约为 96.5%，而使用块的阈值约为 89.7%。他们确定了大 ℓ 阈值的渐近线，但精确值无法进行严格分析。

我们建立了一种方法来确定所有 ℓ ≥ 2 的这些负载阈值，事实上，对于一般k ≥ 2。例如，对于k = ℓ = 2，我们得到 ≈ 96.4995%。我们采用 Leconte、Lelarge 和 Massoulié（2013 年）提出的定理，该定理将统计物理学中的方法应用到超图可定向性领域。实际上，我们图族的可定向性阈值是由某些图限制的置信传播方程决定的。作为旁注，我们提供的实验证据表明，可以使用 Khosla (2013) 算法的改编版本在线性时间内构建布局。