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
Communications of the ACM
Average Cost to Produce Partial Orders
ISAAC '94 Proceedings of the 5th International Symposium on Algorithms and Computation
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
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Tight Results on Minimum Entropy Set Cover
Algorithmica
An efficient algorithm for partial order production
Proceedings of the forty-first annual ACM symposium on Theory of computing
Towards optimal multiple selection
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
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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 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 partial order. Our approach is to replace the target partial order by a weak order (that is, a partial order with a layered structure) extending it, without increasing the information-theoretic lower bound too much. We then solve the problem by applying an efficient multiple selection algorithm. The overall complexity of our algorithm is polynomial. This answers a question of Yao [SIAM J. Comput., 18 (1989), pp. 679-689]. We base our analysis on the entropy of the target partial order, a quantity that can be efficiently computed and provides a good estimate of the information-theoretic lower bound.