Comparison Networks Sorting Sorting binary values Sorting arbitrary numbers Implementing symmetric functions

Sorting Algorithms Example Mergesort(array[1,…,n] of Integers): begin Mergesort(array[1,…,n/2]); Mergesort(array[n/2+1,…,n]); Merge(array[1,…,n/2], array[n/2+1,…,n]); end comparisons required to merge two arrays of size m/2 comparisons to sort n elements Order of comparisons not fixed in advance. Not readily implementable in hardware.

Sorting Networks Sorting Network C D B A A B C D Order of comparisons fixed in advance. Readily implementable in hardware.

Sorting Networks (binary values) inputs outputs 1 sorted Sorting Network 1 1 1 1 1 1 1

Comparator (2-sorter) inputs outputs x min(x, y) C y max(x, y)

Comparator (2-sorter) AON Implementation x min(x, y) max(x, y) y inputs outputs x min(x, y) max(x, y) y

Comparator (2-sorter) inputs outputs x min(x, y) y max(x, y)

Comparison Network 1 1 1 1 width n depth d

Comparison Network 1 1 1 1 n / 2 comparisons per stage d stages

Sorting Network Any ideas?

Sorting Network inputs outputs Sorting Network . n . n 1

Insertion Sort Network inputs outputs depth 2n 3

Batcher Sorting Network Next Lecture

Sorting Arbitrary Numbers inputs outputs x min(x, y) y max(x, y) x, y can be values from any linearly ordered set, e.g., integers, reals, etc.

Integer Comparator X, Y: integers represented as m-bit binary strings. Comparison function: C(X,Y) = 1 if X > Y, 0 otherwise. Idea: use C(X,Y) to select the min and the max of X and Y.

Integer Comparator X C(X, Y) min(X, Y) Y C(X, Y) X C(X, Y) max(X, Y) Y

Sorting Arbitrary Numbers 9 6 2 2 9 6 6 9 2 2 6 9 sorted

Sorting Arbitrary Numbers 1 4 5 1 5 4 4 5 1 1 4 5 sorted

Sorting Arbitrary Numbers 3 7 3 7 7 3 3 7 not sorted How can we verify if a network sorts all possible input sequences?

Sorting Arbitrary Numbers inputs outputs Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 000 000 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 101 011 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 101 011 110 101 not sorted! Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 101 011 110 101 not sorted! 111 111 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers

Sorting Arbitrary Numbers inputs outputs Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 000 000 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 101 011 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 101 011 110 011 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 1 1 1 1 000 000 001 001 010 001 011 011 100 001 101 011 110 011 111 111 Try all possible 0/1 sequences.

Sorting Arbitrary Numbers inputs outputs 000 000 001 001 010 001 011 011 100 001 101 011 110 011 111 111 all sorted! Try all possible 0/1 sequences.

Zero-One Principle If a comparison network sorts all possible sequences of 0’s and 1’s correctly, then it sorts all sequences of arbitrary numbers correctly.

Lemma Given For a monotonically increasing function f,

Lemma Given For a monotonically increasing function f,

Proof: Lemma

Proof: Lemma

Proof: Lemma f is monotonically increasing:

Proof: Lemma f is monotonically increasing:

Proof: Lemma f is monotonically increasing:

Generalization Given

Generalization For a monotonically increasing function f, (by induction)

Proof: Zero-One Principle Suppose b) there exists a sequence that it doesn’t sort, i.e., such that but is placed before in the output. a) the network sorts all sequences of 0’s and 1’s, Define f (x) = 0 if 1 otherwise

Proof: Zero-One Principle Sorting Network .

Proof: Zero-One Principle . Sorting Network . . . .

Proof: Zero-One Principle . Sorting Network . . 1 contradiction! . .

Implementing XOR XOR(X) = 1 if odd 0 otherwise Sorting Network 1 1 1 1

Implementing XOR XOR(X) = 1 if odd 0 otherwise Sorting Network

Symmetric Functions for some subset of f (X) = 1 if 0 otherwise Sorting Network