Worst-Case Efficient Sorting with QuickMergesort
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The two most prominent solutions for the sorting problem are Quicksort and Mergesort. While Quicksort is very fast on average, Mergesort additionally gives worst-case guarantees, but needs extra space for a linear number of elements. Worst-case efficient in-place sorting, however, remains a challenge: the standard solution, Heapsort, suffers from a bad cache behavior and is also not overly fast for in-cache instances. In this work we present median-of-medians QuickMergesort (MoMQuickMergesort), a new variant of QuickMergesort, which combines Quicksort with Mergesort allowing the latter to be implemented in place. Our new variant applies the median-of-medians algorithm for selecting pivots in order to circumvent the quadratic worst case. Indeed, we show that it uses at most n n + 1.6n comparisons for n large enough. We experimentally confirm the theoretical estimates and show that the new algorithm outperforms Heapsort by far and is only around 10 Introsort (std::sort implementation of stdlibc++), which has a rather poor guarantee for the worst case. We also simulate the worst case, which is only around 10 natural candidate to replace Heapsort as a worst-case stopper in Introsort.
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