Abstract
We study the classical problem of sorting when comparison between certain pair of elements are forbidden. Along with the set of elements V, the input to our problem is an undirected graph G(V, E), whose edges represent the pairs that can be directly compared in constant time. We call this the comparison graph. It is also possible that the set of elements forms a partial-order, and not a total-order in which case, the sorting problem is the problem of determining all possible relations in the partial order, i.e. determining the (transitive) orientations of the edges of the graph.
If q is the number of edges missing in the graph, we first give a sorting algorithm that takes 
comparisons improving on the recent upper bound of 
. We also show the first lower bound by giving a graph and an orientation by an adversary where 
comparisons are necessary. Then, we give an 
algorithm (independent of q) when the comparison graph is from a special class of graphs like chordal or comparability graphs. Finally, we make some remarks regarding the complexity of sorting with forbidden comparisons when the elements form a total order.
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Notes
- 1.
The constant used in the paper is 200, instead of 320. We have made the small change to factor in a calculation gap in last equation on page 7 in [1].
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Biswas, A., Jayapaul, V., Raman, V. (2017). Improved Bounds for Poset Sorting in the Forbidden-Comparison Regime. In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_5
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DOI: https://doi.org/10.1007/978-3-319-53007-9_5
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