# Sorting Algorithms

it is sometimes useful to know if a sorting algorithm is stable
A sorting algorithm is stable if it preserves the original order of elements with equal key values (where the key is the value the algorithm sorts by). For example,

#### Bubble Sort

Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent elements if they are in wrong order.
Example:

First Pass:
( 5 1 4 2 8 ) –> ( 1 5 4 2 8 ), Here, algorithm compares the first two elements, and swaps since 5 > 1.
( 1 5 4 2 8 ) –>  ( 1 4 5 2 8 ), Swap since 5 > 4
( 1 4 5 2 8 ) –>  ( 1 4 2 5 8 ), Swap since 5 > 2
( 1 4 2 5 8 ) –> ( 1 4 2 5 8 ), Now, since these elements are already in order (8 > 5), algorithm does not swap them.
Second Pass:
( 1 4 2 5 8 ) –> ( 1 4 2 5 8 )
( 1 4 2 5 8 ) –> ( 1 2 4 5 8 ), Swap since 4 > 2
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –>  ( 1 2 4 5 8 )
Now, the array is already sorted, but our algorithm does not know if it is completed. The algorithm needs one whole pass without any swap to know it is sorted.
Third Pass:
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
template<class T>
void Swap(T* p1,T* p2){
T p3=*p1;
*p1=*p2;
*p2=p3;
return;
}

template<typename T ,int size>
void BubbleSort(T (&t)[size]){
for(int i=1;i<size;i++){
for(int ii=0;ii<size-i;ii++){
if(t[ii]>t[ii+1])
Swap(t+(ii+1),t+(ii));
}
}
}

#### Insertion Sort

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;
}
}

#### Merge Sort

void merge(int array[], int const left, int const mid, int const right)
{
auto const subArrayOne = mid - left + 1;
auto const subArrayTwo = right - mid;

auto *leftArray = new int[subArrayOne],
*rightArray = new int[subArrayTwo];

for (auto i = 0; i < subArrayOne; i++)
leftArray[i] = array[left + i];
for (auto j = 0; j < subArrayTwo; j++)
rightArray[j] = array[mid + 1 + j];

auto indexOfSubArrayOne = 0,
indexOfSubArrayTwo = 0;
int indexOfMergedArray = left;

while (indexOfSubArrayOne < subArrayOne && indexOfSubArrayTwo < subArrayTwo) {
if (leftArray[indexOfSubArrayOne] <= rightArray[indexOfSubArrayTwo]) {
array[indexOfMergedArray] = leftArray[indexOfSubArrayOne];
indexOfSubArrayOne++;
}
else {
array[indexOfMergedArray] = rightArray[indexOfSubArrayTwo];
indexOfSubArrayTwo++;
}
indexOfMergedArray++;
}

while (indexOfSubArrayOne < subArrayOne) {
array[indexOfMergedArray] = leftArray[indexOfSubArrayOne];
indexOfSubArrayOne++;
indexOfMergedArray++;
}

while (indexOfSubArrayTwo < subArrayTwo) {
array[indexOfMergedArray] = rightArray[indexOfSubArrayTwo];
indexOfSubArrayTwo++;
indexOfMergedArray++;
}
}

void mergeSort(int array[], int const begin, int const end)
{
if (begin >= end)
return;
auto mid = begin + (end - begin) / 2;
mergeSort(array, begin, mid);
mergeSort(array, mid + 1, end);
merge(array, begin, mid, end);
}

#### Quick Sort

The key process in quickSort is partition(). Target of partitions is, given an array and an element x of array as pivot, put x at its correct position in sorted array and put all smaller elements (smaller than x) before x, and put all greater elements (greater than x) after x. All this should be done in linear time.

#include <bits/stdc++.h>
using namespace std;

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]; // pivot
int i = (low - 1); // Index of smaller element and indicates the right position of pivot found so far

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);
}
}

#### Counting Sort

void countSort(char arr[])
{
char output[strlen(arr)];
int count[RANGE + 1], i;
memset(count, 0, sizeof(count));
for (i = 0; arr[i]; ++i)
++count[arr[i]];
for (i = 1; i <= RANGE; ++i)
count[i] += count[i - 1];
for (i = 0; arr[i]; ++i) {
output[count[arr[i]] - 1] = arr[i];
--count[arr[i]];
}
for (i = 0; arr[i]; ++i)
arr[i] = output[i];
}
void countSort(vector<int>& arr)
{
int max = *max_element(arr.begin(), arr.end());
int min = *min_element(arr.begin(), arr.end());
int range = max - min + 1;

vector<int> count(range), output(arr.size());
for (int i = 0; i < arr.size(); i++)
count[arr[i] - min]++;

for (int i = 1; i < count.size(); i++)
count[i] += count[i - 1];

for (int i = arr.size() - 1; i >= 0; i--) {
output[count[arr[i] - min] - 1] = arr[i];
count[arr[i] - min]--;
}

for (int i = 0; i < arr.size(); i++)
arr[i] = output[i];
}