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# p - On Sorting an Intransitive Total Ordered Set Using...

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On Sorting an Intransitive Total Ordered Set Using Semi-Heap Jie Wu Department of Computer Science and Engineering Florida Atlantic University Boca Raton, FL 33431 [email protected] Abstract 1 The problem of sorting an intransitive total ordered set, a generalization of regular sorting, is considered. This gen- eralized sorting is based on the fact that there exists a spe- cial linear ordering for any intransitive total ordered set. A new data structure called semi-heap is proposed to con- struct an optimal sorting algorithm. Finally, we propose a cost-optimal parallel algorithm using semi-heap. The run time of this algorithm is with pro- cessors under the EREW PRAM model. 1. Introduction Sorting is one of the fundamental problems in computer science and many different solutions for sorting have been proposed [5, 6]. Basically, given a sequence of numbers as an input, a sorting algorithm generates a permutation (reordering) of the input se- quence such that . We consider a generalization of the sorting problem by replacing with , where is a total order without the transitive property, i.e., it is intransitive. That is, if and , it is not necessary that . The total order requires that for any two elements and , either or , but not both (antisymmetric). The set of elements exhibiting intransitive total or- der can be represented by a directed graph, where represents a directed edge from vertex to vertex . The underlying graph is a complete graph. This graph is also called a tournament [2], representing a tournament of players where every possible pair of players plays one game to decide the winner (and the loser) between them. Sorting on corresponds to finding a Hamiltonian path in the tour- nament. 1 This work was support in part by NSF grant CCR 9900646. Hell and Rosenfeld [4] proved that the bound of find- ing a Hamiltonian path is , the same complex- ity as the regular sorting. They also considered bounds on finding some generalized Hamiltonian paths. It is easy to prove that many regular sorting algorithms can be used to find a Hamiltonian path in a tournament, such as bubble sort, insertion sort, binary insertion sort, and merge sort. Among parallel sorting algorithms, even-odd merge sort can still be applied. However, heapsort and quicksort cannot be used. Bar-Noy and Naor [1] studied different parallel solu- tions based on different models and the number of proces- sors. They showed that under the CRCW PRAM model, the generalized sorting problem can be solved in us- ing processors. Other fast parallel algorithms can be found in [7]. In this paper, we propose a new data structure called semi-heap , which is an extension of a regular heap struc- ture. We introduce an optimal algorithm to de- termine a Hamiltonian path in a tournament based on the semi-heap structure. Then, we propose a cost-optimal par- allel algorithm based on the semi-heap structure that takes in run time using processors in the EREW PRAM model. An implementation of the cost-optimal par- allel algorithm in the network model with a linear array of processors is also shown.

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