O(1) insertion up to the load threshold for d=3 in random-walk cuckoo hashing

Establish that for 3-ary cuckoo hashing with independent uniformly random hash functions and any load factor c below the sharp load threshold c*_3, the random walk insertion algorithm has constant expected insertion time that is independent of the number of keys and table size. Concretely, prove that for d=3 and c<c*_3, the expected number of rehashes required to insert an item is bounded by a constant.

Background

This paper proves that for d≥4 and any load factor c below the known sharp load threshold c*_d, the expected insertion time of the random walk d-ary cuckoo hashing algorithm is O(1). Prior to this work, O(1) insertion up to the threshold was not known for any d≥3, and the best results for d=3 achieved O(1) insertion only up to a load factor below the threshold.

The remaining case d=3 is highlighted by the authors as the main unresolved piece needed to complete the picture for all d≥3. Achieving O(1) insertion at the threshold for d=3 would match the existence threshold for perfect matchings and align the theory with practical performance observations.