NDLI: Time Complexity of Boolean Functions on CREW PRAMs

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Time Complexity of Boolean Functions on CREW PRAMs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kutyowski, Mirosaw
Copyright Year	1991
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. 8797]). It was proved by Parberry and Pei Yuan Yan [SIAM J. Comput., 20 (1991), pp. 8899] 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.
Starting Page	824
Ending Page	833
Page Count	10
File Format	PDF
ISSN	00975397
DOI	10.1137/0220051
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	Modes of computation parallel computation time bounds parallel random access machines
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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