Sorting and Selection in Posets

• Authors: Constantinos Daskalakis;Richard M. Karp;Elchanan Mossel;Samantha J. Riesenfeld;Elad Verbin
• Venue: SIAM Journal on Computing
• Year: 2011

Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, we 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. Our results represent significant progress over previous results from two decades ago by Faigle and Turán. In particular, we present the first algorithm that sorts a width-$w$ poset of size $n$ with query complexity $O(n(w+\log n))$ and prove that this query complexity is asymptotically optimal. We also describe a variant of Mergesort with query complexity $O(wn\log\frac{n}{w})$ and total complexity $O(w^{2}n\log\frac{n}{w})$; an algorithm with the same query complexity was given by Faigle and Turán, but no efficient implementation of that algorithm is known. Both our sorting algorithms can be applied with negligible overhead to the more general problem of reconstructing transitive relations. We also consider two related problems: finding the minimal elements, and its generalization to finding the bottom $k$ “levels,” called the $k$-selection problem. We give efficient deterministic and randomized algorithms for finding the minimal elements with query complexity and total complexity $O(wn)$. We provide matching lower bounds for the query complexity up to a factor of 2 and generalize the results to the $k$-selection problem. Finally, we present efficient algorithms for computing a linear extension of a poset and computing the heights of all elements.