Fast exponentiation using the truncation operation

Abstract

Given an integerk, and anarbitrary integer greater than$2^{2^k }$, we prove a tight bound of$\Theta (\sqrt k )$ on the time required to compute$2^{2^k }$ with operations {+, −, *, /, ⌊·⌋, ≤}, and constants {0, 1}. In contrast, when the floor operation is not available this computation requires Ω(k) time. Using the upper bound, we give an$O(\sqrt {\log n} )$ time algorithm for computing ⌊log loga⌋, for alln-bit integersa. This upper bound matches the lower bound for computing this function given by Mansour, Schieber, and Tiwari. To the best of our knowledge these are the first non-constant tight bounds for computations involving the floor operation.