Abstract
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 left (begin {array}{c}n2end {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(nlog n +kn+k^{2}log n) comparisons and O(nlog 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.
Highlights
In the model of recurring faults in computations, as introduced in [11], operations on certain items yield false results even when they are repeated
One of the problems investigated in [11] is to sort a set of n elements by comparisons, under the assumption Ak that at most k pairs return false comparison results
Recurring comparison faults can origin from software bugs or, most notably, from unreliable floating-point operations in geometric computations [10]
Summary
In the model of recurring faults in computations, as introduced in [11], operations on certain items yield false results even when they are repeated. One building block is a procedure to insert another vertex in an existing order with a minimum number of backward arcs This leads to fault-tolerant sorting algorithms based on Treesort and Quicksort. They essentially need O(n log n) comparisons for any fixed k, which is optimal in a sense. The time is larger by just some “FPT term” in the parameter k These are the first subquadratic algorithms for sorting with recurring comparison faults. Sorting with faults was studied in [10], in their model an algorithm may use expensive and cheap comparisons, where the cheap ones may err for elements whose ranks differ by at most some constant. The number of comparisons needed to decide certain properties of partial orders has been studied in [7]
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