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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

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, even unrealistically, we are to search a list of 700 million items using Binary Search (Algorithm 2.1). What is the maximum number of com parisons that this algorithm must perform before finding a given item or concluding that it is not in the list?

Write a divide-and-conquer algorithm for the Towers of Hanoi Problem. The Towers of Hanoi Problem consists of three pegs and \(n\) disks of different sizes. The object is to move the disks that are stacked, in decreasing order of their size, on one of the three pegs to a new peg using the third one as a temporary peg. The problem should be solved according to the following rules: (1) when a disk is moved, it must be placed on one of the three pegs: (2) only one disk may be moved at a time, and it must be the top disk on one of the pegs; and (3) a larger disk may never be placed on top of a smaller disk. (a) Show for your algorithm that \(S(n)=2^{n}-1 .\) [Here \(S(n)\) denotes the number of steps (moves), given an input of \(n\) disks. (b) Prove that any other algorithm takes at least as many moves as given in part (a).

Show that the recurrence equation for the worst-case time complexity for Mergesort (Algorithms \(2.2 \text { and } 2.4)\) is given by \\[ W(n)=W\left(\left[\frac{n}{2}\right\rfloor\right)+W\left(\left\lceil\frac{n}{2}\right]\right)+n-1 \\] when \(n\) is not restricted to being a power of 2

Write algorithms that perform the operations \(u \times 10^{m}\) \(u\) divide \(10^{n}\) \(u\) rem \(10 "\) where \(u\) represents a large integer, \(m\) is a nonnegative integer, divide returns the quotient in integer division, and rem returns the remainder. Analyze your algorithms, and show that these operations can be done in linear time.
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