Counting Zeros in Random Walks on the Integers and Analysis of Optimal Dual-Pivot Quicksort

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.

Using Non-Linear Difference Equations to Study Quicksort Algorithms

Using non-linear difference equations, combined with symbolic computations, we make a detailed study of the running times of numerous variants of the celebrated Quicksort algorithms, where we consider the variants of single-pivot and multi-pivot Quicksort algorithms as discrete probability problems. With non-linear difference equations, recurrence relations and experimental mathematics techniques, explicit expressions for expectations, variances and even higher moments of their numbers of comparisons and swaps can be obtained. For some variants, Monte Carlo experiments are performed, the numer…

快速排序算法及相关主题

The Quicksort algorithm and related topics

由于排序算法在数学、计算机科学及相关领域有着广泛的应用，因此它引起了人们的广泛关注和研究。在这篇论文中，我们首先处理快速排序算法及其变体的数学分析。具体来说，我们研究的算法的时间复杂度，我们提供了一个完整的演示的方差所需的比较，一个已知的结果，但其详细的证明是不容易读出来的文献。我们还研究了快速排序的变体，其中选择多个枢轴来划分数组。本文的其余部分致力于分析当预先给出与真序相容的偏序时，通过进一步的成对比较来找到真序。我们讨论了一些情况下，偏序集随机出现。为此，我们采用图和信息论的结果。最后，当偏序集来自随机图时，我们得到了线性扩张数量的另一个界，并讨论了Shellsort在合并链中的可能应用。