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Suppose that n elements shall be sorted by comparisons, but some subset of at most k pairs systematically returns false comparison results. This subset is unknown, but the number k is known in advance. Using a connection to feedback arc sets in tournaments (FAST), we characterize the solution space of sorting with recurring comparison faults by a FAST enumeration, which represents all information about the order that can be obtained by doing all n2$\left (\begin {array}{c}n\\2\end {array}\right )$ comparisons. Some optimal parameterized enumeration algorithm for FAST also works for the more general chordal graphs, and this fact contributes to the efficiency of our representation. Next we compute the solution space more efficiently, by fault-tolerant versions of Treesort and Quicksort. We need O(nlogn+kn+k2logn)$O(n\log n +kn+k^{2}\log n)$ comparisons and O(nlogn+kn+k2logn+kF(k2,k))$O(n\log n +kn+k^{2}\log n +kF(k^{2},k))$ time, where F(n, k) is any parameterized time bound for finding a FAST with at most k arcs. Thus, for rare faults the complexity is close to optimal. We also propose directions of further research, revolving around decision diagrams for sorting with recurring faults.

The Solution Space of Sorting with Recurring Comparison Faults