Discrete & Computational Geometry
On the complexity of partial order productions
SIAM Journal on Computing
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
Multiple Quickselect—Hoare's Find algorithm for several elements
Information Processing Letters
Optimal Time Minimal Space Selection Algorithms
Journal of the ACM (JACM)
Algorithm 410: Partial sorting
Communications of the ACM
Some Lower Bounds for Comparison-Based Algorithms
ESA '94 Proceedings of the Second Annual European Symposium on Algorithms
Analysis of multiple quickselect variants
Theoretical Computer Science
Tight Results on Minimum Entropy Set Cover
Algorithmica
Towards optimal multiple selection
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Minimum Entropy Combinatorial Optimization Problems
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Sorting under partial information (without the ellipsoid algorithm)
Proceedings of the forty-second ACM symposium on Theory of computing
An Efficient Algorithm for Partial Order Production
SIAM Journal on Computing
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We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S, by comparing a minimum number of pairs in T. Special cases of this problem include sorting by comparisons, selection, multiple selection, and heap construction. We give an algorithm performing ITLB + o(ITLB) + O(n) comparisons in the worst case. Here, n denotes the size of the ground sets, and ITLB denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target poset. The overall complexity of our algorithm is polynomial. This answers a question of Yao (SICOMP, 1989). Our strategy is to extend the poset S to a weak order W whose corresponding information-theoretic lower bound is provably not much larger than that for S. Taking W instead of S as a target poset, we then solve the problem by applying a multiple selection algorithm that performs not much more than ITLB comparisons. We base our analysis on the entropy of the target poset S, a quantity that can be efficiently computed and provides a good estimate of ITLB.