Time Complexity of Boolean Functions on CREW PRAMs

Abstract

This paper is concerned with parallel random access machines (PRAMS), where each processor can read from and write into a common random access memory. Different processors may read the same memory location at a time, but only one processor is allowed to write into it (the CREW model). Suppose f is a Boolean function of n variables. Let ${\operatorname{CREW}}(f)$ be the number of steps required by CREW PRAMS to compute function f It has been proved that ${\operatorname{CREW}}(f) \geqq \log _b {\operatorname{crit}}(f)$, where $b \approx 4.79$ and $crit(f)$ is the critical complexity of and f (see [S. Cook, C. Dwork, and R. Reischuk, SIAM J. Comput.,15 (1986), pp. 87–97]). It was proved by Parberry and Pei Yuan Yan [SIAM J. Comput., 20 (1991), pp. 88–99] that the same holds for $b = 4$. It follows that the time required by the logical *#8220;or” of n variables is at least $\log _4 n$. This paper presents an essentially different method of estimating PRAM complexity of Boolean functions. Let and $n_f$ be the number of inputs $x \in \{ 0,1 \}^n $ for which $f(x) = 1$. Let $i_f = \max \{ {j:2^j |n_f } \}$. Then ${\operatorname{CREW}}(f) \geqq \log _c (n - i_f )$, where $c \approx 2.618$. Thanks to this result, the time complexity of the logical “or” of n variables is determined exactly. This in turn allows better estimations of time complexity of the threshold functions to be obtained. Another corollary is that for sorting n arbitrary keys, PRAMS require time, which can be determined up to five steps.