On Generalized Comparison-Based Sorting Problems

We survey recent results on comparison-based sorting problems involving partial orders. In particular, we outline recent algorithms for partial order production and sorting under partial information. We emphasize the complementarity of the two problems and the common aspects of the algorithms. We also include open questions on two other related problems, namely partial order identification and sorting with forbidden comparisons.

[1]  László Lovász,et al.  Searching in Trees, Series-Parallel and Interval Orders , 1986, SIAM J. Comput..

[2]  György Turán,et al.  Sorting and Recognition Problems for Ordered Sets , 1988, SIAM J. Comput..

[3]  George Steiner.  Searching in 2-Dimensional Partial Orders , 1987, J. Algorithms.

[4]  Béla Bollobás,et al.  Sorting and Graphs , 1985 .

[5]  Jean Cardinal,et al.  An Efficient Algorithm for Partial Order Production , 2010, SIAM J. Comput..

[6]  Jeff Kahn,et al.  Entropy and sorting , 1992, STOC '92.

[7]  László Lovász,et al.  Entropy splitting for antiblocking corners and perfect graphs , 1990, Comb..

[8]  Jean Cardinal,et al.  Sorting under partial information (without the ellipsoid algorithm) , 2009, STOC '10.

[9]  Gábor Simonyi,et al.  Graph entropy: A survey , 1993, Combinatorial Optimization.

[10]  János Komlós,et al.  Matching nuts and bolts in O(n log n) time , 1996, SODA '96.

[11]  G. Brightwell,et al.  Counting linear extensions , 1991 .

[12]  Michael Saks,et al.  The Information Theoretic Bound for Problems on Ordered Sets and Graphs , 1985 .

[13]  Kurt Mehlhorn,et al.  Searching, Sorting and Randomised Algorithms for Central Elements and Ideal Counting in Posets , 1993, FSTTCS.

[14]  Michael L. Fredman,et al.  How Good is the Information Theory Bound in Sorting? , 1976, Theor. Comput. Sci..

[15]  Gregory J. E. Rawlins.  Compared to what? - an introduction to the analysis of algorithms , 1992, Principles of computer science series.

[16]  Stefan Felsner,et al.  Balancing pairs and the cross product conjecture , 1995 .

[17]  Nathan Linial,et al.  The Information-Theoretic Bound is Good for Merging , 1984, SIAM J. Comput..

[18]  Elchanan Mossel,et al.  Sorting and Selection in Posets , 2007, SIAM J. Comput..

[19]  B. T. Bennett,et al.  Bounds on Optimal Merge Performance, and a Strategy for Optimality , 1972, JACM.

[20]  Noga Alon,et al.  Matching nuts and bolts , 1994, SODA '94.

[21]  Michael E. Saks,et al.  Every Poset Has a Central Element , 1985, J. Comb. Theory, Ser. A.

[22]  Kurt Mehlhorn,et al.  Towards Optimal Multiple Selection , 2005, ICALP.

[23]  Sampath Kannan,et al.  Algorithms for the Generalized Sorting Problem , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[24]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[25]  Graham R. Brightwell,et al.  Balanced pairs in partial orders , 1999, Discret. Math..

[26]  Martin Aigner.  Producing posets , 1981, Discret. Math..

[27]  Andrew Chi-Chih Yao.  On the Complexity of Partial Order Productions , 1989, SIAM J. Comput..

[28]  Nathan Linial,et al.  Balancing extensions via Brunn-Minkowski , 1991, Comb..