A Class of an Almost-Optimal Size-Independent Parallel Prefix Circuits

2012 ◽  
Author(s):  
Hatem M. El-Boghdadi
Keyword(s):  
Optimal Size ◽  
Parallel Prefix ◽  
Prefix Circuits

Size vs. Depth Trade-Off In Parallel Prefix Circuits

1994 ◽  
Author(s):  
S. Lakshmivarahan ◽  
Sudarshan K. Dhall
Keyword(s):  
Lower Bound ◽  
Binary Operation ◽  
Unit Element ◽  
Optimal Size ◽  
Trade Off ◽  
Trade Offs ◽  
The Family ◽  
Constant Operation ◽  
Parallel Prefix ◽  
Prefix Circuits

Ladner-Fischer [1980] was the first one to demonstrate size vs. depth trade-off in parallel prefix circuits. In this Chapter, a lower bound from Snir [1986] on (size + depth) for prefix circuits is first derived. Several designs of parallel prefix circuits with optimal (size + depth) trade-offs are described. In this section, we derive a lower bound from Snir [1986] on the sum of depth and size of a class of prefix circuits. Let x = { x1, x2, • • • , XN }, be a set of N variables. Let D be the domain over which elements of x take their values, and let FN = { f1, f 2, . . . , fN } be a set of functions satisfying the following conditions: (SR1) fi, depends only on the variables x1, x2 , • • • xi, and (SR2) For each variable xi, i = 1, 2, • • • , N, there exist values ai, in D, such that, the set of functions fj |xi = ai = fj(x1,x2, . . . , xi-1, xi = ai, xi+1, . . . , xj), j=1,2, . . . , N, contains a family of (N - 1) functions satisfying these conditions. The family FN satisfying these two conditions is called a self reducible family of functions. Clearly, fi (x1, x2, . . . , xi) = x1 o x2 o . . . o xi, 1 ≤ i ≤ N, satisfies these conditions. Following is an example of a self-reducible family of functions. An associative binary operation θ is said to be non-trivial if the following conditions are satisfied: (i) Let z = x θ y. Then z depends on both x and y, that is, θ is not a projection, nor a constant operation, and (ii) There exists a (right) unit element e of θ, such that, x = x θ e.

Parallel Prefix Circuits

1994 ◽  
Author(s):  
S. Lakshmivarahan ◽  
Sudarshan K. Dhall
Keyword(s):  
Parallel Algorithms ◽  
Parallel Prefix ◽  
Prefix Circuits

This chapter presents a survey of parallel algorithms for computing the prefixes using circuit models. The circuits considered in this Chapter are constrained by the fixed fan-in (equal to two) but are allowed to have arbitrary or unbounded fan-out. Prefix circuits with fixed fan-in and fan-out are described in Chapter 7, and those with unbounded fan-in in Chapter 8. The characteristics of algorithms described in this Chapter are judged according to a number of measures including the size, depth, and fan-out. (Refer to Chapter 2 for definitions.) The layout of the serial circuit, S(N), is given in Figure 1. Clearly, the size, s(N), of this circuit is (N - 1), and the depth, d(N), is (N - 1). The sum of the size and depth for this circuit is. . . s(N) + d(N) = 2N – 2. . . . (1) Each node has a constant fan-out. In Chapter 6, it is shown that the sum of the size and depth of any prefix circuit is lower-bounded by 2N - 2. From this, it follows that the serial circuit is optimal. Thus, any circuit that computes all prefixes in less than serial time must have sizes larger than (N - 1). Perhaps an easy approach to the design of a parallel prefix circuit is to invoke the well known divide-conquer strategy. Let N = 2". According to this strategy, if DC(N) is a circuit that computes the prefixes of N elements, then DC (N) can be designed according to the principle illustrated in Figure 1.

scholarly journals POWER-SPEED TRADE-OFF IN PARALLEL PREFIX CIRCUITS

2005 ◽  
Vol 14 (01) ◽  
pp. 65-98
Author(s):  
S. VANICHAYOBON ◽  
S. K. DHALL ◽  
S. LAKSHMIVARAHAN ◽  
J. K. ANTONIO
Keyword(s):  
Power Consumption ◽  
Comparative Study ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Trade Off ◽  
Trade Offs ◽  
Long Time ◽  
Parallel Prefix ◽  
Capacitance Model ◽  
Prefix Circuits

Optimizing area and speed in parallel prefix circuits has been considered important for a long time. The issue of power consumption in these circuits, however, has not been addressed. This paper presents a comparative study of different parallel prefix circuits from the point of view of power–speed trade-off. An effective circuit capacitance model that is verified through PSpice simulations is used to investigate the power consumption in parallel prefix circuits. The model results in an analytical function for power consumption for each prefix circuit considered. The degrees of freedom studied include different parallel prefix circuits and voltage scaling. The results of the study were applied to evaluate power–speed trade-offs when different prefix circuits are used in Brent's parallel adder.5

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