The document describes sorting algorithms, specifically merge sort and radix sort, detailing their methods of operation and implementation. Merge sort, invented by John von Neumann, utilizes a divide-and-conquer approach to recursively sort data by merging sublists, while radix sort categorizes integers based on their digits and processes them in order. It includes C++ code examples for both algorithms illustrating their execution and sorting capabilities.

2.  A sorting technique that sequences data by
continuously merging items in the list, every
single item in the original unordered list is
merged with another, creating groups of two.
Every two-item group is merged, creating
groups of four and so on until there is one
ordered list. See sort algorithm and merge.

3.  Invented by an American mathematician John
von Neumann in 1945
 One of the first sorting styles proposed for
computers

4. Top-down implementation
 Top down merge sort algorithm that
recursively splits the list (called runs in this
example) into sublists until sublist size is 1,
then merges those sublists to produce a
sorted list. The copy back step is avoided
with alternating the direction of the merge
with each level of recursion.

5. Bottom-up implementation
 Bottom up merge sort algorithm which treats the
list as an array of n sublists (called runs in this
example) of size 1, and iteratively merges sub-
lists back and forth between two buffers:

 Top down merge sort algorithm which recursively
divides the input list into smaller sublists until
the sublists are trivially sorted, and then merges
the sublists while returning up the call chain.

6. To understand merge sort, we take an
unsorted array as the following −
We see here that an array of 6 items is divided
into two arrays of size 3.
23 8 3 13 88 87
23 8 3 13 88 87

7. Now we divide these two arrays into two’s and
one’s.
We further divide these arrays and we achieve
atomic value which can no more be divided.
23 8 3 13 88 87
23 8 3 13 88 87

8. We first compare the element for each list and
then combine them into another list in a sorted
manner. We compare 23 and 8 in the target list
and we put 8 first then 23. Since 3 has no
partner we put it in same place. We can see
that 13 and 88 are already sorted so we put
them sequentially followed by 87.
238 3 13 88 87

9. In the next iteration of the combining phase,
we compare lists of two data values, and merge
them into a list of found data values placing all
in a sorted order.
After the final merging, the list should look like
this −
2383 13 8887
83 13 23 87 88

10. #include <iostream>
using namespace std;
// A function to merge the two half into a sorted data.
void Merge(int *a, int low, int high, int mid)
{
// We have low to mid and mid+1 to high already
sorted.
int i, j, k, temp[high-low+1];
i = low;
k = 0;
j = mid + 1;

11. // Merge the two parts into temp[].
while (i <= mid && j <= high)
{
if (a[i] < a[j])
{
temp[k] = a[i];
k++;
i++;
}

12. else
{
temp[k] = a[j];
k++;
j++;
}
}

13. // Insert all the remaining values from i to mid
into temp[].
while (i <= mid)
{
temp[k] = a[i];
k++;
i++;
}

14. // Insert all the remaining values from j to high
into temp[].
while (i <= mid)
{
temp[k] = a[j];
k++;
j++;
}

15. // Assign sorted data stored in temp[] to a[].
for (i = low; i <= high; i++)
{
a[i] = temp[i-low];
}
}

16. // A function to split array into two parts.
void MergeSort(int *a, int low, int high)
{
int mid;
if (low < high)
{
mid=(low+high)/2;
// Split the data into two half.
MergeSort(a, low, mid);
MergeSort(a, mid+1, high);
// Merge them to get sorted output.
Merge(a, low, high, mid);
}
}

17. int main()
{
int n, i;
cout<<"nEnter the number of data element to
be sorted: ";
cin>>n;
int arr[n];
for(i = 0; i < n; i++)
{
cout<<"Enter element "<<i+1<<": ";
cin>>arr[i];
}

18. MergeSort(arr, 0, n-1);
// Printing the sorted data.
cout<<"nSorted Data ";
for (i = 0; i < n; i++)
cout<<"->"<<arr[i];
return 0;
}

19.  Unsorted Data : 23, 8 ,3 ,13 ,8 ,87

21.  In computer science, radix sort is a non-
comparative integer sorting algorithm that
sorts data with integer keys by grouping keys
by the individual digits which share the same
significant position and value.

22.  Radix Sort dates back as far as 1887 to the
work of Herman Hollerith on tabulating
machines.
 Radix Sort is an algorithm that sorts a list of
numbers and comes under the category of
distribution sort

23. Least Significant Digit (LSD)
 A least significant digit (LSD) Radix Sort is a fast
stable sorting algorithm which can be used to
sort keys in integer representation order.
 Keys may be a string of characters, or numerical
digits in a given ‘radix’.
 The processing of the keys begins at the least
significant digit (the rightmost digit), and
proceeds to the most significant digit (the
leftmost digit).

24. Most Significant Digit (MSD)
 A most significant digit (MSD) radix sort can
be used to sort keys in lexicographic order.
 Unlike a least significant digit (LSD) radix
sort, a most significant digit radix sort does
not necessarily preserve the original order of
duplicate keys.
 An MSD radix sort starts processing the keys
from the most significant digit, leftmost digit,
to the least significant digit, rightmost digit

25. Radix sort works by sorting each digit from least
significant digit to most significant digit. Consider this
example:
88, 8, 23, 3, 2, 29, 177, 40
Our task is to sort them in ascending order. Since there
are 10 digits from 0 to 9 so we need to label arrays
from 0 to 9. So 177 is the biggest number among the
unsorted array and it has 3 digits. We will fill all the
numbers with 0s that are smaller than 177. It goes like
this:
088, 008, 023, 003, 002, 029, 177, 040

26. Since all of them have 3 digits this will require
3 pass.
Pass 1: sort the arrays using first digit from the
right most part.
088, 008, 023, 003, 002, 029, 177, 040
Pass 1 is complete.
04
0
00
2
00
3
02
3
17
7
00
8
08
8
02
9

27. Take the numbers from the bucket and arrange
them from bottom to top.
040, 002, 023, 003, 177, 088, 008, 029
Pass 2: Sort the numbers by using the 2nd digit
from right.
040, 002, 023, 003, 177, 088, 008, 029
Pass 2 is complete.
00
8
00
3
00
2
02
9
02
3
04
0
17
7
08
8

28. Take the numbers from the bucket and arrange them from
bottom to top. It should be arranged like this.
002, 003, 008, 023, 029, 040, 177, 088
Pass 3: Sort the numbers by using the 3rd digit
from left.
002, 003, 008, 023, 029, 040, 177, 088
088
040
029
023
008
003
002
177

29. Pass 3 is complete.
Take the numbers from the bucket and
arrange them from bottom to top.
002, 003, 008, 023, 029, 040, 088, 177
We have completed the 3 passes so we will
now stop here. Remove now all the leading
0s.
2, 3, 8, 23, 29, 40, 88, 177
So the numbers are now sorted in
ascending order.

30. #include <iostream>
using namespace std;
// A utility function to get maximum value in arr[]
int getMax(int arr[], int n)
{
int mx = arr[0];
for (int i = 1; i < n; i++)
if (arr[i] > mx)
mx = arr[i];
return mx;
}

31. // A function to do counting sort of arr[] according
to
// the digit represented by exp.
void countSort(int arr[], int n, int exp)
{
int output[n]; // output array
int i, count[10] = {0};
// Store count of occurrences in count[]
for (i = 0; i < n; i++)
count[ (arr[i]/exp)%10 ]++;

32. // Change count[i] so that count[i] now contains
actual
// position of this digit in output[]
for (i = 1; i < 10; i++)
count[i] += count[i - 1];
// Build the output array
for (i = n - 1; i >= 0; i--)
{
output[count[ (arr[i]/exp)%10 ] - 1] = arr[i];
count[ (arr[i]/exp)%10 ]--;
}

33. // Copy the output array to arr[], so that arr[] now
// contains sorted numbers according to current
digit
for (i = 0; i < n; i++)
arr[i] = output[i];
}
// The main function to that sorts arr[] of size n
using
// Radix Sort
void radixsort(int arr[], int n)
{

34. // Find the maximum number to know number of
digits
int m = getMax(arr, n);
// Do counting sort for every digit. Note that
instead
// of passing digit number, exp is passed. exp
is 10^i
// where i is current digit number
for (int exp = 1; m/exp > 0; exp *= 10)
countSort(arr, n, exp);
}

35. // A utility function to print an array
void print(int arr[], int n)
{
for (int i = 0; i < n; i++)
cout << arr[i] << " ";
}

36. // Driver program to test above functions
int main()
{
// arr[] = {170, 45, 75, 90, 802, 24, 2, 66};
//int n = sizeof(arr)/sizeof(arr[0]);
int n;
cout << "How many numbers? "; cin >> n;
int arr[n];

37. cout << "Enter the numbers:n";
for (int i = 0;i < n;i++) {
cout << "t";
cin >> arr[i];
}
radixsort(arr, n);
print(arr, n);
return 0;
}

38. Unsorted array a = {3, 10, 9, 8, 53, 2, 62, 71}