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Sorting Under Forbidden Comparisons

Authors Indranil Banerjee, Dana Richards

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Indranil Banerjee
Dana Richards

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Indranil Banerjee and Dana Richards. Sorting Under Forbidden Comparisons. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.22

Abstract

In this paper we study the problem of sorting under forbidden comparisons where some pairs of elements may not be compared (forbidden pairs). Along with the set of elements V the input to our problem is a graph G(V, E), whose edges represents the pairs that we can compare in constant time. Given a graph with n vertices and m = binom(n)(2) - q edges we propose the first non-trivial deterministic algorithm which makes O((q + n)*log(n)) comparisons with a total complexity of O(n^2 + q^(omega/2)), where omega is the exponent in the complexity of matrix multiplication. We also propose a simple randomized algorithm for the problem which makes widetilde O(n^2/sqrt(q+n) + nsqrt(q)) probes with high probability. When the input graph is random we show that widetilde O(min(n^(3/2), pn^2)) probes suffice, where p is the edge probability.

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Keywords
  • Sorting
  • Random Graphs
  • Complexity

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