Comparison Networks

Comparison networks: Sorting networks are comparison networks that always sort their inputs, so it makes sense to begin our discussion with comparison networks and their characteristics. A comparison network is composed solely of wires and comparators. A comparator, shown in Figure 27.1(a), is a device with two inputs, x and y, and two outputs, x' and y', that performs the following function:

Figure 27.1: (a) A comparator with inputs x and y and outputs x' and y'. (b) The same comparator, drawn as a single vertical line. Inputs x = 7, y = 3 and outputs x' = 3, y' = 7 are shown.

x'	=	min(x, y),
y'	=	max(x, y).

Because the pictorial representation of a comparator in Figure 27.1(a) is too bulky for our purposes, we shall adopt the convention of drawing comparators as single vertical lines, as shown in Figure 27.1(b). Inputs appear on the left and outputs on the right, with the smaller input value appearing on the top output and the larger input value appearing on the bottom output. We can thus think of a comparator as sorting its two inputs.

We shall assume that each comparator operates in O(1) time. In other words, we assume that the time between the appearance of the input values x and y and the production of the output values x' and y' is a constant.

A wire transmits a value from place to place. Wires can connect the output of one comparator to the input of another, but otherwise they are either network input wires or network output wires. Throughout this chapter, we shall assume that a comparison network contains n input wires a₁, a₂,...,a_n, through which the values to be sorted enter the network, and n output wires b₁, b₂,...,b_n, which produce the results computed by the network. Also, we shall speak of the input sequence 〈a₁, a₂,...,a_n〉 and the output sequence 〈b₁, b₂,...,b_n〉, referring to the values on the input and output wires. That is, we use the same name for both a wire and the value it carries. Our intention will always be clear from the context.

Figure 27.2 shows a comparison network, which is a set of comparators interconnected by wires. We draw a comparison network on n inputs as a collection of n horizontal lines with comparators stretched vertically. Note that a line does not represent a single wire, but rather a sequence of distinct wires connecting various comparators. The top line in Figure 27.2, for example, represents three wires: input wire a₁, which connects to an input of comparator A; a wire connecting the top output of comparator A to an input of comparator C; and output wire b₁, which comes from the top output of comparator C. Each comparator input is connected to a wire that is either one of the network's n input wires a₁, a₂,...,a_n or is connected to the output of another comparator. Similarly, each comparator output is connected to a wire that is either one of the network's n output wires b₁, b₂,...,b_n or is connected to the input of another comparator. The main requirement for interconnecting comparators is that the graph of interconnections must be acyclic: if we trace a path from the output of a given comparator to the input of another to an output to an input, etc., the path we trace must never cycle back on itself and go through the same comparator twice. Thus, as in Figure 27.2, we can draw a comparison network with network inputs on the left and network outputs on the right; data move through the network from left to right.

Figure 27.2: (a) A 4-input, 4-output comparison network, which is in fact a sorting network. At time 0, the input values shown appear on the four input wires. (b) At time 1, the values shown appear on the outputs of comparators A and B, which are at depth 1. (c) At time 2, the values shown appear on the outputs of comparators C and D, at depth 2. Output wires b₁ and b₄ now have their final values, but output wires b₂ and b₃ do not. (d) At time 3, the values shown appear on the outputs of comparator E, at depth 3. Output wires b₂ and b₃ now have their final values.

Each comparator produces its output values only when both of its input values are available to it. In Figure 27.2(a), for example, suppose that the sequence 〈9, 5, 2, 6〉 appears on the input wires at time 0. At time 0, then, only comparators A and B have all their input values available. Assuming that each comparator requires one time unit to compute its output values, comparators A and B produce their outputs at time 1; the resulting values are shown in Figure 27.2(b). Note that comparators A and B produce their values at the same time, or "in parallel." Now, at time 1, comparators C and D, but not E, have all their input values available. One time unit later, at time 2, they produce their outputs, as shown in Figure 27.2(c). Comparators C and D operate in parallel as well. The top output of comparator C and the bottom output of comparator D connect to output wires b₁ and b₄, respectively, of the comparison network, and these network output wires therefore carry their final values at time 2. Meanwhile, at time 2, comparator E has its inputs available, and Figure 27.2(d) shows that it produces its output values at time 3. These values are carried on network output wires b₂ and b₃, and the output sequence 〈2, 5, 6, 9〉 is now complete.