Jean Cardinal
ABSTRACT
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.
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Index Terms
- An efficient algorithm for partial order production
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