O(1) insertion up to the load thresholds in multi-capacity cuckoo hashing

Establish that in the cuckoo hashing model where each table slot has capacity b>1 (with independent uniformly random hash functions), for any capacity b>1 and any number of hash functions, the random walk insertion algorithm achieves constant expected insertion time for all load factors c below the corresponding sharp load thresholds. Concretely, prove O(1) expected insertion up to the known thresholds in the multi-capacity bucket model.

Background

The paper discusses an alternative generalization of cuckoo hashing in which each slot (bucket) can hold more than one item. For this model, the sharp load thresholds for the existence of feasible assignments are known for both two hash functions and for d≥3.

While O(1) expected insertion time for random walk insertion has been established for some load factors below the threshold in this multi-capacity setting, matching O(1) insertion up to the threshold has not yet been proven for any capacity greater than one.