Adaptive Binary Sorting Schemes and Associated Interconnection Networks

Many routing problems in parallel processing such as concentration and permutation problems can be cast as sorting problems. In this paper we consider the problem of sorting on a new model, called an adaptive sorting network. We show that any sequence of n bits can be sorted on this model in O(lg n) bit-level delay using O(n) constant fanin gates. This improves the cost complexity of Batcher’s binary sorters by a factor of O(lg n) while matching their sorting time. The only other network that can sort binary sequences in O(n) cost is the network version of columnsort algorithm but this requires excessive pipelining. We also show that any sequence of n numbers can be sorted on the same model in O(lg n) comparator-level delay using O(n lg n lg lg n) comparators. In addition, using binary sorters, we construct permutation networks with O(n lg n) bit-level cost, and O(lg n) bit-level delay. These results provide the asymptotically least-cost sorting and permutation networks to date. We, of course, note that the well-known AKS sorting network has O(lg n) sorting time and O(n lg n) cost, but the constants hidden in these complexities are so large that our complexities outperform those of the AKS sorting network until n becomes extremely large.