 # Lecture 4 Sorting Networks Comparator comparator A Sorting

• Slides: 42 Lecture 4 Sorting Networks Comparator comparator A Sorting Network 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9 A sorting network is a comparison network which output monotone nondecreasing sequence for every input. Depth 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9 Depth is the maximum number of comparators on a path from an input wire to an output wire. Depth = parallel time 9 5 2 2 5 9 6 5 2 2 5 6 6 6 9 9 Depth is the maximum number of comparators on a path from an input wire to an output wire. Insertion Sort key Sorting network constructed from insertion sort. How to construct a sorting network from merging sort? Divide and Conquer • Divide the problem into subproblems. • Conquer the subproblems by solving them recursively. • Combine the solutions to subproblems into the solution for original problem. Merge Sort Procedure Structure Sorting network Merging network Sorting network Construction of Merging Network • 0 -1 principal. • Bitonic sorter. • Merging network. 0 -1 principal Lemma Proof of 0 -1 Principal Bitonic Sequence Bitonic 0 -1 Sequence Some Properties The half-cleaner bitonic 0 0 1 0 1 1 0 0 0 1 bitonic clean bitonic The half-cleaner bitonic 0 0 1 1 1 0 1 bitonic clean Lemma (a)One of two halfs is bitonic clean. (b) every number in the 1 st half ≤ any element in the 2 nd half. Proof (case 1) 0 0 0 1 0 1 0 Proof (case 2) 0 0 1 0 0 0 1 Proof (case 3) 0 0 0 1 1 1 0 0 1 Proof (case 4) 0 0 0 1 1 0 0 0 1 Proof (case 5) 1 0 1 0 1 1 1 Proof (case 6) 1 1 0 1 0 1 1 Proof (case 7) 0 1 1 0 0 1 1 Proof (case 8) 0 1 1 0 1 bitonic sorted Half cleaners sorted 0 0 0 1 0 0 0 sorted 0 1 1 1 0 1 1 Half cleaners Merging Network Structure Sorting network Merging network Sorting network Sorting Network Merging Networks What we learnt in this lecture? • What is sorting network? • Depth = parallel time. • Sorting network from Merge sort. Permutation Network • Switching network • Rearrangeability • Nework with 2 x 2 crossbars Crossbar Switch A crossbar switch can realize any matching between Inputs and outputs. 3 -stage Clos Network 1 n n m Rearrangeability Theorem Network with 2 x 2 crossbars Puzzle