Static dictionaries on AC0 RAMs: query time &thetas;(√log n/log log n) is necessary and sufficient

In this paper we consider solutions to the static dictionary problem on AC⁰ RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are in AC⁰. Our main result is a tight upper and lower bound of θ(√log n/log log n) on the time for answering membership queries in a set of size n when reasonable space is used for the data structure storing the set; the upper bound can be obtained using O(n) space, and the lower bound holds even if we allow space 2^{polylog n}. Several variations of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unit-cost assumption: any RAM instruction set which permits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth Ω(log w/log log to), where w is the word size of the machine (and log the size of the universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by M.L. Fredman, J. Komlos and E. Szemeredi (1984)