The Spatial Complexity of Oblivious k -Probe Hash Functions

The problem of constructing a dense static hash-based lookup table T for a set of n elements belonging to a universe U = \ 0, 1, 2,⋯ , m -1 \ is considered. Nearly tight bounds on the spatial complexity of oblivious $O(1)$-probe hash functions, which are defined to depend solely on their search key argument, are provided. This establishes a significant gap between oblivious and nonoblivious search. In particular, the results include the following: • A lower bound showing that oblivious k-probe hash functions require a program size of Ω((n / k²)e⁻ᵏ+log log m) bits, on average. • A probabilistic construction of a family of oblivious k-probe hash functions that can be specified in O(n e⁻ᵏ +log log m) bits, which nearly matches the above lower bound. • A variation of an explicit $O(1)$ time 1-probe (perfect) hash function family that can be specified in O(n+log log m) bits, which is tight to within a constant factor of the lower bound.