Q.5.
Write algorithms and analyse (Time and Space) the following sor ng algorithms: Bubble Sort,
Selec on Sort, Inser on Sort, Merge Sort, Quick Sort, and Heap Sort. Implement each of the
algorithms in C language.
ANS :-
BUBBLE SORT
ALGORITHM –
BubbleSort(array, n)
for i = 0 to n-1
for j = 0 to n-i-2
if array[j] > array[j+1]
swap(array[j], array[j+1])
end for
end BubbleSort
Time Complexity:
Best Case: O(n) (When the array is already sorted)
Average Case: O(n^2)
Worst Case: O(n^2)
Space Complexity:
Auxiliary Space: O(1) (In-place sorting)
C Implimentation
#include <stdio.h>
void bubbleSort(int arr[], int n) {
int i, j, temp;
for (i = 0; i < n-1; i++) {
for (j = 0; j < n-i-1; j++) {
if (arr[j] > arr[j+1]) {
temp = arr[j];
arr[j] = arr[j+1];
arr[j+1] = temp;
}
}
}
}
void printArray(int arr[], int n) {
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
printf("\n");
}
int main() {
int arr[] = {64, 34, 25, 12, 22, 11, 90};
int n = sizeof(arr) / sizeof(arr[0]);
bubbleSort(arr, n);
printf("Sorted array: \n");
printArray(arr, n);
return 0;
}
SELECTION SORT
ALGORITHM -
SelectionSort(array, n)
for i = 0 to n-1
minIndex = i
for j = i+1 to n-1
if array[j] < array[minIndex]
minIndex = j
end for
swap(array[minIndex], array[i])
end for
end SelectionSort
Time Complexity:
Best, Average, and Worst Case: O(n^2)
Space Complexity:
Auxiliary Space: O(1) (In-place sorting)
C Implementation –
#include <stdio.h>
void selectionSort(int arr[], int n) {
int i, j, min_idx, temp;
for (i = 0; i < n-1; i++) {
min_idx = i;
for (j = i+1; j < n; j++)
if (arr[j] < arr[min_idx])
min_idx = j;
temp = arr[min_idx];
arr[min_idx] = arr[i];
arr[i] = temp;
}
}
void printArray(int arr[], int n) {
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
printf("\n");
}
int main() {
int arr[] = {64, 25, 12, 22, 11};
int n = sizeof(arr) / sizeof(arr[0]);
selectionSort(arr, n);
printf("Sorted array: \n");
printArray(arr, n);
return 0;
}
INSERTION SORT
ALGORITHM -
InsertionSort(array, n)
for i = 1 to n-1
key = array[i]
j = i-1
while j >= 0 and array[j] > key
array[j+1] = array[j]
j = j-1
end while
array[j+1] = key
end for
end InsertionSort
Time Complexity:
Best Case: O(n)
Average Case: O(n^2)
Worst Case: O(n^2)
Space Complexity:
Auxiliary Space: O(1)
C Implementation –
#include <stdio.h>
void insertionSort(int arr[], int n) {
int i, key, j;
for (i = 1; i < n; i++) {
key = arr[i];
j = i - 1;
while (j >= 0 && arr[j] > key) {
arr[j + 1] = arr[j];
j = j - 1;
}
arr[j + 1] = key;
}
}
void printArray(int arr[], int n) {
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
printf("\n");
}
int main() {
int arr[] = {12, 11, 13, 5, 6};
int n = sizeof(arr) / sizeof(arr[0]);
insertionSort(arr, n);
printf("Sorted array: \n");
printArray(arr, n);
return 0;
}
MERGE SORT
ALGORITHM -
MergeSort(array, left, right)
if left < right
mid = (left + right) / 2
MergeSort(array, left, mid)
MergeSort(array, mid + 1, right)
Merge(array, left, mid, right)
end if
end MergeSort
Merge(array, left, mid, right)
create temp arrays L and R
copy elements into L and R from array
i, j, k = 0
while L and R have elements
if L[i] <= R[j]
array[k] = L[i]
i++
else
array[k] = R[j]
j++
end if
k++
end while
copy remaining elements of L and R into array
end Merge
Time Complexity:
Best, Average, and Worst Case: O(nlogn)
Space Complexity:
Auxiliary Space: O(n)
C Implementation –
#include <stdio.h>
#include <stdlib.h>
void merge(int arr[], int l, int m, int r) {
int i, j, k;
int n1 = m - l + 1;
int n2 = r - m;
int L[n1], R[n2];
for (i = 0; i < n1; i++)
L[i] = arr[l + i];
for (j = 0; j < n2; j++)
R[j] = arr[m + 1 + j];
i = 0; j = 0; k = l;
while (i < n1 && j < n2) {
if (L[i] <= R[j]) {
arr[k] = L[i];
i++;
} else {
arr[k] = R[j];
j++;
}
k++;
}
while (i < n1) {
arr[k] = L[i];
i++;
k++;
}
while (j < n2) {
arr[k] = R[j];
j++;
k++;
}
}
void mergeSort(int arr[], int l, int r) {
if (l < r) {
int m = l + (r - l) / 2;
mergeSort(arr, l, m);
mergeSort(arr, m + 1, r);
merge(arr, l, m, r);
}
}
void printArray(int arr[], int n) {
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
printf("\n");
}
int main() {
int arr[] = {12, 11, 13, 5, 6, 7};
int arr_size = sizeof(arr) / sizeof(arr[0]);
mergeSort(arr, 0, arr_size - 1);
printf("Sorted array: \n");
printArray(arr, arr_size);
return 0;
}
QUICK SORT
ALGORITHM -
QuickSort(array, low, high)
if low < high
pivotIndex = Partition(array, low, high)
QuickSort(array, low, pivotIndex - 1)
QuickSort(array, pivotIndex + 1, high)
end if
end QuickSort
Partition(array, low, high)
pivot = array[high]
i = low - 1
for j = low to high-1
if array[j] < pivot
i++
swap(array[i], array[j])
end if
end for
swap(array[i+1], array[high])
return i+1
end Partition
Time Complexity:
Best and Average Case: O(nlogn)
Worst Case: O(n^2)
Space Complexity:
Auxiliary Space: O(logn)
C Implementation –
#include <stdio.h>
void swap(int* a, int* b) {
int t = *a;
*a = *b;
*b = t;
}
int partition(int arr[], int low, int high) {
int pivot = arr[high];
int i = (low - 1);
for (int j = low; j <= high - 1; j++) {
if (arr[j] < pivot) {
i++;
swap(&arr[i], &arr[j]);
}
}
swap(&arr[i + 1], &arr[high]);
return (i + 1);
}
void quickSort(int arr[], int low, int high) {
if (low < high) {
int pi = partition(arr, low, high);
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
void printArray(int arr[], int size) {
for (int i = 0; i < size; i++)
printf("%d ", arr[i]);
printf("\n");
}
int main() {
int arr[] = {10, 7, 8, 9, 1, 5};
int n = sizeof(arr) / sizeof(arr[0]);
quickSort(arr, 0, n - 1);
printf("Sorted array: \n");
printArray(arr, n);
return 0;
}
HEAP SORT
ALGORITHM -
HeapSort(array, n)
BuildMaxHeap(array, n)
for i = n-1 to 1
swap(array[0], array[i])
MaxHeapify(array, 0, i)
end for
end HeapSort
BuildMaxHeap(array, n)
for i = n/2 - 1 to 0
MaxHeapify(array, i, n)
end for
end BuildMaxHeap
MaxHeapify(array, i, n)
largest = i
left = 2*i + 1
right = 2*i + 2
if left < n and array[left] > array[largest]
largest = left
end if
if right < n and array[right] > array[largest]
largest = right
end if
if largest != i
swap(array[i], array[largest])
MaxHeapify(array, largest, n)
end if
end MaxHeapify
Overall Time Complexity:
Building the heap: O(n)
Extracting elements: O(nlogn)
Total Time Complexity: O(nlogn)
Space Complexity:
Auxiliary Space:O(1) because Heap Sort is an in-place sorting algorithm.
C Implementation –
#include <stdio.h>
void swap(int *a, int *b) {
int temp = *a;
*a = *b;
*b = temp;
}
void heapify(int arr[], int n, int i) {
int largest = i; // Initialize largest as root
int left = 2 * i + 1; // left child
int right = 2 * i + 2; // right child
if (left < n && arr[left] > arr[largest])
largest = left;
if (right < n && arr[right] > arr[largest])
largest = right;
if (largest != i) {
swap(&arr[i], &arr[largest]);
heapify(arr, n, largest);
}
}
void heapSort(int arr[], int n) {
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
for (int i = n - 1; i > 0; i--) {
swap(&arr[0], &arr[i]);
heapify(arr, i, 0);
}
}
void printArray(int arr[], int n) {
for (int i = 0; i < n; ++i)
printf("%d ", arr[i]);
printf("\n");
}
int main() {
int arr[] = {12, 11, 13, 5, 6, 7};
int n = sizeof(arr) / sizeof(arr[0]);
printf("Original array:\n");
printArray(arr, n);
heapSort(arr, n);
printf("Sorted array using Heap Sort:\n");
printArray(arr, n);
return 0;
}
