Galloping in natural merge sorts
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We study the algorithm TimSort and the sub-routine it uses to merge monotonic (non-decreasing) sub-arrays, hereafter called runs. More precisely, we look at the impact on the number of element comparisons performed of using this sub-routine instead of a naive routine. In this article, we introduce a new object for measuring the complexity of arrays. This notion dual to the notion of runs on which TimSort built its success so far, hence we call it dual runs. It induces complexity measures that are dual to those induced by runs. We prove, for this new complexity measure, results that are similar to those already known when considering standard run-induced measures. Although our new results do not lead to any improvement on the number of element moves performed, they may lead to dramatic improvements on the number of element comparisons performed by the algorithm. In order to do so, we introduce new notions of fast- and middle-growth for natural merge sorts, which allow deriving the same upper bounds. After using these notions successfully on TimSort, we prove that they can be applied to a wealth of variants of TimSort and other natural merge sorts.
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