Literature Review on Parallel Sorting of Intransitive Total Ordered Sets

Sorting of transitive total ordered sets is one of the best investigated topics in Computer Science. A generalization of this is sorting an intransitive total ordered set which is more complex. While a transitive total ordered set (e. g. a subset of the real numbers) holds a ≥ b∧b ≥ a⇒ a ≥ c, this does not have to be true for a intransitive total ordered set. Only a → b ∨̇ b → a is claimed. Nevertheless it can be proved that there is always an ordering that satisfies a1 → a2 → a3 . . .→ an. A good example for such a set is a tournament (e. g. in sports). Each competitor either wins or loses against every other competitor. But even if it is known that Player A defeats Player B who defeats Player C in turn, nothing can be said safely about the result of the match Player A has against Player C. Actually, the relation in such a intransitive ordered set is often referred to as a tournament. The symbol for the relation be ≺. The tournament can be represented as a directed graph where every player corresponds to a vertex. Then a b is represented by an edge from vertex a to vertex b. If so, there must not exist a directed edge from b to a since this would violate the assumptions for an intransitive total ordered set. When represented as a graph, sorting a tournament is equivalent to finding a Hamiltonian path in this graph. Given a graph, a Hamiltonian path is a path through which the graph visits each vertex precisely once. I has been proven by Redei [4] that there is always such a path in every tournament. Unlike the regular sorting problem, there might exist more than one appropriate solution.