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Chapter 2: Problem 23

Write a nonrecursive algorithm for Quicksort (Algorithm 2.6). Analyze your algorithm, and give the results using order notation.

Short Answer

Expert verified
A non-recursive Quicksort algorithm can be implemented by using an explicit 'stack' data structure. The average, best case and worst-case time complexities of this implementation are \(O(n \log n)\), \(O(n \log n)\) and \(O(n^2)\) respectively. The space complexity is \(O(n)\).

Step by step solution

01

Implement Non-Recursive Quicksort

Create a non-recursive Quicksort algorithm. This can be done using an iterative version of the Quicksort algorithm with explicit 'stack' data structure that substitutes for the execution stack of the recursive version. In your version of Quicksort, use a partition operation which puts all elements less than a chosen pivot element to its left and all greater elements to its right.
02

Analyze the Algorithm

Next, perform the analysis of the non-recursive Quicksort algorithm. The best case time complexity of Quicksort is \(O(n \log n)\) which is achieved when the partition process always picks the middle element as pivot. The worst case time complexity of Quicksort is \(O(n^2)\). This worst case behavior occurs when the partition process always picks greatest or smallest element as pivot. The space complexity of the algorithm is \(O(n)\) because the maximum auxiliary space required is \(n\).
03

Present Results Using Order Notation

Present the results using order notation. The average, best case and worst-case time complexities of Quicksort are \(O(n \log n)\), \(O(n \log n)\) and \(O(n^2)\) respectively. The auxiliary space or the space complexity of the algorithm is \(O(n)\).

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Most popular questions from this chapter

Assuming that Quicksort uses the first item in the list as the pivot item: (a) Give a list of \(n\) items (for example, an array of 10 integers) representing the worst-case scenario. (b) Give a list of \(n\) items (for example, an array of 10 integers) representing the best-case scenario.

When a divide-and-conquer algorithm divides an instance of size \(n\) of a problem into subinstances each of size \(n / c\), the recurrence relation is typically given by \\[ \begin{array}{l} T(n)=a T\left(\frac{n}{c}\right)+g(n) \quad \text { for } n>1 \\ T(1)=d \end{array} \\] where \(g(n)\) is the cost of the dividing and combining processes, and \(d\) is a constant. Let \(n=c^{k}\) (a) Show that \\[ T\left(c^{k}\right)=d \times d^{k}+\sum_{j=1}^{k}\left[a^{k-j} \times g\left(c^{j}\right)\right] \\] (b) Solve the recurrence relation given that \(g(n) \in \Theta(n)\)

How many multiplications would be performed in finding the product of two \(64 \times 64\) matrices using the standard algorithm?

Use Mergesort (Algorithms 2.2 and 2.4 ) to sort the following list. Show the actions step by step. \\[ 123 \quad 34 \quad 189 \quad 56 \quad 150 \quad 12 \quad 9 \quad 240 \\]

Suppose that, in a divide-and-conquer algorithm, we always divide an in stance of size \(n\) of a problem into 10 subinstances of size \(n / 3,\) and the dividing and combining steps take a time in \(\Theta\left(n^{2}\right)\). Write a recumence equation for the running time \(T(n),\) and solve the equation for \(T(n)\).
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