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Chapter 7: Problem 1

Implement the Insertion Sort algorithm (Algorithm 7.1), run it on your system, and study its best-case, average-case, and worst-case time complexities using several problem instances.

Short Answer

Expert verified
Insertion Sort Algorithm has been implemented and executed on several problem instances. Upon analyzing the time complexities, it has been observed that in the best-case scenario, the time complexity is \(O(n)\). For the average and worst-case scenarios, the time complexity is \(O(n^2)\).

Step by step solution

01

Implement the Insertion Sort Algorithm

Begin by understanding the Insertion Sort Methodology. It is a simple sorting algorithm that iteratively builds a sorted list by continuously picking unsorted items and placing them in the correct location within the sorted portion. The algorithm can be implemented using a loop within a loop. The outer loop ranges from 1 to the length of the array, while the inner loop compares the j-th element with the (j-1)-th element, swapping them if they are in the wrong order.
02

Run the Algorithm

Now that the algorithm is implemented, it is time to run it. Test the algorithm using several problem instances, including different ordered lists (ascending, descending, random), and lists of varying sizes.
03

Study Time Complexities

Time complexity of a function quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input. The time complexity of best, average, and worst cases of Insertion Sort are: Best case- \(O(n)\) which happens when the input is an already sorted array; Average case: time complexity is \(O(n^2)\) when elements are in random order; Worst case: time complexity is \(O(n^2)\) which happens when the input is a reversely sorted array.

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Most popular questions from this chapter

Among Selection Sort, Insertion Sort, Mergesort, Quicksort, and Heapsort, which algorithm would you choose in each list-sorting situation below? Justify your answers. (a) The list has several hundred records. The records are quite long, but the keys are very short. (b) The list has about 45,000 records. It is necessary that the sort be completed reasonably quickly in all cases. There is barely enough memory to hold the 45,000 records. (c) The list has about 45,000 records, but it starts off only slightly out of order. (d) The list has about 25,000 records. It is desirable to complete the sort as quickly as possible on the average, but it is not critical that the sort be completed quickly in every single case.

Show that there are \(n(n-1) / 2\) inversions in a permutation of \(n\) distinct ordered elements with respect to its transpose.

Write a linear-time sorting algorithm that sorts a permutation of integers 1 through \(n\), inclusive.

Write an algorithm that sorts a list of \(n\) elements in nonincreasing order by finding the largest and smallest elements and exchanges those elements with the elements in the first and last positions. Then the size of the list is reduced by \(2,\) excluding the two elements that are already in the proper positions, and the process is repeated on the remaining part of the list until the cntire list is sorted, Analyze your algorithm, and show the results using order notation.

Is Exchange Sort (Algorithm 1.3 ) or Insertion Sort (Algorithm 7.1 ) more appropriate when we need to find in nonincreasing order the \(k\) largest (or in nondecreasing order the \(k\) smallest) keys in a list of \(n\) keys? Justify your answer.
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