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Number of ways to reach the end of matrix with non-zero AND value

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  • Difficulty Level : Expert
  • Last Updated : 26 Nov, 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[0][0] 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 <bits/stdc++.h>
#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[0][0], 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




<script>
 
// Javascript implementation of the approach
var n = 3;
var maxV = 20;
 
// 3d array to store
// states of dp
var dp = new Array(n);
 
for(var i = 0; i<n; i++)
{
    dp[i] = new Array(n);
    for(var j =0; j<n;j++)
    {
        dp[i][j] = new Array(maxV);
    }
}
 
var v = new Array(n);
 
// Array to determine whether
// a state has been solved before
for(var i = 0; i<n; i++)
{
    v[i] = new Array(n);
    for(var j =0; j<n;j++)
    {
        v[i][j] = new Array(maxV);
    }
}
 
// Function to return the count of required paths
function countWays(i, j, x, 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])
        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
var arr = [ [ 1, 2, 1 ],
                  [ 1, 1, 0 ],
                  [ 2, 1, 1 ] ];
document.write( countWays(0, 0, arr[0][0], arr));
 
 
</script>

Output: 

1

 

Time Complexity: O(n2)

Auxiliary Space: O(n4 * maxV)


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