Sorting and Selection in Posets
Constantinos Daskalakis (1), Richard M. Karp (1), Elchanan Mossel (1),, Samantha Riesenfeld (1), Elad Verbin (2) ((1) U.C. Berkeley, (2) Tel Aviv, University)

TL;DR
This paper develops efficient algorithms for sorting and selecting elements in partially ordered sets, extending classical sorting to handle incomparability with near-optimal query complexities.
Contribution
It introduces new algorithms for sorting and selection in posets with proven near-optimal query and total complexities, extending prior work from two decades ago.
Findings
New sorting algorithm with query complexity O(wn + nlog(n))
Efficient algorithms for minimal element determination with O(wn) complexity
Generalization to k-selection problem with derived upper bounds
Abstract
Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, we study a more general setting, in which some pairs of objects are incomparable. This generalization is relevant in applications related to rankings in sports, college admissions, or conference submissions. It also has potential applications in biology, such as comparing the evolutionary fitness of different strains of bacteria, or understanding input-output relations among a set of metabolic reactions or the causal influences among a set of interacting genes or proteins. Our results improve and extend results from two decades ago of Faigle and Tur\'{a}n. A measure of complexity of a partially ordered set (poset) is its width. Our algorithms obtain information about a poset by queries that compare two elements. We present an algorithm that sorts, i.e.…
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Taxonomy
TopicsRough Sets and Fuzzy Logic · Advanced Algebra and Logic · semigroups and automata theory
