Discrete & Computational Geometry
Fredman-Kolmo´s bounds and information theory
SIAM Journal on Algebraic and Discrete Methods
SIAM Journal on Discrete Mathematics
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)
Balanced pairs in partial orders
Discrete Mathematics - Special issue on partial ordered sets
Bounds on Optimal Merge Performance, and a Strategy for Optimality
Journal of the ACM (JACM)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Convex Optimization
Graph entropy and quantum sorting problems
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Sorting and selection in posets
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
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 revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (J. Comput. System Sci. 51 (3), 390---399, 1995) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical.We develop efficient algorithms for sorting under partial information. Like Kahn and Kim, our approach relies on graph entropy. However, our algorithms differ in essential ways from theirs. Rather than resorting to convex programming for computing the entropy, we approximate the entropy, or make sure it is computed only once, in a restricted class of graphs, permitting the use of a simpler algorithm.