# Count pair of nodes with greater Bitwise AND than Bitwise XOR in given Linked List

• Difficulty Level : Medium
• Last Updated : 20 Jun, 2022

Given a singly linked list, the task is to Count the pairs of nodes with greater Bitwise AND than Bitwise XOR.

Examples:

Input: list: 1->4->2->6->3
Output: 2
Explanation: 1st List Node Pair: (4, 6 ), Bitwise AND = 4, Bitwise XOR = 2
2nd List Node Pair: (2, 3), Bitwise AND = 2, Bitwise XOR = 1

Input: list: 17->34->62->46->30->51
Output: 7
Explanation: Valid List Node Pairs are (17, 30 ), (34, 62), (34, 46), (34, 51), (62, 46), (62, 51), (46, 51).

Naive Approach: The naive approach is to iterate the linked list and for each node find all other possible nodes forming a pair such that the bitwise AND is greater than bitwise XOR.

Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach: The problem can be solved efficiently by using the below observation:

If First Set bit (Most significant bit) of two number is at same position, then the Bitwise AND or that numbers is always greater than the XOR because the XOR of two 1 is 0 and AND of two 1s is 1.
For any other cases, XOR will be always greater than the AND

Follow the below steps to solve the problem:

• Traverse the linked list and Store the MSB position for each Node value in an array.
• Initialize a variable ans to store the total possible pairs.
• Create a hash map to store the count of nodes that have the same value of MSB (Most significant bit).
• Traverse the array containing the MSB position and in each iteration:
• Get the count of nodes that have the same position of MSB.
• Add the count of possible pairs from these nodes into ans.

Below is the implementation of the above approach:

## C++

 `// C++ program for above approach`   `#include ` `using` `namespace` `std;`   `// Structure of node of singly linked list` `struct` `Node {` `    ``int` `data;` `    ``Node* next;` `    ``Node(``int` `x)` `    ``{` `        ``data = x;` `        ``next = NULL;` `    ``}` `};`   `// Inserting new node` `// at the  beginning of the linked list` `void` `push(``struct` `Node** head_ref,` `          ``int` `new_data)` `{` `    ``// Create a new node with the given data.` `    ``struct` `Node* new_node` `        ``= ``new` `Node(new_data);`   `    ``// Make the new node point to the head.` `    ``new_node->next = (*head_ref);`   `    ``// Make the new node as the head node.` `    ``(*head_ref) = new_node;` `}`   `// Function to find the` `// count of all possible pairs` `int` `PerfectPair(Node* head, ``int` `K)` `{` `    ``int` `ans = 0, size = 0;` `    ``unordered_map<``int``, ``int``> mp;` `    ``vector<``int``> firstSetBit;`   `    ``// Iterate Linked List and store the` `    ``// firstSetBit position` `    ``// for each Node Data` `    ``while` `(head != NULL) {` `        ``firstSetBit.push_back(` `            ``log2(head->data));` `        ``size++;` `        ``head = head->next;` `    ``}`   `    ``// Check all previous node` `    ``// which can make` `    ``// pair with current node` `    ``for` `(``int` `i = 0; i < size; i++) {` `        ``ans += mp[firstSetBit[i]];` `        ``mp[firstSetBit[i]]++;` `    ``}` `    ``return` `ans;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `K = 4;`   `    ``// Create an empty singly linked list` `    ``struct` `Node* head = NULL;`   `    ``// Insert values in Linked List` `    ``push(&head, 51);` `    ``push(&head, 30);` `    ``push(&head, 46);` `    ``push(&head, 62);` `    ``push(&head, 34);` `    ``push(&head, 17);`   `    ``// Call PerfectPair function` `    ``cout << PerfectPair(head, K);` `    ``return` `0;` `}`

## Java

 `// Java program for above approach`   `import` `java.util.ArrayList;` `import` `java.util.HashMap;`   `public` `class` `GFG {`   `    ``// Structure of node of singly linked list` `    ``static` `class` `Node {` `        ``int` `data;` `        ``Node next;` `        ``Node(``int` `x) { data = x; }` `    ``};`   `    ``// Inserting new node` `    ``// at the beginning of the linked list` `    ``static` `void` `push(``int` `new_data)` `    ``{` `        ``// Create a new node with the given data.` `        ``Node new_node = ``new` `Node(new_data);`   `        ``// Make the new node point to the head.` `        ``if` `(head == ``null``) {` `            ``head = new_node;` `            ``return``;` `        ``}`   `        ``new_node.next = head;`   `        ``// Make the new node as the head node.` `        ``head = new_node;` `    ``}` `    ``static` `Node head;`   `    ``// Function to find the` `    ``// count of all possible pairs` `    ``static` `int` `PerfectPair(``int` `K)` `    ``{` `        ``int` `ans = ``0``, size = ``0``;` `        ``HashMap mp` `            ``= ``new` `HashMap();` `        ``ArrayList firstSetBit` `            ``= ``new` `ArrayList();`   `        ``// Iterate Linked List and store the` `        ``// firstSetBit position` `        ``// for each Node Data` `        ``while` `(head != ``null``) {` `            ``firstSetBit.add(log2(head.data));` `            ``size++;` `            ``head = head.next;` `        ``}`   `        ``// Check all previous node` `        ``// which can make` `        ``// pair with current node` `        ``for` `(``int` `i = ``0``; i < size; i++) {` `            ``int` `val` `                ``= mp.getOrDefault(firstSetBit.get(i), ``0``);` `            ``ans += val;` `            ``mp.put(firstSetBit.get(i), val + ``1``);` `        ``}` `        ``return` `ans;` `    ``}`   `    ``// Function to calculate the` `    ``// log base 2 of an integer` `    ``public` `static` `int` `log2(``int` `N)` `    ``{` `        ``// calculate log2 N indirectly` `        ``// using log() method` `        ``int` `result = (``int``)(Math.log(N) / Math.log(``2``));`   `        ``return` `result;` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``int` `K = ``4``;`   `        ``// Create an empty singly linked list` `        ``head = ``null``;`   `        ``// Insert values in Linked List` `        ``push(``51``);` `        ``push(``30``);` `        ``push(``46``);` `        ``push(``62``);` `        ``push(``34``);` `        ``push(``17``);`   `        ``// Call PerfectPair function` `        ``System.out.println(PerfectPair(K));` `    ``}` `}`   `// This code is contributed by jainlovely450`

## Python3

 `# Python program for above approach` `import` `math`   `class` `GFG:` `  `  `    ``# Structure of node of singly linked list` `    ``class` `Node:` `        ``data ``=` `0` `        ``next` `=` `None`   `        ``def` `__init__(``self``, x):` `            ``self``.data ``=` `x` `            `  `    ``# Inserting new node` `    ``# at the beginning of the linked list` `    ``@staticmethod` `    ``def` `push(new_data):` `      `  `        ``# Create a new node with the given data.` `        ``new_node ``=` `GFG.Node(new_data)` `        `  `        ``# Make the new node point to the head.` `        ``if` `(GFG.head ``=``=` `None``):` `            ``GFG.head ``=` `new_node` `            ``return` `        ``new_node.``next` `=` `GFG.head` `        `  `        ``# Make the new node as the head node.` `        ``GFG.head ``=` `new_node` `    ``head ``=` `None` `    `  `    ``# Function to find the` `    ``# count of all possible pairs` `    ``@staticmethod` `    ``def` `PerfectPair(K):` `        ``ans ``=` `0` `        ``size ``=` `0` `        ``mp ``=` `dict``()` `        ``firstSetBit ``=` `[]` `        `  `        ``# Iterate Linked List and store the` `        ``# firstSetBit position` `        ``# for each Node Data` `        ``while` `(GFG.head !``=` `None``):` `            ``firstSetBit.append(GFG.log2(GFG.head.data))` `            ``size ``+``=` `1` `            ``GFG.head ``=` `GFG.head.``next` `            `  `        ``# Check all previous node` `        ``# which can make` `        ``# pair with current node` `        ``i ``=` `0` `        ``while` `(i < size):` `            ``try``:` `                ``val ``=` `mp[firstSetBit[i]]` `            ``except``:` `                ``val ``=` `0` `            ``ans ``+``=` `val` `            ``mp[firstSetBit[i]] ``=` `val ``+` `1` `            ``i ``+``=` `1` `        ``return` `ans` `      `  `    ``# Function to calculate the` `    ``# log base 2 of an integer` `    ``@staticmethod` `    ``def` `log2(N):` `      `  `        ``# calculate log2 N indirectly` `        ``# using log() method` `        ``result ``=` `int``((math.log(N) ``/` `math.log(``2``)))` `        ``return` `result` `      `  `    ``# Driver code` `    ``@staticmethod` `    ``def` `main(args):` `        ``K ``=` `4` `        `  `        ``# Create an empty singly linked list` `        ``GFG.head ``=` `None` `        `  `        ``# Insert values in Linked List` `        ``GFG.push(``51``)` `        ``GFG.push(``30``)` `        ``GFG.push(``46``)` `        ``GFG.push(``62``)` `        ``GFG.push(``34``)` `        ``GFG.push(``17``)`   `        ``# Call PerfectPair function` `        ``print``(GFG.PerfectPair(K))`   `if` `__name__ ``=``=` `"__main__"``:` `    ``GFG.main([])` `    `  `'''This Code is written by Rajat Kumar'''`

## C#

 `// C# program for above approach`   `using` `System;` `using` `System.Collections.Generic;` `using` `System.Collections;` `static` `class` `GFG` `{`   `    ``public` `class` `Node` `    ``{` `        ``public` `int` `data;` `        ``public`  `Node next;` `        ``public` `Node(``int` `x) { data = x; }` `    ``};` `    ``// Structure of node of singly linked list`     `    ``// Inserting new node` `    ``// at the beginning of the linked list` `    ``static` `void` `push(``int` `new_data)` `    ``{` `        ``// Create a new node with the given data.` `        ``Node new_node = ``new` `Node(new_data);`   `        ``// Make the new node point to the head.` `        ``if` `(head == ``null``)` `        ``{` `            ``head = new_node;` `            ``return``;` `        ``}`   `        ``new_node.next = head;`   `        ``// Make the new node as the head node.` `        ``head = new_node;` `    ``}` `    ``static` `Node head;`   `    ``// Function to find the` `    ``// count of all possible pairs` `    ``static` `int` `PerfectPair(``int` `K)` `    ``{` `        ``int` `ans = 0, size = 0;` `        ``Dictionary<``int``, ``int``> mp = ``new` `Dictionary<``int``, ``int``>();` `        ``ArrayList firstSetBit = ``new` `ArrayList();`   `        ``// Iterate Linked List and store the` `        ``// firstSetBit position` `        ``// for each Node Data` `        ``while` `(head != ``null``)` `        ``{` `            ``firstSetBit.Add(log2(head.data));` `            ``size++;` `            ``head = head.next;` `        ``}`   `        ``// Check all previous node` `        ``// which can make` `        ``// pair with current node` `        ``for` `(``int` `i = 0; i < size; i++)` `        ``{` `            ``int` `val = mp.GetValueOrDefault((``int``)firstSetBit[i], 0);` `            ``ans += val;` `            ``mp[(``int``)firstSetBit[i]] = val + 1;` `        ``}` `        ``return` `ans;` `    ``}`   `    ``// Function to calculate the` `    ``// log base 2 of an integer` `    ``public` `static` `int` `log2(``int` `N)` `    ``{` `        ``// calculate log2 N indirectly` `        ``// using log() method` `        ``int` `result = (``int``)(Math.Log(N) / Math.Log(2));`   `        ``return` `result;` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `Main()` `    ``{` `        ``int` `K = 4;`   `        ``// Create an empty singly linked list` `        ``head = ``null``;`   `        ``// Insert values in Linked List` `        ``push(51);` `        ``push(30);` `        ``push(46);` `        ``push(62);` `        ``push(34);` `        ``push(17);`   `        ``// Call PerfectPair function` `        ``Console.Write(PerfectPair(K));` `    ``}` `}`   `// This code is contributed by Saurabh Jaiswal`

Output

`7`

Time Complexity: O(N * log N)
Auxiliary Space: O(N)

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