Given a Binary Matrix. The task is to find the pair of row in the Binary matrix that has maximum bit difference

Examples:

Input: mat[][] = {{1, 1, 1, 1}, {1, 1, 0, 1}, {0, 0, 0, 0}}; Output : (1, 3) Bit difference between row numbers 1 and 3 is maximum with value 4. Bit difference between 1 and 2 is 1 and between 2 and 3 is 3. Input: mat[][] = {{1 ,1 ,1 ,1 } {1 ,0, 1 ,1 } {0 ,1 ,0 ,0 } {0, 0 ,0 ,0 }} Output : (2, 3) Bit difference between rows 2 and 3 is maximum which is 4. Input: mat[][] = {{1 ,0 ,1 ,1 } {1 ,1 ,1 ,1 } {0 ,1 ,0 ,1 } {1, 0 ,0 ,0 }} Output : (1, 3) or (2 ,4 ) or (3 ,4 ) They all are having maximum bit difference that is 3

**Simple solution** of this problem is that pick each row of binary matrix one -by -one and compute maximum bit difference with rest of the rows of matrix .at last return rows those have maximum bit difference .

An **Efficient solution** using Trie Data Structure. Below is algorithm.

1). Create an empty Trie. Every node of Trie contains two children for 0 and 1 bits. 2). Insert First Row of Binary matrix into Trie 3).Traverse rest of the rows of given Binary Matrix a). For Each Row First we search maximum bit difference with rows that we insert before that in Trie and count bit difference b). For every search we update maximum bit_diff count if needed else not store pair of index that have maximum bit difference c). At Last Print Pair

`// C++ program to Find Pair of row in Binary matrix` `// that has maximum Bit difference` `#include<bits/stdc++.h>` `using` `namespace` `std;` ` ` `// Maximum size of matrix` `const` `int` `MAX = 100;` ` ` `struct` `TrieNode` `{` ` ` `int` `leaf; ` `//store index of visited row` ` ` `struct` `TrieNode *Child[2];` `};` ` ` `// Utility function to create a new Trie node` `TrieNode * getNode()` `{` ` ` `TrieNode * newNode = ` `new` `TrieNode;` ` ` `newNode->leaf = 0;` ` ` `newNode->Child[0] = newNode->Child[1] = NULL;` ` ` `return` `newNode;` `}` ` ` `// utility function insert new row in trie` `void` `insert(TrieNode *root, ` `int` `Mat[][MAX], ` `int` `n,` ` ` `int` `row_index)` `{` ` ` `TrieNode * temp = root;` ` ` ` ` `for` `(` `int` `i=0; i<n; i++)` ` ` `{` ` ` `// Add a new Node into trie` ` ` `if` `(temp->Child[ Mat[row_index][i] ] == NULL)` ` ` `temp->Child[ Mat[row_index][i] ] = getNode();` ` ` ` ` `// move current node to point next node in trie` ` ` `temp = temp->Child[ Mat[row_index][i] ];` ` ` `}` ` ` ` ` `// store index of currently inserted row` ` ` `temp->leaf = row_index +1 ;` `}` ` ` `// utility function calculate maximum bit difference of` `// current row with previous visited row of binary matrix` `pair<` `int` `, ` `int` `> maxBitDiffCount(TrieNode * root,` ` ` `int` `Mat[][MAX], ` `int` `n, ` `int` `row_index)` `{` ` ` `TrieNode * temp = root;` ` ` `int` `count = 0;` ` ` ` ` `// Find previous visited row of binary matrix` ` ` `// that has starting bit same as current row` ` ` `for` `(` `int` `i= 0 ; i < n ; i++)` ` ` `{` ` ` `// First look for same bit in trie` ` ` `if` `(temp->Child[ Mat[row_index][i] ] != NULL)` ` ` `temp = temp->Child[ Mat[row_index][i] ];` ` ` ` ` `// Else looking for opposite bit` ` ` `else` `if` `(temp->Child[1 - Mat[row_index][i]] != NULL)` ` ` `{` ` ` `temp = temp->Child[1- Mat[row_index][i]];` ` ` `count++;` ` ` `}` ` ` `}` ` ` ` ` `int` `leaf_index = temp->leaf;` ` ` `int` `count1 = 0 ;` ` ` `temp = root;` ` ` ` ` `// Find previous visited row of binary matrix` ` ` `// that has starting bit opposite to current row` ` ` `for` `(` `int` `i= 0 ; i < n ; i++)` ` ` `{` ` ` `// First looking for opposite bit` ` ` `if` `(temp->Child[ 1 - Mat[row_index][i] ] !=NULL)` ` ` `{` ` ` `temp = temp->Child[ 1- Mat[row_index][i] ];` ` ` `count1++;` ` ` `}` ` ` ` ` `// Else look for same bit in trie` ` ` `else` `if` `(temp->Child[ Mat[row_index][i] ] != NULL)` ` ` `temp = temp->Child[ Mat[row_index][i] ];` ` ` `}` ` ` ` ` `pair <` `int` `,` `int` `> P = count1 > count ?` ` ` `make_pair(count1, temp->leaf):` ` ` `make_pair(count, leaf_index);` ` ` ` ` `// return pair that contain both bit difference` ` ` `// count and index of row with we get bit` ` ` `// difference` ` ` `return` `P;` `}` ` ` `// Returns maximum bit difference pair of row` `void` `maxDiff( ` `int` `mat[][MAX], ` `int` `n, ` `int` `m)` `{` ` ` `TrieNode * root = getNode();` ` ` ` ` `// Insert first matrix row in trie` ` ` `insert(root, mat, m, 0);` ` ` ` ` `int` `max_bit_diff = INT_MIN;` ` ` `pair <` `int` `,` `int` `> P, temp ;` ` ` ` ` `// Traverse all rest row of binary matrix` ` ` `for` `(` `int` `i = 1 ; i < n; i++)` ` ` `{` ` ` `// compute bit difference with previous visited` ` ` `// rows of matrix` ` ` `temp = maxBitDiffCount(root, mat, m ,i);` ` ` ` ` `// update maximum bit difference` ` ` `if` `(max_bit_diff < temp.first )` ` ` `{` ` ` `max_bit_diff = temp.first;` ` ` `P = make_pair( temp.second, i+1);` ` ` `}` ` ` ` ` `// insert current row value into Trie` ` ` `insert(root, mat, m, i );` ` ` `}` ` ` ` ` `// print maximum bit difference pair in row` ` ` `cout << ` `"("` `<< P.first <<` `", "` `<< P.second << ` `")"` `;` `}` ` ` `// Driver program` `int` `main()` `{` ` ` `int` `mat[][MAX] = {{0 ,1 ,0 ,1, 0 },` ` ` `{1, 0, 1 ,1 ,0 },` ` ` `{0 ,0 ,1 ,0, 1 }` ` ` `};` ` ` `maxDiff(mat, 3, 5) ;` ` ` `return` `0;` `}` |

Output:

1 , 3

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