  Dietzfelbinger and Weidling [DW07] proposed a natural variation of cuckoo
hashing where each of $cn$ objects is assigned $k = 2$ intervals of size $\ell$
in a linear (or cyclic) hash table of size $n$ and both start 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 $\ell$. In particular, the load threshold
is higher, i.e. the load $c$ that can be achieved with high probability. For
instance, Lehman and Panigrahy [LP09] empirically observed the threshold for
$\ell = 2$ to be around $96.5\%$ as compared to roughly $89.7\%$ using blocks.
They managed to pin down the asymptotics of the thresholds for large $\ell$,
but the precise values resisted rigorous analysis.
  We establish a method to determine these load thresholds for all $\ell \geq
2$, and, in fact, for general $k \geq 2$. For instance, for $k = \ell = 2$ we
get $\approx 96.4995\%$. The key tool we employ is an insightful and general
theorem due to Leconte, Lelarge, and Massouli\'e [LLM13], 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 with loads close to the threshold using an adapted version of an algorithm
by Khosla [Kho13].
