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Number of ways to reach the end of matrix with non-zero AND value
• Difficulty Level : Expert
• Last Updated : 13 May, 2021

Given an N * N matrix arr[][] consisting of non-negative integers, the task is to find the number of ways to reach arr[N – 1][N – 1] with a non-zero AND value starting from the arr by going down or right in every move. Whenever a cell arr[i][j] is reached, ‘AND’ value is updated as currentVal & arr[i][j].

Examples:

Input: arr[][] = {
{1, 1, 1},
{1, 1, 1},
{1, 1, 1}}
Output:
All the paths will give non-zero and value.
Thus, number of ways equals 6.
Input: arr[][] = {
{1, 1, 2},
{1, 2, 1},
{2, 1, 1}}
Output: 0

Approach: This problem can be solved using dynamic programming. First, we need to decide the states of the DP. For every cell arr[i][j] and a number X, we will store the number of ways to reach the arr[N – 1][N – 1] from arr[i][j] with non-zero AND where X is the AND value of path till now. Thus, our solution will use 3-dimensional dynamic programming, two for the coordinates of the cells and one for X.
The required recurrence relation is:

dp[i][j][X] = dp[i][j + 1][X & arr[i][j]] + dp[i + 1][j][X & arr[i][j]]

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#define n 3``#define maxV 20``using` `namespace` `std;` `// 3d array to store``// states of dp``int` `dp[n][n][maxV];` `// Array to determine whether``// a state has been solved before``int` `v[n][n][maxV];` `// Function to return the count of required paths``int` `countWays(``int` `i, ``int` `j, ``int` `x, ``int` `arr[][n])``{` `    ``// Base cases``    ``if` `(i == n || j == n)``        ``return` `0;` `    ``x = (x & arr[i][j]);``    ``if` `(x == 0)``        ``return` `0;` `    ``if` `(i == n - 1 && j == n - 1)``        ``return` `1;` `    ``// If a state has been solved before``    ``// it won't be evaluated again``    ``if` `(v[i][j][x])``        ``return` `dp[i][j][x];` `    ``v[i][j][x] = 1;` `    ``// Recurrence relation``    ``dp[i][j][x] = countWays(i + 1, j, x, arr)``                  ``+ countWays(i, j + 1, x, arr);` `    ``return` `dp[i][j][x];``}` `// Driver code``int` `main()``{``    ``int` `arr[n][n] = { { 1, 2, 1 },``                      ``{ 1, 1, 0 },``                      ``{ 2, 1, 1 } };` `    ``cout << countWays(0, 0, arr, arr);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``class` `GFG {` `    ``static` `int` `n = ``3``;``    ``static` `int` `maxV = ``20``;` `    ``// 3d array to store``    ``// states of dp``    ``static` `int``[][][] dp = ``new` `int``[n][n][maxV];` `    ``// Array to determine whether``    ``// a state has been solved before``    ``static` `int``[][][] v = ``new` `int``[n][n][maxV];` `    ``// Function to return the count of required paths``    ``static` `int` `countWays(``int` `i, ``int` `j,``                         ``int` `x, ``int` `arr[][])``    ``{` `        ``// Base cases``        ``if` `(i == n || j == n) {``            ``return` `0``;``        ``}` `        ``x = (x & arr[i][j]);``        ``if` `(x == ``0``) {``            ``return` `0``;``        ``}` `        ``if` `(i == n - ``1` `&& j == n - ``1``) {``            ``return` `1``;``        ``}` `        ``// If a state has been solved before``        ``// it won't be evaluated again``        ``if` `(v[i][j][x] == ``1``) {``            ``return` `dp[i][j][x];``        ``}` `        ``v[i][j][x] = ``1``;` `        ``// Recurrence relation``        ``dp[i][j][x] = countWays(i + ``1``, j, x, arr)``                      ``+ countWays(i, j + ``1``, x, arr);` `        ``return` `dp[i][j][x];``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `arr[][] = { { ``1``, ``2``, ``1` `},``                        ``{ ``1``, ``1``, ``0` `},``                        ``{ ``2``, ``1``, ``1` `} };` `        ``System.out.println(countWays(``0``, ``0``, arr[``0``][``0``], arr));``    ``}``}` `// This code contributed by Rajput-Ji`

## Python3

 `# Python3 implementation of the approach``n ``=` `3``maxV ``=` `20` `# 3d array to store states of dp``dp ``=` `[[[``0` `for` `i ``in` `range``(maxV)]``          ``for` `i ``in` `range``(n)]``          ``for` `i ``in` `range``(n)]` `# Array to determine whether``# a state has been solved before``v ``=` `[[[``0` `for` `i ``in` `range``(maxV)]``         ``for` `i ``in` `range``(n)]``         ``for` `i ``in` `range``(n)]` `# Function to return``# the count of required paths``def` `countWays(i, j, x, arr):` `    ``# Base cases``    ``if` `(i ``=``=` `n ``or` `j ``=``=` `n):``        ``return` `0` `    ``x ``=` `(x & arr[i][j])``    ``if` `(x ``=``=` `0``):``        ``return` `0` `    ``if` `(i ``=``=` `n ``-` `1` `and` `j ``=``=` `n ``-` `1``):``        ``return` `1` `    ``# If a state has been solved before``    ``# it won't be evaluated again``    ``if` `(v[i][j][x]):``        ``return` `dp[i][j][x]` `    ``v[i][j][x] ``=` `1` `    ``# Recurrence relation``    ``dp[i][j][x] ``=` `countWays(i ``+` `1``, j, x, arr) ``+` `\``                  ``countWays(i, j ``+` `1``, x, arr);` `    ``return` `dp[i][j][x]` `# Driver code``arr ``=` `[[``1``, ``2``, ``1` `],``       ``[``1``, ``1``, ``0` `],``       ``[``2``, ``1``, ``1` `]]` `print``(countWays(``0``, ``0``, arr[``0``][``0``], arr))` `# This code is contributed by Mohit Kumar`

## C#

 `// C# implementation of the approach``using` `System;` `class` `GFG``{` `    ``static` `int` `n = 3;``    ``static` `int` `maxV = 20;` `    ``// 3d array to store``    ``// states of dp``    ``static` `int``[,,] dp = ``new` `int``[n, n, maxV];` `    ``// Array to determine whether``    ``// a state has been solved before``    ``static` `int``[,,] v = ``new` `int``[n, n, maxV];` `    ``// Function to return the count of required paths``    ``static` `int` `countWays(``int` `i, ``int` `j,``                        ``int` `x, ``int` `[,]arr)``    ``{` `        ``// Base cases``        ``if` `(i == n || j == n)``        ``{``            ``return` `0;``        ``}` `        ``x = (x & arr[i, j]);``        ``if` `(x == 0)``        ``{``            ``return` `0;``        ``}` `        ``if` `(i == n - 1 && j == n - 1)``        ``{``            ``return` `1;``        ``}` `        ``// If a state has been solved before``        ``// it won't be evaluated again``        ``if` `(v[i, j, x] == 1)``        ``{``            ``return` `dp[i, j, x];``        ``}` `        ``v[i, j, x] = 1;` `        ``// Recurrence relation``        ``dp[i, j, x] = countWays(i + 1, j, x, arr)``                    ``+ countWays(i, j + 1, x, arr);` `        ``return` `dp[i, j, x];``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``int` `[,]arr = { { 1, 2, 1 },``                        ``{ 1, 1, 0 },``                        ``{ 2, 1, 1 } };` `    ``Console.WriteLine(countWays(0, 0, arr[0,0], arr));``    ``}``}` `// This code is contributed by AnkitRai01`

## Javascript

 ``
Output:
`1`

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