Sorting Under Forbidden Comparisons

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