Swap nodes in a linked list without swapping data
Last Updated :
20 Sep, 2024
Given a singly linked list with two values x and y, the task is to swap two nodes having values x and y without swapping data.
Examples:
Input: 1->2->3->4->5, x = 2, y = 4
Output: 1->4->3->2->5
Input: 10->15->12->13->20->14, x = 10, y = 20
Output: 20->15->12->13->10->14
Possible cases to handle:
When swapping 2 nodes, x and y, in a linked list, there are a few cases that can arise:
- x and y are adjacent to each other, and
- One of x or y is the head node of the list.
- None of x or y is either a head node or a tail node.
- x and y are not adjacent to each other, and:
- One of x or y is the head node of the list.
- None of x or y is either a head node or a tail node.
- x or y don’t even exist in the linked list.
Therefore, our solution must handle all these cases.
[Naive Approach] By Keeping Track of Previous Node - O(n) Time and O(1) Space:
The idea is to first search x and y nodes in the given linked list. If any of them is not present, then return. While searching for x and y, keep track of current and previous pointers. First change next of previous pointers, then change next of current pointers.
Below is the diagram of the case where either Node X or Node Y is the head node of the list.
Swapping when Node X is the head of the List.
Below is the diagram of the case where Nodes X and Node Y are both not the head node of the list.
Swapping when neither Node X or Node Y is the head of the list.
Below is the implementation of the above approach:
C++
// C++ program to swap the nodes of a linked list rather
// than swapping the data from the nodes.
#include <iostream>
using namespace std;
class Node {
public:
int data;
Node* next;
Node(int x) {
data = x;
next = nullptr;
}
};
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
Node* swapNodes(Node* head, int x, int y) {
// Nothing to do if x and y are the same
if (x == y) return head;
Node *prevX = nullptr, *currX = nullptr;
Node *prevY = nullptr, *currY = nullptr;
Node *curr = head;
// First loop to find x
while (curr != nullptr) {
if (curr->data == x) {
currX = curr;
break;
}
prevX = curr;
curr = curr->next;
}
curr = head;
// Second loop to find y
while (curr != nullptr) {
if (curr->data == y) {
currY = curr;
break;
}
prevY = curr;
curr = curr->next;
}
// If either x or y is not present, nothing to do
if (currX == nullptr || currY == nullptr) return head;
// If x is not head of the linked list
if (prevX != nullptr) {
prevX->next = currY;
} else {
head = currY;
}
// If y is not head of the linked list
if (prevY != nullptr) {
prevY->next = currX;
} else {
head = currX;
}
// Swap next pointers
Node* temp = currY->next;
currY->next = currX->next;
currX->next = temp;
return head;
}
void print(Node* curr) {
while (curr != nullptr) {
cout << curr->data << " ";
curr = curr->next;
}
cout << endl;
}
int main() {
// Constructed linked list:
// 1->2->3->4->5
Node* head = new Node(1);
head->next = new Node(2);
head->next->next = new Node(3);
head->next->next->next = new Node(4);
head->next->next->next->next = new Node(5);
head = swapNodes(head, 4, 3);
print(head);
return 0;
}
// C program to swap the nodes of a linked list rather
// than swapping the data from the nodes.
#include <stdio.h>
#include <stdlib.h>
struct Node {
int data;
struct Node* next;
};
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
struct Node* swapNodes(struct Node* head, int x, int y) {
// Nothing to do if x and y are the same
if (x == y)
return head;
struct Node *prevX = NULL, *currX = NULL;
struct Node *prevY = NULL, *currY = NULL;
struct Node *curr = head;
// First loop to find x
while (curr != NULL) {
if (curr->data == x) {
currX = curr;
break;
}
prevX = curr;
curr = curr->next;
}
curr = head;
// Second loop to find y
while (curr != NULL) {
if (curr->data == y) {
currY = curr;
break;
}
prevY = curr;
curr = curr->next;
}
// If either x or y is not present, nothing to do
if (currX == NULL || currY == NULL)
return head;
// If x is not head of the linked list
if (prevX != NULL)
prevX->next = currY;
else
head = currY;
// If y is not head of the linked list
if (prevY != NULL)
prevY->next = currX;
else
head = currX;
// Swap next pointers
struct Node* temp = currY->next;
currY->next = currX->next;
currX->next = temp;
return head;
}
void print(struct Node* curr) {
while (curr != NULL) {
printf("%d ", curr->data);
curr = curr->next;
}
printf("\n");
}
struct Node* createNode(int x) {
struct Node* node =
(struct Node*)malloc(sizeof(struct Node));
node->data = x;
node->next = NULL;
return node;
}
int main() {
// Constructed linked list: 1->2->3->4->5
struct Node* head = createNode(1);
head->next = createNode(2);
head->next->next = createNode(3);
head->next->next->next = createNode(4);
head->next->next->next->next = createNode(5);
head = swapNodes(head, 4, 3);
print(head);
return 0;
}
// Java program to swap the nodes of a linked list rather
// than swapping the data from the nodes.
class Node {
int data;
Node next;
Node(int data) {
this.data = data;
this.next = null;
}
}
class GfG {
// Function to swap nodes x and y in linked list by changing links
static Node swapNodes(Node head, int x, int y) {
// Nothing to do if x and y are the same
if (x == y) {
return head;
}
Node prevX = null, currX = null;
Node prevY = null, currY = null;
Node curr = head;
// First loop to find x
while (curr != null) {
if (curr.data == x) {
currX = curr;
break;
}
prevX = curr;
curr = curr.next;
}
curr = head;
// Second loop to find y
while (curr != null) {
if (curr.data == y) {
currY = curr;
break;
}
prevY = curr;
curr = curr.next;
}
// If either x or y is not present, nothing to do
if (currX == null || currY == null) {
return head;
}
// If x is not head of the linked list
if (prevX != null) {
prevX.next = currY;
} else {
head = currY;
}
// If y is not head of the linked list
if (prevY != null) {
prevY.next = currX;
} else {
head = currX;
}
// Swap next pointers
Node temp = currY.next;
currY.next = currX.next;
currX.next = temp;
return head;
}
static void printList(Node curr) {
while (curr != null) {
System.out.print(curr.data + " ");
curr = curr.next;
}
System.out.println();
}
public static void main(String[] args) {
// Constructed linked list: 1->2->3->4->5
Node head = new Node(1);
head.next = new Node(2);
head.next.next = new Node(3);
head.next.next.next = new Node(4);
head.next.next.next.next = new Node(5);
head = swapNodes(head, 4, 3);
printList(head);
}
}
# Python program to swap the nodes of a linked list rather
# than swapping the data from the nodes.
class Node:
def __init__(self, data):
self.data = data
self.next = None
# Function to swap nodes x and y in linked
# list by changing links
def swapNodes(head, x, y):
# Nothing to do if x and y are the same
if x == y:
return head
prevX = None
currX = head
while currX and currX.data != x:
prevX = currX
currX = currX.next
prevY = None
currY = head
while currY and currY.data != y:
prevY = currY
currY = currY.next
# If either x or y is not present, nothing to do
if currX is None or currY is None:
return head
# If x is not head of the linked list
if prevX:
prevX.next = currY
else:
head = currY
# If y is not head of the linked list
if prevY:
prevY.next = currX
else:
head = currX
# Swap next pointers
currX.next, currY.next = currY.next, currX.next
return head
def printList(curr):
while curr:
print(curr.data, end=" ")
curr = curr.next
print()
if __name__ == "__main__":
# Constructed linked list: 1->2->3->4->5
head = Node(1)
head.next = Node(2)
head.next.next = Node(3)
head.next.next.next = Node(4)
head.next.next.next.next = Node(5)
head = swapNodes(head, 4, 3)
printList(head)
// C# program to swap the nodes of a linked list rather
// than swapping the data from the nodes.
using System;
class Node {
public int data;
public Node next;
public Node(int data) {
this.data = data;
this.next = null;
}
}
class GfG {
// Function to swap nodes x and y in linked list by changing links
static Node swapNodes(Node head, int x, int y) {
// Nothing to do if x and y are the same
if (x == y)
return head;
Node prevX = null;
Node currX = head;
// Find currX and prevX
while (currX != null && currX.data != x) {
prevX = currX;
currX = currX.next;
}
Node prevY = null;
Node currY = head;
// Find currY and prevY
while (currY != null && currY.data != y) {
prevY = currY;
currY = currY.next;
}
// If either x or y is not present, nothing to do
if (currX == null || currY == null)
return head;
// If x is not head of the linked list
if (prevX != null)
prevX.next = currY;
else
head = currY;
// If y is not head of the linked list
if (prevY != null)
prevY.next = currX;
else
head = currX;
// Swap next pointers
Node temp = currY.next;
currY.next = currX.next;
currX.next = temp;
return head;
}
static void printList(Node curr) {
while (curr != null) {
Console.Write(curr.data + " ");
curr = curr.next;
}
Console.WriteLine();
}
static void Main(string[] args) {
// Constructed linked list: 1->2->3->4->5
Node head = new Node(1);
head.next = new Node(2);
head.next.next = new Node(3);
head.next.next.next = new Node(4);
head.next.next.next.next = new Node(5);
head = swapNodes(head, 4, 3);
printList(head);
}
}
// Javascript program to swap the nodes of a linked list rather
// than swapping the data from the nodes.
class Node {
constructor(data) {
this.data = data;
this.next = null;
}
}
// Function to swap nodes x and y in linked
// list by changing links
function swapNodes(head, x, y) {
// Nothing to do if x and y are the same
if (x === y) return head;
let prevX = null, currX = head;
// Find currX and prevX
while (currX && currX.data !== x) {
prevX = currX;
currX = currX.next;
}
let prevY = null, currY = head;
// Find currY and prevY
while (currY && currY.data !== y) {
prevY = currY;
currY = currY.next;
}
// If either x or y is not present, nothing to do
if (!currX || !currY) return head;
// If x is not head of the linked list
if (prevX) prevX.next = currY;
else head = currY;
// If y is not head of the linked list
if (prevY) prevY.next = currX;
else head = currX;
// Swap next pointers
let temp = currY.next;
currY.next = currX.next;
currX.next = temp;
return head;
}
function printList(curr) {
while (curr) {
console.log(curr.data + " ");
curr = curr.next;
}
console.log();
}
// Constructed linked list:
// 1->2->3->4->5
let head = new Node(1);
head.next = new Node(2);
head.next.next = new Node(3);
head.next.next.next = new Node(4);
head.next.next.next.next = new Node(5);
head = swapNodes(head, 4, 3);
printList(head);
Time Complexity: O(n), where n is the number of nodes in linked list.
Auxiliary Space: O(1)
[Efficient Approach] Using Single traversal - O(n) Time and O(1) Space:
The above code can be optimized to search x and y in a single traversal. Two loops are used to keep the program simple.
Follow the steps below to solve the problem:
- If x and y are the same, return as no need to swap.
- Start traversing the List to search for x and y nodes by keeping track of the previous nodes prevX and prevY.
- If either x or y is not found in the list, return without making any changes.
- else, update connections:
- If x is not the head, set prevX->next to currY. Otherwise, set currY as the new head.
- If y is not the head, set prevY->next to currX. Otherwise, set currX as the new head.
- Swap the next pointers of currX and currY to complete the swap.
Below is the implementation of the above approach:
C++
// C++ program to swaps the nodes of linked list rather
// than swapping the data from the nodes.
#include <iostream>
using namespace std;
class Node {
public:
int data;
Node* next;
Node(int x) {
data = x;
next = nullptr;
}
};
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
Node* swapNodes(Node* head, int x, int y) {
// Nothing to do if x and y are same
if (x == y)
return head;
Node *prevX = nullptr, *currX = nullptr;
Node *prevY = nullptr, *currY = nullptr;
Node *prev = nullptr, *curr = head;
// Single loop to find both x and y
while (curr != nullptr) {
if (curr->data == x) {
prevX = prev;
currX = curr;
} else if (curr->data == y) {
prevY = prev;
currY = curr;
}
prev = curr;
curr = curr->next;
}
// If either x or y is not present, nothing to do
if (currX == nullptr || currY == nullptr)
return head;
// If x is not head of the linked list
if (prevX != nullptr)
prevX->next = currY;
else // Else make y the new head
head = currY;
// If y is not head of the linked list
if (prevY != nullptr)
prevY->next = currX;
else
// Else make x the new head
head = currX;
// Swap next pointers
Node* temp = currY->next;
currY->next = currX->next;
currX->next = temp;
return head;
}
void print(Node* curr) {
while (curr != nullptr) {
cout << curr->data << " ";
curr = curr->next;
}
}
int main() {
// Constructed linked list:
// 1->2->3->4->5
Node* head = new Node(1);
head->next = new Node(2);
head->next->next = new Node(3);
head->next->next->next = new Node(4);
head->next->next->next->next = new Node(5);
head = swapNodes(head, 4, 3);
print(head);
return 0;
}
// C program to swaps the nodes of linked list rather
// than swapping the data from the nodes.
#include <stdio.h>
#include <stdlib.h>
struct Node {
int data;
struct Node* next;
};
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
struct Node* swapNodes(struct Node* head, int x, int y) {
// Nothing to do if x and y are same
if (x == y)
return head;
struct Node *prevX = NULL, *currX = NULL;
struct Node *prevY = NULL, *currY = NULL;
struct Node *prev = NULL, *curr = head;
// Single loop to find both x and y
while (curr != NULL) {
if (curr->data == x) {
prevX = prev;
currX = curr;
} else if (curr->data == y) {
prevY = prev;
currY = curr;
}
prev = curr;
curr = curr->next;
}
// If either x or y is not present, nothing to do
if (currX == NULL || currY == NULL)
return head;
// If x is not head of the linked list
if (prevX != NULL)
prevX->next = currY;
else
head = currY;
// If y is not head of the linked list
if (prevY != NULL)
prevY->next = currX;
else
head = currX;
// Swap next pointers
struct Node* temp = currY->next;
currY->next = currX->next;
currX->next = temp;
return head;
}
void print(struct Node* curr) {
while (curr != NULL) {
printf("%d ", curr->data);
curr = curr->next;
}
}
struct Node* createNode(int x) {
struct Node* node =
(struct Node*)malloc(sizeof(struct Node));
node->data = x;
node->next = NULL;
return node;
}
int main() {
// Constructed linked list:
// 1->2->3->4->5
struct Node* head = createNode(1);
head->next = createNode(2);
head->next->next = createNode(3);
head->next->next->next = createNode(4);
head->next->next->next->next = createNode(5);
head = swapNodes(head, 4, 3);
print(head);
return 0;
}
// Java program to swaps the nodes of linked list rather
// than swapping the data from the nodes.
class Node {
int data;
Node next;
Node(int data) {
this.data = data;
this.next = null;
}
}
class GfG {
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
static Node swapNodes(Node head, int x, int y) {
// Nothing to do if x and y are same
if (x == y)
return head;
Node prevX = null, currX = null;
Node prevY = null, currY = null;
Node prev = null, curr = head;
// Single loop to find both x and y
while (curr != null) {
if (curr.data == x) {
prevX = prev;
currX = curr;
} else if (curr.data == y) {
prevY = prev;
currY = curr;
}
prev = curr;
curr = curr.next;
}
// If either x or y is not present, nothing to do
if (currX == null || currY == null)
return head;
// If x is not head of the linked list
if (prevX != null)
prevX.next = currY;
else
// Else make y the new head
head = currY;
// If y is not head of the linked list
if (prevY != null)
prevY.next = currX;
else
// Else make x the new head
head = currX;
// Swap next pointers
Node temp = currY.next;
currY.next = currX.next;
currX.next = temp;
return head;
}
static void printList(Node curr) {
while (curr != null) {
System.out.print(curr.data + " ");
curr = curr.next;
}
}
public static void main(String[] args) {
// Constructed linked list:
// 1->2->3->4->5
Node head = new Node(1);
head.next = new Node(2);
head.next.next = new Node(3);
head.next.next.next = new Node(4);
head.next.next.next.next = new Node(5);
head = swapNodes(head, 4, 3);
printList(head);
}
}
# Python program to swaps the nodes of linked list rather
# han swapping the data from the nodes.
class Node:
def __init__(self, data):
self.data = data
self.next = None
# Function to swap nodes x and y in linked list by
# changing links and return the updated head
def swapNodes(head, x, y):
# Nothing to do if x and y are same
if x == y:
return head
prevX = None
currX = None
prevY = None
currY = None
prev = None
curr = head
# Single loop to find both x and y
while curr:
if curr.data == x:
prevX = prev
currX = curr
elif curr.data == y:
prevY = prev
currY = curr
prev = curr
curr = curr.next
# If either x or y is not present, nothing to do
if currX is None or currY is None:
return head
# If x is not head of the linked list
if prevX is not None:
prevX.next = currY
else:
head = currY
# If y is not head of the linked list
if prevY is not None:
prevY.next = currX
else:
# Else make x the new head
head = currX
# Swap next pointers
temp = currY.next
currY.next = currX.next
currX.next = temp
return head
def printList(curr):
while curr:
print(curr.data, end=" ")
curr = curr.next
if __name__ == "__main__":
# Constructed linked list:
# 1->2->3->4->5
head = Node(1)
head.next = Node(2)
head.next.next = Node(3)
head.next.next.next = Node(4)
head.next.next.next.next = Node(5)
head = swapNodes(head, 4, 3)
printList(head)
// C# program to swaps the nodes of linked list rather
// than swapping the data from the nodes.
using System;
class Node {
public int data;
public Node next;
public Node(int data) {
this.data = data;
this.next = null;
}
}
class GfG {
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
static Node swapNodes(Node head, int x, int y) {
// Nothing to do if x and y are same
if (x == y)
return head;
Node prevX = null, currX = null;
Node prevY = null, currY = null;
Node prev = null, curr = head;
// Single loop to find both x and y
while (curr != null) {
if (curr.data == x) {
prevX = prev;
currX = curr;
} else if (curr.data == y) {
prevY = prev;
currY = curr;
}
prev = curr;
curr = curr.next;
}
// If either x or y is not present, nothing to do
if (currX == null || currY == null)
return head;
// If x is not head of the linked list
if (prevX != null)
prevX.next = currY;
else
// Else make y the new head
head = currY;
// If y is not head of the linked list
if (prevY != null)
prevY.next = currX;
else
// Else make x the new head
head = currX;
// Swap next pointers
Node temp = currY.next;
currY.next = currX.next;
currX.next = temp;
return head;
}
static void printList(Node curr) {
while (curr != null) {
Console.Write(curr.data + " ");
curr = curr.next;
}
}
static void Main(string[] args) {
// Constructed linked list:
// 1->2->3->4->5
Node head = new Node(1);
head.next = new Node(2);
head.next.next = new Node(3);
head.next.next.next = new Node(4);
head.next.next.next.next = new Node(5);
head = swapNodes(head, 4, 3);
printList(head);
}
}
// Javascript program to swaps the nodes of linked list rather
// than swapping the data from the nodes.
class Node {
constructor(data) {
this.data = data;
this.next = null;
}
}
// Function to swap nodes x and y in linked list by
// changing links and return the updated head
function swapNodes(head, x, y) {
// Nothing to do if x and y are same
if (x === y) return head;
let prevX = null, currX = null;
let prevY = null, currY = null;
let prev = null, curr = head;
// Single loop to find both x and y
while (curr) {
if (curr.data === x) {
prevX = prev;
currX = curr;
} else if (curr.data === y) {
prevY = prev;
currY = curr;
}
prev = curr;
curr = curr.next;
}
// If either x or y is not present, nothing to do
if (!currX || !currY) return head;
// If x is not head of the linked list
if (prevX) prevX.next = currY;
else head = currY;
// If y is not head of the linked list
if (prevY) prevY.next = currX;
else head = currX;
// Swap next pointers
let temp = currY.next;
currY.next = currX.next;
currX.next = temp;
return head;
}
function printList(curr) {
while (curr) {
console.log(curr.data + " ");
curr = curr.next;
}
}
// Constructed linked list:
// 1->2->3->4->5
let head = new Node(1);
head.next = new Node(2);
head.next.next = new Node(3);
head.next.next.next = new Node(4);
head.next.next.next.next = new Node(5);
head = swapNodes(head, 4, 3);
printList(head);
Time Complexity: O(n), where n is the number of nodes in linked list.
Auxiliary Space: O(1)
Swap nodes in a linked list without swapping data
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Basics & Prerequisites
Data Structures
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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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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