Following is a typical recursive implementation of Quick Sort that uses last element as pivot.
C++
// CPP code for recursive function of Quicksort
#include <bits/stdc++.h>
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 l, int h)
{
int x = arr[h];
int i = (l - 1);
for (int j = l; j <= h - 1; j++) {
if (arr[j] <= x) {
i++;
swap(arr[i], arr[j]);
}
}
swap(arr[i + 1], arr[h]);
return (i + 1);
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
void quickSort(int A[], int l, int h)
{
if (l < h) {
/* Partitioning index */
int p = partition(A, l, h);
quickSort(A, l, p - 1);
quickSort(A, p + 1, h);
}
}
// Driver code
int main()
{
int n = 5;
int arr[n] = { 4, 2, 6, 9, 2 };
quickSort(arr, 0, n - 1);
for (int i = 0; i < n; i++) {
cout << arr[i] << " ";
}
return 0;
}
// Java program for implementation of QuickSort
import java.util.*;
class QuickSort {
/* 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 */
static int partition(int arr[], int low, int high)
{
int pivot = arr[high];
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++;
// 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 */
static void qSort(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
qSort(arr, low, pi - 1);
qSort(arr, pi + 1, high);
}
}
// Driver code
public static void main(String args[])
{
int n = 5;
int arr[] = { 4, 2, 6, 9, 2 };
qSort(arr, 0, n - 1);
for (int i = 0; i < n; i++) {
System.out.print(arr[i] + " ");
}
}
}
# A typical recursive Python
# implementation of QuickSort
# 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
def partition(arr, low, high):
i = (low - 1) # index of smaller element
pivot = arr[high] # pivot
for j in range(low, high):
# If current element is smaller
# than or equal to pivot
if arr[j] <= pivot:
# increment index of
# smaller element
i += 1
arr[i], arr[j] = arr[j], arr[i]
arr[i + 1], arr[high] = arr[high], arr[i + 1]
return (i + 1)
# The main function that implements QuickSort
# arr[] --> Array to be sorted,
# low --> Starting index,
# high --> Ending index
# Function to do Quick sort
def quickSort(arr, low, high):
if low < high:
# pi is partitioning index, arr[p] is now
# at right place
pi = partition(arr, low, high)
# Separately sort elements before
# partition and after partition
quickSort(arr, low, pi-1)
quickSort(arr, pi + 1, high)
# Driver Code
if __name__ == '__main__' :
arr = [4, 2, 6, 9, 2]
n = len(arr)
# Calling quickSort function
quickSort(arr, 0, n - 1)
for i in range(n):
print(arr[i], end = " ")
// C# program for implementation of
// QuickSort
using System;
class GFG {
/* 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 */
static int partition(int[] arr,
int low, int high)
{
int temp;
int pivot = arr[high];
// index of smaller element
int i = (low - 1);
for (int j = low; j <= high - 1; j++) {
// If current element is
// smaller than or
// equal to pivot
if (arr[j] <= pivot) {
i++;
// swap arr[i] and arr[j]
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
// swap arr[i+1] and arr[high]
// (or pivot)
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 */
static void qSort(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
qSort(arr, low, pi - 1);
qSort(arr, pi + 1, high);
}
}
// Driver code
public static void Main()
{
int n = 5;
int[] arr = { 4, 2, 6, 9, 2 };
qSort(arr, 0, n - 1);
for (int i = 0; i < n; i++)
Console.Write(arr[i] + " ");
}
}
// This code is contributed by nitin mittal.
<script>
// JavaScript program for implementation of QuickSort
/* 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 */
function partition(arr, low, high)
{
let temp;
let pivot = arr[high];
// index of smaller element
let i = (low - 1);
for (let j = low; j <= high - 1; j++) {
// If current element is
// smaller than or
// equal to pivot
if (arr[j] <= pivot) {
i++;
// swap arr[i] and arr[j]
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
// swap arr[i+1] and arr[high]
// (or pivot)
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 */
function qSort(arr, low, high)
{
if (low < high) {
/* pi is partitioning index,
arr[pi] is now at right place */
let pi = partition(arr, low, high);
// Recursively sort elements
// before partition and after
// partition
qSort(arr, low, pi - 1);
qSort(arr, pi + 1, high);
}
}
let n = 5;
let arr = [ 4, 2, 6, 9, 2 ];
qSort(arr, 0, n - 1);
for (let i = 0; i < n; i++)
document.write(arr[i] + " ");
</script>
<?php
// PHP code for recursive function
// of Quicksort
// Function to swap numbers
function swap(&$a, &$b)
{
$temp = $a;
$a = $b;
$b = $temp;
}
/* This function takes last element as pivot,
places the pivot element at its correct
position in sorted array, and places
all smaller (smaller than pivot) to left
of pivot and all greater elements to
right of pivot */
function partition (&$arr, $l, $h)
{
$x = $arr[$h];
$i = ($l - 1);
for ($j = $l; $j <= $h - 1; $j++)
{
if ($arr[$j] <= $x)
{
$i++;
swap ($arr[$i], $arr[$j]);
}
}
swap ($arr[$i + 1], $arr[$h]);
return ($i + 1);
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
function quickSort(&$A, $l, $h)
{
if ($l < $h)
{
/* Partitioning index */
$p = partition($A, $l, $h);
quickSort($A, $l, $p - 1);
quickSort($A, $p + 1, $h);
}
}
// Driver code
$n = 5;
$arr = array(4, 2, 6, 9, 2);
quickSort($arr, 0, $n - 1);
for($i = 0; $i < $n; $i++)
{
echo $arr[$i] . " ";
}
// This code is contributed by
// rathbhupendra
?>
Output:
2 2 4 6 9
The above implementation can be optimized in many ways
1) The above implementation uses the last index as a pivot. This causes worst-case behavior on already sorted arrays, which is a commonly occurring case. The problem can be solved by choosing either a random index for the pivot or choosing the middle index of the partition or choosing the median of the first, middle, and last element of the partition for the pivot. (See this for details)
2) To reduce the recursion depth, recur first for the smaller half of the array, and use a tail call to recurse into the other.
3) Insertion sort works better for small subarrays. Insertion sort can be used for invocations on such small arrays (i.e. where the length is less than a threshold t determined experimentally). For example, this library implementation of Quicksort uses insertion sort below size 7.
Despite the above optimizations, the function remains recursive and uses function call stack to store intermediate values of l and h. The function call stack stores other bookkeeping information together with parameters. Also, function calls involve overheads like storing activation records of the caller function and then resuming execution. The above function can be easily converted to an iterative version with the help of an auxiliary stack. Following is an iterative implementation of the above recursive code.
C++
// An iterative implementation of quick sort
#include <bits/stdc++.h>
using namespace std;
/* This function is same in both iterative and recursive*/
int partition(int arr[], int l, int h)
{
int x = arr[h];
int i = (l - 1);
for (int j = l; j <= h - 1; j++) {
if (arr[j] <= x) {
i++;
swap(arr[i], arr[j]);
}
}
swap(arr[i + 1], arr[h]);
return (i + 1);
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
void quickSortIterative(int arr[], int l, int h)
{
// Create an auxiliary stack
int stack[h - l + 1];
// initialize top of stack
int top = -1;
// push initial values of l and h to stack
stack[++top] = l;
stack[++top] = h;
// Keep popping from stack while is not empty
while (top >= 0) {
// Pop h and l
h = stack[top--];
l = stack[top--];
// Set pivot element at its correct position
// in sorted array
int p = partition(arr, l, h);
// If there are elements on left side of pivot,
// then push left side to stack
if (p - 1 > l) {
stack[++top] = l;
stack[++top] = p - 1;
}
// If there are elements on right side of pivot,
// then push right side to stack
if (p + 1 < h) {
stack[++top] = p + 1;
stack[++top] = h;
}
}
}
// A utility function to print contents of arr
void printArr(int arr[], int n)
{
int i;
for (i = 0; i < n; ++i)
cout << arr[i] << " ";
}
// Driver code
int main()
{
int arr[] = { 4, 3, 5, 2, 1, 3, 2, 3 };
int n = sizeof(arr) / sizeof(*arr);
quickSortIterative(arr, 0, n - 1);
printArr(arr, n);
return 0;
}
// This is code is contributed by rathbhupendra
// An iterative implementation of quick sort
#include <stdio.h>
// A utility function to swap two elements
void swap(int* a, int* b)
{
int t = *a;
*a = *b;
*b = t;
}
/* This function is same in both iterative and recursive*/
int partition(int arr[], int l, int h)
{
int x = arr[h];
int i = (l - 1);
for (int j = l; j <= h - 1; j++) {
if (arr[j] <= x) {
i++;
swap(&arr[i], &arr[j]);
}
}
swap(&arr[i + 1], &arr[h]);
return (i + 1);
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
void quickSortIterative(int arr[], int l, int h)
{
// Create an auxiliary stack
int stack[h - l + 1];
// initialize top of stack
int top = -1;
// push initial values of l and h to stack
stack[++top] = l;
stack[++top] = h;
// Keep popping from stack while is not empty
while (top >= 0) {
// Pop h and l
h = stack[top--];
l = stack[top--];
// Set pivot element at its correct position
// in sorted array
int p = partition(arr, l, h);
// If there are elements on left side of pivot,
// then push left side to stack
if (p - 1 > l) {
stack[++top] = l;
stack[++top] = p - 1;
}
// If there are elements on right side of pivot,
// then push right side to stack
if (p + 1 < h) {
stack[++top] = p + 1;
stack[++top] = h;
}
}
}
// A utility function to print contents of arr
void printArr(int arr[], int n)
{
int i;
for (i = 0; i < n; ++i)
printf("%d ", arr[i]);
}
// Driver program to test above functions
int main()
{
int arr[] = { 4, 3, 5, 2, 1, 3, 2, 3 };
int n = sizeof(arr) / sizeof(*arr);
quickSortIterative(arr, 0, n - 1);
printArr(arr, n);
return 0;
}
// Java program for implementation of QuickSort
import java.util.*;
class QuickSort {
/* 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 */
static int partition(int arr[], int low, int high)
{
int pivot = arr[high];
// index of smaller element
int i = (low - 1);
for (int j = low; j <= high - 1; 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;
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
static void quickSortIterative(int arr[], int l, int h)
{
// Create an auxiliary stack
int[] stack = new int[h - l + 1];
// initialize top of stack
int top = -1;
// push initial values of l and h to stack
stack[++top] = l;
stack[++top] = h;
// Keep popping from stack while is not empty
while (top >= 0) {
// Pop h and l
h = stack[top--];
l = stack[top--];
// Set pivot element at its correct position
// in sorted array
int p = partition(arr, l, h);
// If there are elements on left side of pivot,
// then push left side to stack
if (p - 1 > l) {
stack[++top] = l;
stack[++top] = p - 1;
}
// If there are elements on right side of pivot,
// then push right side to stack
if (p + 1 < h) {
stack[++top] = p + 1;
stack[++top] = h;
}
}
}
// Driver code
public static void main(String args[])
{
int arr[] = { 4, 3, 5, 2, 1, 3, 2, 3 };
int n = 8;
// Function calling
quickSortIterative(arr, 0, n - 1);
for (int i = 0; i < n; i++) {
System.out.print(arr[i] + " ");
}
}
}
# Python program for implementation of Quicksort
# This function is same in both iterative and recursive
def partition(arr, l, h):
i = ( l - 1 )
x = arr[h]
for j in range(l, h):
if arr[j] <= x:
# increment index of smaller element
i = i + 1
arr[i], arr[j] = arr[j], arr[i]
arr[i + 1], arr[h] = arr[h], arr[i + 1]
return (i + 1)
# Function to do Quick sort
# arr[] --> Array to be sorted,
# l --> Starting index,
# h --> Ending index
def quickSortIterative(arr, l, h):
# Create an auxiliary stack
size = h - l + 1
stack = [0] * (size)
# initialize top of stack
top = -1
# push initial values of l and h to stack
top = top + 1
stack[top] = l
top = top + 1
stack[top] = h
# Keep popping from stack while is not empty
while top >= 0:
# Pop h and l
h = stack[top]
top = top - 1
l = stack[top]
top = top - 1
# Set pivot element at its correct position in
# sorted array
p = partition( arr, l, h )
# If there are elements on left side of pivot,
# then push left side to stack
if p-1 > l:
top = top + 1
stack[top] = l
top = top + 1
stack[top] = p - 1
# If there are elements on right side of pivot,
# then push right side to stack
if p + 1 < h:
top = top + 1
stack[top] = p + 1
top = top + 1
stack[top] = h
# Driver code to test above
arr = [4, 3, 5, 2, 1, 3, 2, 3]
n = len(arr)
quickSortIterative(arr, 0, n-1)
print ("Sorted array is:")
for i in range(n):
print ("% d" % arr[i]),
# This code is contributed by Mohit Kumra
// C# program for implementation of QuickSort
using System;
class GFG {
/* 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 */
static int partition(int[] arr, int low,
int high)
{
int temp;
int pivot = arr[high];
// index of smaller element
int i = (low - 1);
for (int j = low; j <= high - 1; j++) {
// If current element is smaller
// than or equal to pivot
if (arr[j] <= pivot) {
i++;
// swap arr[i] and arr[j]
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
// swap arr[i+1] and arr[high]
// (or pivot)
temp = arr[i + 1];
arr[i + 1] = arr[high];
arr[high] = temp;
return i + 1;
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
static void quickSortIterative(int[] arr,
int l, int h)
{
// Create an auxiliary stack
int[] stack = new int[h - l + 1];
// initialize top of stack
int top = -1;
// push initial values of l and h to
// stack
stack[++top] = l;
stack[++top] = h;
// Keep popping from stack while
// is not empty
while (top >= 0) {
// Pop h and l
h = stack[top--];
l = stack[top--];
// Set pivot element at its
// correct position in
// sorted array
int p = partition(arr, l, h);
// If there are elements on
// left side of pivot, then
// push left side to stack
if (p - 1 > l) {
stack[++top] = l;
stack[++top] = p - 1;
}
// If there are elements on
// right side of pivot, then
// push right side to stack
if (p + 1 < h) {
stack[++top] = p + 1;
stack[++top] = h;
}
}
}
// Driver code
public static void Main()
{
int[] arr = { 4, 3, 5, 2, 1, 3, 2, 3 };
int n = 8;
// Function calling
quickSortIterative(arr, 0, n - 1);
for (int i = 0; i < n; i++)
Console.Write(arr[i] + " ");
}
}
// This code is contributed by anuj_67.
<script>
// Javascript program for implementation of QuickSort
/* 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 */
function partition(arr, low, high)
{
let temp;
let pivot = arr[high];
// index of smaller element
let i = (low - 1);
for (let j = low; j <= high - 1; j++) {
// If current element is smaller
// than or equal to pivot
if (arr[j] <= pivot) {
i++;
// swap arr[i] and arr[j]
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
// swap arr[i+1] and arr[high]
// (or pivot)
temp = arr[i + 1];
arr[i + 1] = arr[high];
arr[high] = temp;
return i + 1;
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
function quickSortIterative(arr, l, h)
{
// Create an auxiliary stack
let stack = new Array(h - l + 1);
stack.fill(0);
// initialize top of stack
let top = -1;
// push initial values of l and h to
// stack
stack[++top] = l;
stack[++top] = h;
// Keep popping from stack while
// is not empty
while (top >= 0) {
// Pop h and l
h = stack[top--];
l = stack[top--];
// Set pivot element at its
// correct position in
// sorted array
let p = partition(arr, l, h);
// If there are elements on
// left side of pivot, then
// push left side to stack
if (p - 1 > l) {
stack[++top] = l;
stack[++top] = p - 1;
}
// If there are elements on
// right side of pivot, then
// push right side to stack
if (p + 1 < h) {
stack[++top] = p + 1;
stack[++top] = h;
}
}
}
let arr = [ 4, 3, 5, 2, 1, 3, 2, 3 ];
let n = 8;
// Function calling
quickSortIterative(arr, 0, n - 1);
for (let i = 0; i < n; i++)
document.write(arr[i] + " ");
// This code is contributed by mukesh07.
</script>
<?php
// An iterative implementation of quick sort
// A utility function to swap two elements
function swap ( &$a, &$b )
{
$t = $a;
$a = $b;
$b = $t;
}
/* This function is same in both iterative and recursive*/
function partition (&$arr, $l, $h)
{
$x = $arr[$h];
$i = ($l - 1);
for ($j = $l; $j <= $h- 1; $j++)
{
if ($arr[$j] <= $x)
{
$i++;
swap ($arr[$i], $arr[$j]);
}
}
swap ($arr[$i + 1], $arr[$h]);
return ($i + 1);
}
/* A[] --> Array to be sorted,
l --> Starting index,
h --> Ending index */
function quickSortIterative (&$arr, $l, $h)
{
// Create an auxiliary stack
$stack=array_fill(0, $h - $l + 1, 0);
// initialize top of stack
$top = -1;
// push initial values of l and h to stack
$stack[ ++$top ] = $l;
$stack[ ++$top ] = $h;
// Keep popping from stack while is not empty
while ( $top >= 0 )
{
// Pop h and l
$h = $stack[ $top-- ];
$l = $stack[ $top-- ];
// Set pivot element at its correct position
// in sorted array
$p = partition( $arr, $l, $h );
// If there are elements on left side of pivot,
// then push left side to stack
if ( $p-1 > $l )
{
$stack[ ++$top ] = $l;
$stack[ ++$top ] = $p - 1;
}
// If there are elements on right side of pivot,
// then push right side to stack
if ( $p+1 < $h )
{
$stack[ ++$top ] = $p + 1;
$stack[ ++$top ] = $h;
}
}
}
// A utility function to print contents of arr
function printArr( $arr, $n )
{
for ( $i = 0; $i < $n; ++$i )
echo $arr[$i]." ";
}
// Driver code
$arr = array(4, 3, 5, 2, 1, 3, 2, 3);
$n = count($arr);
quickSortIterative($arr, 0, $n - 1 );
printArr($arr, $n );
// This is code is contributed by chandan_jnu
?>
Output:
1 2 2 3 3 3 4 5
Time Complexity: O(n*log(n))
Auxiliary Space: O(n)
The above-mentioned optimizations for recursive quicksort can also be applied to the iterative version.
1) Partition process is the same in both recursive and iterative. The same techniques to choose optimal pivot can also be applied to the iterative version.
2) To reduce the stack size, first push the indexes of smaller half.
3) Use insertion sort when the size reduces below an experimentally calculated threshold.
References:
https://en.wikipedia.org/wiki/Quicksort
This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team.
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Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
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String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
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Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
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Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
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Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
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Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
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Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
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Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
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Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
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Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
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Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
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Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
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Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
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Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
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Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
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Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
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Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
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Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
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GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
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