Tables should be sorted (on random access machines) - Fich, Faith; Miltersen, Peter Bro (BRICS Report Series, 1995)
We consider the problem of storing an n element subset S of a universe of size m, so that membership queries (is x in S?) can be answered efficiently. The model of computation is a random access machine with the standard instruction set (direct and indirect addressing, conditional branching, addition, subtraction, and multiplication). We show that if s memory registers are used to store S, where n <= s <= m/n^epsilon, then query time Omega(log n) is necessary in the worst case. That is, under these conditions, the solution consisting of storing S as a sorted table and doing binary search is optimal. The condition s <= m/n^epsilon is essentially optimal; we show that if n + m/n^o(1) registers may be used, query time o(log n) is possible.