# QuickSort using Random Pivoting

In this article we will discuss how to implement QuickSort using random pivoting. In QuickSort we first partition the array in place such that all elements to the left of the pivot element are smaller, while all elements to the right of the pivot are greater that the pivot. Then we recursively call the same procedure for left and right subarrays.

Unlike merge sort we don’t need to merge the two sorted arrays. Thus Quicksort requires lesser auxiliary space than Merge Sort, which is why it is often preferred to Merge Sort.Using a randomly generated pivot we can further improve the time complexity of QuickSort.

We have discussed at two popular methods for partitioning the arrays-Hoare’s vs Lomuto partition scheme
It is advised that the reader has read that article or knows how to implement the QuickSort using either of the two partition schemes.

Algorithm for random pivoting using Lomuto Partitioning

```
partition(arr[], lo, hi)
pivot = arr[hi]
i = lo     // place for swapping
for j := lo to hi – 1 do
if arr[j] <= pivot then
swap arr[i] with arr[j]
i = i + 1
swap arr[i] with arr[hi]
return i

partition_r(arr[], lo, hi)
r = Random Number from lo to hi
Swap arr[r] and arr[hi]
return partition(arr, lo, hi)

quicksort(arr[], lo, hi)
if lo < hi
p = partition_r(arr, lo, hi)
quicksort(arr, p-1, hi)
quicksort(arr, p+1, hi)

```

Algorithm for random pivoting using Hoare Partitioning

```partition(arr[], lo, hi)
pivot = arr[lo]
i = lo - 1  // Initialize left index
j = hi + 1  // Initialize right index

// Find a value in left side greater
// than pivot
do
i = i + 1
while arr[i]  pivot

if i >= j then
return j

swap arr[i] with arr[j]

partition_r(arr[], lo, hi)
r = Random number from lo to hi
Swap arr[r] and arr[lo]
return partition(arr, lo, hi)

quicksort(arr[], lo, hi)
if lo < hi
p = partition_r(arr, lo, hi)
quicksort(arr, p, hi)
quicksort(arr, p+1, hi)

```

## Lomuto (C++)

 `/* C++ implementation QuickSort using Lomuto's partition ` `   ``Scheme.*/` `#include ` `#include ` `using` `namespace` `std; ` ` `  `/* This function takes last element as pivot, places ` `  ``the pivot element at its correct position in sorted ` `  ``array, and places all smaller (smaller than pivot) ` `  ``to left of pivot and all greater elements to right ` `  ``of pivot */` `int` `partition(``int` `arr[], ``int` `low, ``int` `high) ` `{ ` `    ``int` `pivot = arr[high]; ``// pivot ` `    ``int` `i = (low - 1); ``// Index of smaller element ` ` `  `    ``for` `(``int` `j = low; j <= high - 1; j++) { ` ` `  `        ``// If current element is smaller than or ` `        ``// equal to pivot ` `        ``if` `(arr[j] <= pivot) { ` ` `  `            ``i++; ``// increment index of smaller element ` `            ``swap(arr[i], arr[j]); ` `        ``} ` `    ``} ` `    ``swap(arr[i + 1], arr[high]); ` `    ``return` `(i + 1); ` `} ` ` `  `// Generates Random Pivot, swaps pivot with ` `// end element and calls the partition function ` `int` `partition_r(``int` `arr[], ``int` `low, ``int` `high) ` `{ ` `    ``// Generate a random number in between ` `    ``// low .. high ` `    ``srand``(``time``(NULL)); ` `    ``int` `random = low + ``rand``() % (high - low); ` ` `  `    ``// Swap A[random] with A[high] ` `    ``swap(arr[random], arr[high]); ` ` `  `    ``return` `partition(arr, low, high); ` `} ` ` `  `/* The main function that implements QuickSort ` `arr[] --> Array to be sorted, ` `low --> Starting index, ` `high --> Ending index */` `void` `quickSort(``int` `arr[], ``int` `low, ``int` `high) ` `{ ` `    ``if` `(low < high) { ` ` `  `        ``/* pi is partitioning index, arr[p] is now ` `        ``at right place */` `        ``int` `pi = partition_r(arr, low, high); ` ` `  `        ``// Separately sort elements before ` `        ``// partition and after partition ` `        ``quickSort(arr, low, pi - 1); ` `        ``quickSort(arr, pi + 1, high); ` `    ``} ` `} ` ` `  `/* Function to print an array */` `void` `printArray(``int` `arr[], ``int` `size) ` `{ ` `    ``int` `i; ` `    ``for` `(i = 0; i < size; i++) ` `        ``printf``(``"%d "``, arr[i]); ` `    ``printf``(``"\n"``); ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``int` `arr[] = { 10, 7, 8, 9, 1, 5 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``quickSort(arr, 0, n - 1); ` `    ``printf``(``"Sorted array: \n"``); ` `    ``printArray(arr, n); ` `    ``return` `0; ` `} `

## Hoare (C++)

 `/* C++ implementation of QuickSort using Hoare's ` `   ``partition scheme. */` `#include ` `#include ` `using` `namespace` `std; ` ` `  `/* This function takes last element as pivot, places ` `   ``the pivot element at its correct position in sorted ` `    ``array, and places all smaller (smaller than pivot) ` `   ``to left of pivot and all greater elements to right ` `   ``of pivot */` `int` `partition(``int` `arr[], ``int` `low, ``int` `high) ` `{ ` `    ``int` `pivot = arr[low]; ` `    ``int` `i = low - 1, j = high + 1; ` ` `  `    ``while` `(``true``) { ` ` `  `        ``// Find leftmost element greater than ` `        ``// or equal to pivot ` `        ``do` `{ ` `            ``i++; ` `        ``} ``while` `(arr[i] < pivot); ` ` `  `        ``// Find rightmost element smaller than ` `        ``// or equal to pivot ` `        ``do` `{ ` `            ``j--; ` `        ``} ``while` `(arr[j] > pivot); ` ` `  `        ``// If two pointers met. ` `        ``if` `(i >= j) ` `            ``return` `j; ` ` `  `        ``swap(arr[i], arr[j]); ` `    ``} ` `} ` ` `  `// Generates Random Pivot, swaps pivot with ` `// end element and calls the partition function ` `// In Hoare partition the low element is selected ` `// as first pivot ` `int` `partition_r(``int` `arr[], ``int` `low, ``int` `high) ` `{ ` `    ``// Generate a random number in between ` `    ``// low .. high ` `    ``srand``(``time``(NULL)); ` `    ``int` `random = low + ``rand``() % (high - low); ` ` `  `    ``// Swap A[random] with A[high] ` `    ``swap(arr[random], arr[low]); ` ` `  `    ``return` `partition(arr, low, high); ` `} ` ` `  `/* The main function that implements QuickSort ` ` ``arr[] --> Array to be sorted, ` `  ``low  --> Starting index, ` `  ``high  --> Ending index */` `void` `quickSort(``int` `arr[], ``int` `low, ``int` `high) ` `{ ` `    ``if` `(low < high) { ` `        ``/* pi is partitioning index, arr[p] is now ` `           ``at right place */` `        ``int` `pi = partition_r(arr, low, high); ` ` `  `        ``// Separately sort elements before ` `        ``// partition and after partition ` `        ``quickSort(arr, low, pi); ` `        ``quickSort(arr, pi + 1, high); ` `    ``} ` `} ` ` `  `/* Function to print an array */` `void` `printArray(``int` `arr[], ``int` `n) ` `{ ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``printf``(``"%d "``, arr[i]); ` `    ``printf``(``"\n"``); ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``int` `arr[] = { 10, 7, 8, 9, 1, 5 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``quickSort(arr, 0, n - 1); ` `    ``printf``(``"Sorted array: \n"``); ` `    ``printArray(arr, n); ` `    ``return` `0; ` `} `

## Java

 `// Java program to illustrate Randomised Quick Sort  ` `import` `java.util.*; ` ` `  `class` `RandomisedQuickSort  ` `{  ` `    ``public` `static` `int` `N = ``5``; ` `    ``public` `static` `int``[] arr = ``new` `int``[N]; ` `     `  `    ``// This Function helps in calculating random  ` `    ``// numbers between low(inclusive) and high(inclusive) ` `    ``void` `random(``int` `low,``int` `high) ` `    ``{ ` `     `  `        ``Random rand= ``new` `Random(); ` `        ``int` `pivot = rand.nextInt(high-low) + low; ` `         `  `        ``int` `temp1=arr[pivot]; ` `        ``arr[pivot]=arr[high]; ` `        ``arr[high]=temp1; ` `    ``} ` `     `  `    ``/* This function takes last element as pivot,  ` `    ``places the pivot element at its correct  ` `    ``position in sorted array, and places all  ` `    ``smaller (smaller than pivot) to left of  ` `    ``pivot and all greater elements to right  ` `    ``of pivot */` `     `  `    ``int` `partition(``int` `arr[], ``int` `low, ``int` `high)  ` `    ``{  ` `        ``// pivot is choosen randomly ` `        ``int` `pivot = arr[high];  ` `     `  ` `  `        ``int` `i = (low-``1``); ``// index of smaller element  ` `        ``for` `(``int` `j = low; j < high; j++)  ` `        ``{  ` `            ``// If current element is smaller than or  ` `            ``// equal to pivot  ` `            ``if` `(arr[j] <= pivot)  ` `            ``{  ` `                ``i++;  ` ` `  `                ``// swap arr[i] and arr[j]  ` `                ``int` `temp = arr[i];  ` `                ``arr[i] = arr[j];  ` `                ``arr[j] = temp;  ` `            ``}  ` `        ``}  ` ` `  `        ``// swap arr[i+1] and arr[high] (or pivot)  ` `        ``int` `temp = arr[i+``1``];  ` `        ``arr[i+``1``] = arr[high];  ` `        ``arr[high] = temp;  ` ` `  `        ``return` `i+``1``;  ` `    ``}  ` ` `  ` `  `    ``/* The main function that implements QuickSort()  ` `    ``arr[] --> Array to be sorted,  ` `    ``low --> Starting index,  ` `    ``high --> Ending index */` `    ``void` `sort(``int` `arr[], ``int` `low, ``int` `high)  ` `    ``{  ` `        ``if` `(low < high)  ` `        ``{  ` `            ``/* pi is partitioning index, arr[pi] is  ` `            ``now at right place */` `            ``int` `pi = partition(arr, low, high);  ` ` `  `            ``// Recursively sort elements before  ` `            ``// partition and after partition  ` `            ``sort(arr, low, pi-``1``);  ` `            ``sort(arr, pi+``1``, high);  ` `        ``}  ` `    ``}  ` ` `  `    ``/* A utility function to print array of size n */` `    ``static` `void` `printArray(``int` `arr[])  ` `    ``{  ` `        ``int` `n = arr.length;  ` `        ``for` `(``int` `i = ``0``; i < n; ++i)  ` `            ``System.out.print(arr[i]+``" "``);  ` `        ``System.out.println();  ` `    ``}  ` ` `  `    ``// Driver code  ` `    ``public` `static` `void` `main(String args[])  ` `    ``{  ` `        ``int` `arr[] = {``10``, ``7``, ``8``, ``9``, ``1``, ``5``};  ` `        ``int` `n = arr.length;  ` ` `  `        ``RandomisedQuickSort ob = ``new` `RandomisedQuickSort();  ` `        ``ob.sort(arr, ``0``, n-``1``);  ` ` `  `        ``System.out.println(``"sorted array"``);  ` `        ``printArray(arr);  ` `    ``}  ` `}  ` ` `  `// This code is contributed by Ritika Gupta. `

## Lomuto(Python)

 `# Python implementation QuickSort using  ` `# Lomuto's partition Scheme. ` `import` `random ` ` `  `''' ` `The function which implements QuickSort. ` `arr :- array to be sorted. ` `start :- starting index of the array. ` `stop :- ending index of the array. ` `'''` `def` `quicksort(arr, start , stop): ` `    ``if``(start < stop): ` `         `  `        ``# pivotindex is the index where  ` `        ``# the pivot lies in the array ` `        ``pivotindex ``=` `partitionrand(arr, start, stop) ` `         `  `        ``# At this stage the array is partially sorted  ` `        ``# around the pivot. Separately sorting the  ` `        ``# left half of the array and the right half of the array. ` `        ``quicksort(arr , start , pivotindex ``-` `1``) ` `        ``quicksort(arr, pivotindex ``+` `1``, stop) ` ` `  `# This function generates random pivot, swaps the first ` `# element with the pivot and calls the partition fucntion. ` `def` `partitionrand(arr , start, stop): ` ` `  `    ``# Generating a random number between the  ` `    ``# starting index of the array and the ` `    ``# ending index of the array. ` `    ``randpivot ``=` `random.randrange(start, stop) ` ` `  `    ``# Swapping the starting element of the array and the pivot ` `    ``arr[start], arr[randpivot] ``=` `arr[randpivot], arr[start] ` `    ``return` `partition(arr, start, stop) ` ` `  `''' ` `This function takes the first element as pivot,  ` `places the pivot element at the correct position  ` `in the sorted array. All the elements are re-arranged  ` `according to the pivot, the elements smaller than the ` `pivot is places on the left and the elements ` `greater than the pivot is placed to the right of pivot. ` `'''` `def` `partition(arr,start,stop): ` `    ``pivot ``=` `start ``# pivot ` `    ``i ``=` `start ``+` `1` `# a variable to memorize where the  ` `                  ``# partition in the array starts from. ` `    ``for` `j ``in` `range``(start ``+` `1``, stop ``+` `1``): ` `         `  `        ``# if the current element is smaller or equal to pivot, ` `        ``# shift it to the left side of the partition. ` `        ``if` `arr[j] <``=` `arr[pivot]: ` `            ``arr[i] , arr[j] ``=` `arr[j] , arr[i] ` `            ``i ``=` `i ``+` `1` `    ``arr[pivot] , arr[i ``-` `1``] ``=` `arr[i ``-` `1``] , arr[pivot] ` `    ``pivot ``=` `i ``-` `1` `    ``return` `(pivot) ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"``: ` `    ``array ``=` `[``10``, ``7``, ``8``, ``9``, ``1``, ``5``] ` `    ``quicksort(array, ``0``, ``len``(array) ``-` `1``) ` `    ``print``(array) ` ` `  `# This code is contributed by soumyasaurav `

## Haore(Python)

 `# Python implementation QuickSort using  ` `# Hoare's partition Scheme. ` ` `  `import` `random ` ` `  `''' ` `The function which implements randomised QuickSort, ` `using Haore's partition scheme. ` `arr :- array to be sorted. ` `start :- starting index of the array. ` `stop :- ending index of the array. ` `'''` `def` `quicksort(arr, start, stop): ` `    ``if``(start < stop): ` `         `  `        ``# pivotindex is the index where ` `        ``# the pivot lies in the array ` `        ``pivotindex ``=` `partitionrand(arr, start, stop) ` `         `  `        ``# At this stage the array is partially sorted  ` `        ``# around the pivot. separately sorting the  ` `        ``# left half of the array and the right half of the array. ` `        ``quicksort(arr , start , pivotindex) ` `        ``quicksort(arr, pivotindex ``+` `1``, stop) ` ` `  `# This function generates random pivot, swaps the first ` `# element with the pivot and calls the partition function. ` `def` `partitionrand(arr , start, stop): ` ` `  `    ``# Generating a random number between  ` `    ``# the starting index of the array and  ` `    ``# the ending index of the array. ` `    ``randpivot ``=` `random.randrange(start, stop) ` ` `  `    ``# Swapping the starting element of  ` `    ``# the array and the pivot ` `    ``arr[start], arr[randpivot] ``=` `arr[randpivot], arr[start] ` `    ``return` `partition(arr, start, stop) ` ` `  `''' ` `This function takes the first element as pivot,  ` `places the pivot element at the correct position  ` `in the sorted array. All the elements are re-arranged  ` `according to the pivot, the elements smaller than  ` `the pivot is places on the left and the elements ` `greater than the pivot is placed to the right of pivot. ` `'''` `def` `partition(arr,start,stop): ` `    ``pivot ``=` `start ``# pivot ` `    ``i ``=` `start ``-` `1` `    ``j ``=` `stop ``+` `1` `    ``while` `True``: ` `        ``while` `True``: ` `            ``i ``=` `i ``+` `1` `            ``if` `arr[i] >``=` `arr[pivot]: ` `                ``break` `        ``while` `True``: ` `            ``j ``=` `j ``-` `1` `            ``if` `arr[j] <``=` `arr[pivot]: ` `                ``break` `        ``if` `i >``=` `j: ` `            ``return` `j ` `        ``arr[i] , arr[j] ``=` `arr[j] , arr[i] ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"``: ` `    ``array ``=` `[``10``, ``7``, ``8``, ``9``, ``1``, ``5``] ` `    ``quicksort(array, ``0``, ``len``(array) ``-` `1``) ` `    ``print``(array) ` ` `  `# This code is contributed by soumyasaurav `

Output:

`Sorted array: 1 5 7 8 9 10`

Analysis of Randomized Quick Sort

Notes

• Using random pivoting we improve the expected or average time complexity to O (N log N). The Worst Case complexity is still O ( N^2 ).

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