On sorting an intransitive total ordered set using semi-heap

The problem of sorting an intransitive total ordered set, a generalization of regular sorting, is considered. This generalized sorting is based on the fact that there exists a special linear ordering for any intransitive total ordered set. A new data structure called semi-heap is proposed to construct an optimal Θ(n log n) sorting algorithm. Finally, we propose a cost-optimal parallel algorithm using semi-heap. The run time of this algorithm is Θ(n) with Θ(log n) processors under the EREW PRAM model