Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths

We present an average case analysis of a variant of dual-pivot quicksort. We show that the used algorithmic partitioning strategy is optimal, i.e., it minimizes the expected number of key comparisons. For the analysis, we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear, logarithmic, and constant terms. An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proven.Comment: This article supersedes arXiv:1602.0403

Analysis of pivot sampling in dual-pivot Quicksort: A holistic analysis of Yaroslavskiy's partitioning scheme

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-015-0041-7The new dual-pivot Quicksort by Vladimir Yaroslavskiy-used in Oracle's Java runtime library since version 7-features intriguing asymmetries. They make a basic variant of this algorithm use less comparisons than classic single-pivot Quicksort. In this paper, we extend the analysis to the case where the two pivots are chosen as fixed order statistics of a random sample. Surprisingly, dual-pivot Quicksort then needs more comparisons than a corresponding version of classic Quicksort, so it is clear that counting comparisons is not sufficient to explain the running time advantages observed for Yaroslavskiy's algorithm in practice. Consequently, we take a more holistic approach and give also the precise leading term of the average number of swaps, the number of executed Java Bytecode instructions and the number of scanned elements, a new simple cost measure that approximates I/O costs in the memory hierarchy. We determine optimal order statistics for each of the cost measures. It turns out that the asymmetries in Yaroslavskiy's algorithm render pivots with a systematic skew more efficient than the symmetric choice. Moreover, we finally have a convincing explanation for the success of Yaroslavskiy's algorithm in practice: compared with corresponding versions of classic single-pivot Quicksort, dual-pivot Quicksort needs significantly less I/Os, both with and without pivot sampling.Peer ReviewedPostprint (author's final draft

Engineering Java 7's Dual Pivot Quicksort Using MaLiJAn

Wild S, Nebel M, Reitzig R, Laube U. Engineering Java 7's Dual Pivot Quicksort Using MaLiJan. In: Proceedings of the 15th Meeting on Algorithm Engineering and Experiments, ALENEX 2013, New Orleans, Louisiana, USA, January 7, 2013. Philadelphia, PA: Society for Industrial and Applied Mathematics; 2013: 55--69

Average Case and Distributional Analysis of Dual-Pivot Quicksort

In 2009, Oracle replaced the long-serving sorting algorithm in its Java 7 runtime library by a new dual-pivot Quicksort variant due to Vladimir Yaroslavskiy. The decision was based on the strikingly good performance of Yaroslavskiy's implementation in running time experiments. At that time, no precise investigations of the algorithm were available to explain its superior performance—on the contrary: previous theoretical studies of other dual-pivot Quicksort variants even discouraged the use of two pivots. In 2012, two of the authors gave an average case analysis of a simplified version of Yaroslavskiy's algorithm, proving that savings in the number of comparisons are possible. However, Yaroslavskiy's algorithm needs more swaps, which renders the analysis inconclusive. To force the issue, we herein extend our analysis to the fully detailed style of Knuth: we determine the exact number of executed Java Bytecode instructions. Surprisingly, Yaroslavskiy's algorithm needs sightly more Bytecode instructions than a simple implementation of classic Quicksort—contradicting observed running times. As in Oracle's library implementation, we incorporate the use of Insertionsort on small subproblems and show that it indeed speeds up Yaroslavskiy's Quicksort in terms of Bytecodes; but even with optimal Insertionsort thresholds, the new Quicksort variant needs slightly more Bytecode instructions on average. Finally, we show that the (suitably normalized) costs of Yaroslavskiy's algorithm converge to a random variable whose distribution is characterized by a fixed-point equation. From that, we compute variances of costs and show that for large n, costs are concentrated around their mean

Quicksort Is Optimal For Many Equal Keys

I prove that the average number of comparisons for median-of-$k$ Quicksort (with fat-pivot a.k.a. three-way partitioning) is asymptotically only a constant $\alpha_k$ times worse than the lower bound for sorting random multisets with $\Omega(n^\varepsilon)$ duplicates of each value (for any $\varepsilon>0$). The constant is $\alpha_k = \ln(2) / \bigl(H_{k+1}-H_{(k+1)/2} \bigr)$, which converges to 1 as $k\to\infty$, so Quicksort is asymptotically optimal for inputs with many duplicates. This resolves a conjecture by Sedgewick and Bentley (1999, 2002) and constitutes the first progress on the analysis of Quicksort with equal elements since Sedgewick's 1977 article

The Quicksort algorithm and related topics

Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quicksort algorithm and its variants. Specifically, we study the time complexity of the algorithm and we provide a complete demonstration of the variance of the number of comparisons required, a known result but one whose detailed proof is not easy to read out of the literature. We also examine variants of Quicksort, where multiple pivots are chosen for the partitioning of the array. The rest of this work is dedicated to the analysis of finding the true order by further pairwise comparisons when a partial order compatible with the true order is given in advance. We discuss a number of cases where the partially ordered sets arise at random. To this end, we employ results from Graph and Information Theory. Finally, we obtain an alternative bound on the number of linear extensions when the partially ordered set arises from a random graph, and discuss the possible application of Shellsort in merging chains