Sorting Network - Optimal Sorting

Optimal Sorting

The efficiency of a sorting network can be measured by its total size (the number of comparators used), or by its depth (the maximum number of comparators along any path from an input to an output). The asymptotically best known sorting network, called AKS network after its discoverers Ajtai, Komlós, and Szemerédi, achieves depth O(log n) and size O(n log n) for n inputs, which is asymptotically optimal. A simplified version of the AKS network was described by Paterson. While an important theoretical discovery, the AKS network has little or no practical application because of the large linear constants hidden by the Big-O notation. These are partly due to a construction of an expander graph. Finding sorting networks with size cn log n for small c remains a fundamental open problem.

Some important progress in designing optimal sorting network is done using genetic algorithm technique as well. (M. Mitchell, 1998)

For 1 to 8 inputs optimal sorting networks are known. They have respectively 0, 1, 3, 5, 9, 12, 16 and 19 comparators (Knuth, 1997). The optimal depths for up to 10 inputs are known and they are respectively 0, 1, 3, 3, 5, 5, 6, 6, 7, 7.