SORTING AND SELECTION IN POSETS

Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, the authors study these problems in the context of partially ordered sets, in which some pairs of objects are incomparable. This generalization is interesting from a combinatorial perspective, and it has immediate applications in ranking scenarios where there is no underlying linear ordering, e.g., conference submissions. It also has applications in reconstructing certain types of networks, including biological networks. In particular, the authors present the first algorithm that sorts a width-w poset of size n with query complexity O(n(w + logn)) and prove that this query complexity is asymptotically optimal. The authors also describe a variant of Mergesort with query complexity O(wnlog n/w ) and total complexity O(w^2n log n/w ). Both the sorting algorithms can be applied with negligible overhead to the more general problem of reconstructing transitive relations.