Systolic algorithms, example: sorting networks

We take a simple model of parallel computation that is, most of the operations are executed simultaneously. The sorting network is made up of parallel wires on which the numbers travel from left to right. At places pointed by a vertical line, a comparator gate guides smaller (or lighter) number to the upper output wire and the bigger (or heavier) number to the lower output wire. Clearly, adequate comparators in right positions will cause the network to sort correctly - however how many gates does it take, where must they be positioned?

Lemma: Given that f monotonic, that is, x ≤ y ⇒ f(x) ≤ f(y).

When a network of comparators transforms x = x1, .., xn to y = y1, .., yn, then it transforms the f(x) to the f(y).

Theorem (0-1 principle): When a network S with n input lines sorts all the 2n vectors of 0s and 1s to non-decreasing order, then S sorts any vector of n arbitrary numbers properly.

Proof by contraposition: If S fails to sort some of the vector x = x1, .., xn of arbitrary numbers, the it as well fails to sort some of the binary vector f(x). Assume that S transform x into y with a ‘sorting error’, that is, yi > yi+1 for some index i. Made a binary-valued monotonic function f: f(x) = 0 for x < yi, f(x) = 1 for x ≥ yi. S transforms f(x) = f(x1),.., f(xn) to  f(y) = f(y1), .., f(yn). As f(yi) = 1 > f(yi+1) = 0, S fails to sort binary vector x. QED.

Theorem (testing proves correctness): When a sorting network S which uses adjacent comparisons just sorts the ‘inverse vector’ x1 > x2 > ... > xn, then it sorts any arbitrary vector.

Note: Serial and parallel computation lead to completely distinct resource characteristics.

Example: Prove that a sorting network using just adjacent comparisons should have ≥ n-choose-2 comparators.

Solution: The transposition in a sequence x1, x2, ... , xn  is a pair (i, j) with i < j and xi > xj . The inverse vector consists of n-choose-2 transpositions. Each and every comparator decreases the number of transpositions by utmost 1.

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