Trans-dichotomous algorithms without multiplication — Some upper and lower bounds

Abstract

We show that on a RAM with addition, subtraction, bitwise Boolean operations and shifts, but no multiplication, there is a transdichotomous solution to the static dictionary problem using linear space and with query time √log n(log log n)1+0(1). On the way, we show that two w-bit words can be multiplied in time (log w)1+0(1) and that time Ω(log w) is necessary, and that θ(log log w) time is necessary and sufficient for identifying the least significant set bit of a word.