(a)
To show that the insertion sort can sort the
(a)
Explanation of Solution
The procedure of insertion sort in non-increasing order is as below:
INSERTION-SORT(A)
For
while
The j loop is running from 2 to the length of the array and then value in the index of array is stored in the key variable. After that the variable j is decreased by 1 and stored in the variable i.
Now, while loop is used in which two conditions are given: first condition is variable i is greater than 0 and the value of the array at index i must be less than key value.
The value at index i is swapped at the index
Analysis:
The worst case running time of the insertion sort to sort a list of length n is
sub lists of length k , the time taken by insertion sort is calculated as follows:
Hence, the worst case running time of insertion sort to sort the
(b)
To show that the worst case running time to merge the sub lists is
(b)
Explanation of Solution
There are
For this, take two sublists and merge them simultaneously. The time taken by this process is
Hence, the worst case running time to merge the sub listsis
(c)
To find the largest value of k as a function of n so that the modified
(c)
Explanation of Solution
Merge sort is a sorting algorithm that uses divide and conquer method. In merge sort, divide the input sequence in two halves and then sort them, Finally, merge the sorted halves.
The modified algorithm has same running time as standard merge sort if
Hence, the running time of the modified algorithm has same as standard merge sort if
(d)
To choose the value of k in practice.
(d)
Explanation of Solution
Merge sort is a sorting algorithm that uses the divide and conquer method. In merge sort, divide the input sequence in two halves and then sort them. Finally, merge the sorted halves.
For considering the value of k, choose the largest length of sub list as kso that insertion sort becomes faster than merge sort.
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Chapter 2 Solutions
Introduction to Algorithms
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