An Efficient Algorithm for Partial Order Production

TL;DR

This paper introduces an efficient algorithm for partial order production that nearly matches the information-theoretic lower bound, using a layered approach and multiple selection techniques, and addresses a longstanding open problem.

Contribution

It presents a polynomial-time algorithm that approximates the optimal comparison count for partial order production, extending the target order to a weak order to improve efficiency.

Findings

Algorithm performs close to the information-theoretic lower bound

Uses entropy-based analysis for partial order complexity

Addresses an open problem posed by Yao in 1989

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 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). 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.