Asymptotically Optimal Circuit for a Storage Access Function

Let gk:{0,1}n+k → {0,1}, where n = 2k, be the binary function defined by gk(a1,···, ak, X0,···, xn-1) = x(a) where (a) is the natural number with binary representation a1,···, ak. This function models the reading operation in a random-access storage. In [1] Paul proved a 2n lower bound to the combinational complexity of gk. This correspondence derives a realization for gk in a circuit with 2n + 0(√n) gates and a depth asymptotic to k.