Introduction to Randomized Algorithms: Quick Sort and Quick Selection

Introduction to Randomized Algorithms: Quick Sort and Quick Selection

.1 Introduction 1.2 Why Randomized Algorithms

Example: Sorting algorithms. The Merge-Sort algorithm is asymptotically best deterministic algorithm. It is not too hard to describe. However, the same asymptotic running time can be achieved by the simple randomized Quick-sort algorithm. The algorithm picks a random element as a pivot and partitions the rest of the elements: those smaller than the pivot and those bigger than the pivot. Recurse...

An Introduction to Randomized Algorithms

In this section, we present the classic quick sort algorithm and compute the expected running time of the algorithm. We assume that the elements of the set are all distinct. Below is the randomized quick sort algorithm. Algorithm RandQuickSort(S) Choose a pivot element xi u.a.r from S = {x1, x2, · · · , xn} Split the set S into two subsets S1 = {xj |xj < xi} and S2 = {xj |xj > xi} by comparing ...

An Efficient Quicksort using Value based Pivot Selection an Bidirectional Partitioning

Quick sort is generally considered to be the best internal sorting algorithm, and is often used as a yardstick by which the efficiency of other sorting algorithms is compared. It is, therefore essential that its performance is studied thoroughly. This includes studying the worst case behaviour of the algorithm, and especially when the algorithm is experimentally evaluated. The worst case runnin...

Automated Recurrence Analysis for Almost-Linear Expected-Runtime Bounds

Abstract. We consider the problem of developing automated techniques for solving recurrence relations to aid the expected-runtime analysis of programs. Several classical textbook algorithms have quite efficient expected-runtime complexity, whereas the corresponding worst-case bounds are either inefficient (e.g., QUICK-SORT), or completely ineffective (e.g., COUPON-COLLECTOR). Since the main foc...