Number of paths with exactly k coins

Given a matrix where every cell has some number of coins. Count number of ways to reach bottom right from top left with exactly k coins. We can move to (i+1, j) and (i, j+1) from a cell (i, j).

Example:

Input:  k = 12
        mat[][] = { {1, 2, 3},
                    {4, 6, 5},
                    {3, 2, 1}
                  };
Output:  2
There are two paths with 12 coins
1 -> 2 -> 6 -> 2 -> 1
1 -> 2 -> 3 -> 5 -> 1

We strongly recommend that you click here and practice it, before moving on to the solution.

The above problem can be recursively defined as below:  

pathCount(m, n, k):   Number of paths to reach mat[m][n] from mat[0][0] 
                      with exactly k coins

If (m == 0 and n == 0)
   return 1 if mat[0][0] == k else return 0
Else:
    pathCount(m, n, k) = pathCount(m-1, n, k - mat[m][n]) + 
                         pathCount(m, n-1, k - mat[m][n]) 

Below is the implementation of above recursive algorithm. 

C++14

// A Naive Recursive C++ program
// to count paths with exactly
// 'k' coins
#include <bits/stdc++.h>
#define R 3
#define C 3
using namespace std;

// Recursive function to count paths with sum k from
// (0, 0) to (m, n)
int pathCountRec(int mat[][C], int m, int n, int k)
{
    // Base cases
    if (m < 0 || n < 0 || k < 0) return 0;
    if (m==0 && n==0) return (k == mat[m][n]);

    // (m, n) can be reached either through (m-1, n) or
    // through (m, n-1)
    return pathCountRec(mat, m-1, n, k-mat[m][n]) +
           pathCountRec(mat, m, n-1, k-mat[m][n]);
}

// A wrapper over pathCountRec()
int pathCount(int mat[][C], int k)
{
    return pathCountRec(mat, R-1, C-1, k);
}

// Driver program
int main()
{
    int k = 12;
    int mat[R][C] = { {1, 2, 3},
                      {4, 6, 5},
                      {3, 2, 1}
                  };
    cout << pathCount(mat, k);
    return 0;
}

Java

// A Naive Recursive Java program to
// count paths with exactly 'k' coins

class GFG {

    static final int R = 3;
    static final int C = 3;

// Recursive function to count paths with sum k from
// (0, 0) to (m, n)
    static int pathCountRec(int mat[][], int m, int n, int k) {
        // Base cases
        if (m < 0 || n < 0 || k < 0) {
            return 0;
        }
        if (m == 0 && n == 0 && (k == mat[m][n])) {
            return 1;
        }

        // (m, n) can be reached either through (m-1, n) or
        // through (m, n-1)
        return pathCountRec(mat, m - 1, n, k - mat[m][n])
                + pathCountRec(mat, m, n - 1, k - mat[m][n]);
    }

// A wrapper over pathCountRec()
    static int pathCount(int mat[][], int k) {
        return pathCountRec(mat, R - 1, C - 1, k);
    }

    // Driver code
    public static void main(String[] args) {
        int k = 12;
        int mat[][] = {{1, 2, 3},
        {4, 6, 5},
        {3, 2, 1}
        };
        System.out.println(pathCount(mat, k));
    }
}

/* This Java code is contributed by Rajput-Ji*/

Python3

# A Naive Recursive Python program to
# count paths with exactly 'k' coins

R = 3
C = 3

# Recursive function to count paths
# with sum k from (0, 0) to (m, n)
def pathCountRec(mat, m, n, k):

    # Base cases
    if m < 0 or n < 0 or k < 0:
        return 0
    elif m == 0 and n == 0:
        return k == mat[m][n]

    # #(m, n) can be reached either
    # through (m-1, n) or through
    # (m, n-1)
    return (pathCountRec(mat, m-1, n, k-mat[m][n])
         + pathCountRec(mat, m, n-1, k-mat[m][n]))

# A wrapper over pathCountRec()
def pathCount(mat, k):
    return pathCountRec(mat, R-1, C-1, k)

# Driver Program
k = 12
mat = [[1, 2, 3],
       [4, 6, 5],
       [3, 2, 1]]

print(pathCount(mat, k))

# This code is contributed by Shrikant13.

C#

using System;

// A Naive Recursive c# program to
// count paths with exactly 'k' coins

public class GFG
{

    public const int R = 3;
    public const int C = 3;

// Recursive function to count paths with sum k from
// (0, 0) to (m, n)
    public static int pathCountRec(int[][] mat, int m, int n, int k)
    {
        // Base cases
        if (m < 0 || n < 0 || k < 0)
        {
            return 0;
        }
        if (m == 0 && n == 0 && (k == mat[m][n]))
        {
            return 1;
        }

        // (m, n) can be reached either through (m-1, n) or
        // through (m, n-1)
        return pathCountRec(mat, m - 1, n, k - mat[m][n])
                + pathCountRec(mat, m, n - 1, k - mat[m][n]);
    }

// A wrapper over pathCountRec()
    public static int pathCount(int[][] mat, int k)
    {
        return pathCountRec(mat, R - 1, C - 1, k);
    }

    // Driver code
    public static void Main(string[] args)
    {
        int k = 12;
        int[][] mat = new int[][]
        {
            new int[] {1, 2, 3},
            new int[] {4, 6, 5},
            new int[] {3, 2, 1}
        };
        Console.WriteLine(pathCount(mat, k));
    }
}

// This code is contributed by Shrikant13

PHP

<?php
// A Naive Recursive PHP program to
// count paths with exactly 'k' coins

$R = 3;
$C = 3;

// Recursive function to count paths
// with sum k from (0, 0) to (m, n)
function pathCountRec( $mat, $m, $n, $k)
{

    // Base cases
    if ($m < 0 or $n < 0 or $k < 0)
        return 0;
    if ($m == 0 and $n == 0)
        return ($k == $mat[$m][$n]);

    // (m, n) can be reached either
    // through (m-1, n) or through
    // (m, n-1)
    return pathCountRec($mat, $m - 1,
                $n, $k - $mat[$m][$n])
              + pathCountRec($mat, $m,
           $n - 1, $k - $mat[$m][$n]);
}

// A wrapper over pathCountRec()
function pathCount($mat, $k)
{
    global $R, $C;
    return pathCountRec($mat, $R-1,
                            $C-1, $k);
}

// Driver program

    $k = 12;
    $mat = array(array(1, 2, 3),
                 array(4, 6, 5),
                 array(3, 2, 1) );

    echo pathCount($mat, $k);

// This code is contributed by anuj_67.
?>

Javascript

<script>

// A Naive Recursive Javascript program
// to count paths with exactly
// 'k' coins
let R = 3;
let C = 3;

// Recursive function to count paths with
// sum k from (0, 0) to (m, n)
function pathCountRec(mat, m, n, k)
{

    // Base cases
    if (m < 0 || n < 0 || k < 0)
        return 0;
    if (m == 0 && n == 0)
        return (k == mat[m][n]);

    // (m, n) can be reached either through (m-1, n) or
    // through (m, n-1)
    return pathCountRec(mat, m - 1, n, k - mat[m][n]) +
           pathCountRec(mat, m, n - 1, k - mat[m][n]);
}

// A wrapper over pathCountRec()
function pathCount(mat, k)
{
    return pathCountRec(mat, R - 1, C - 1, k);
}

// Driver code
let k = 12;
let mat = [ [ 1, 2, 3 ],
            [ 4, 6, 5 ],
            [ 3, 2, 1 ] ];

document.write(pathCount(mat, k));

// This code is contributed by divyesh072019

</script>

Output

2

Time Complexity: O(2^m+n)
Auxiliary Space: O(m*n), since m*n extra space has been taken.

The time complexity of above solution recursive is exponential. We can solve this problem in Pseudo Polynomial Time (time complexity is dependent on numeric value of input) using Dynamic Programming. The idea is to use a 3 dimensional table dp[m][n][k] where m is row number, n is column number and k is number of coins. Below is Dynamic Programming based the implementation.

C++14

// A Dynamic Programming based C++ program to count paths with
// exactly 'k' coins
#include <bits/stdc++.h>
#define R 3
#define C 3
#define MAX_K 1000
using namespace std;

int dp[R][C][MAX_K];

int pathCountDPRecDP(int mat[][C], int m, int n, int k)
{
    // Base cases
    if (m < 0 || n < 0 || k < 0) return 0;
    if (m==0 && n==0) return (k == mat[m][n]);

    // If this subproblem is already solved
    if (dp[m][n][k] != -1) return dp[m][n][k];

    // (m, n) can be reached either through (m-1, n) or
    // through (m, n-1)
    dp[m][n][k] = pathCountDPRecDP(mat, m-1, n, k-mat[m][n]) +
                  pathCountDPRecDP(mat, m, n-1, k-mat[m][n]);

    return dp[m][n][k];
}

// This function mainly initializes dp[][][] and calls
// pathCountDPRecDP()
int pathCountDP(int mat[][C], int k)
{
    memset(dp, -1, sizeof dp);
    return pathCountDPRecDP(mat, R-1, C-1, k);
}

// Driver Program to test above functions
int main()
{
    int k = 12;
    int mat[R][C] = { {1, 2, 3},
                      {4, 6, 5},
                      {3, 2, 1}
                  };
    cout << pathCountDP(mat, k);
    return 0;
}

Java

// A Dynamic Programming based JAVA program to count paths
// with exactly 'k' coins

class GFG {

    static final int R = 3;
    static final int C = 3;
    static final int MAX_K = 100;

    static int[][][] dp = new int[R][C][MAX_K];

    static int pathCountDPRecDP(int[][] mat, int m, int n,
                                int k)
    {
        // Base cases
        if (m < 0 || n < 0 || k < 0)
            return 0;
        if (m == 0 && n == 0)
            return (k == mat[m][n] ? 1 : 0);

        // If this subproblem is already solved
        if (dp[m][n][k] != -1)
            return dp[m][n][k];

        // (m, n) can be reached either through (m-1, n) or
        // through (m, n-1)
        dp[m][n][k]
            = pathCountDPRecDP(mat, m - 1, n, k - mat[m][n])
              + pathCountDPRecDP(mat, m, n - 1,
                                 k - mat[m][n]);

        return dp[m][n][k];
    }

    // This function mainly initializes dp[][][] and calls
    // pathCountDPRecDP()
    static int pathCountDP(int[][] mat, int k)
    {
        for (int i = 0; i < R; i++)
            for (int j = 0; j < C; j++)
                for (int l = 0; l < MAX_K; l++)
                    dp[i][j][l] = -1;

        return pathCountDPRecDP(mat, R - 1, C - 1, k);
    }

    // Driver Program to test above functions
    public static void main(String[] args)
    {
        int k = 12;
        int[][] mat = new int[][] { new int[] { 1, 2, 3 },
                                    new int[] { 4, 6, 5 },
                                    new int[] { 3, 2, 1 } };

        System.out.println(pathCountDP(mat, k));
    }
}

// This code is contributed by ihritik

Python3

# A Dynamic Programming based Python3 program to
# count paths with exactly 'k' coins
R = 3
C = 3
MAX_K = 1000

def pathCountDPRecDP(mat, m, n, k):

    # Base cases
    if m < 0 or n < 0 or k < 0:
        return 0
    elif m == 0 and n == 0:
        return k == mat[m][n]

    # If this subproblem is already solved
    if (dp[m][n][k] != -1):
        return dp[m][n][k]

    # #(m, n) can be reached either
    # through (m-1, n) or through
    # (m, n-1)
    dp[m][n][k] = (pathCountDPRecDP(mat, m - 1, n, k - mat[m][n]) +
                   pathCountDPRecDP(mat, m, n - 1, k - mat[m][n]))

    return dp[m][n][k]

# A wrapper over pathCountDPRecDP()
def pathCountDP(mat, k):
    return pathCountDPRecDP(mat, R - 1, C - 1, k)

# Driver Code
k = 12

# Initialising dp[][][]
dp = [[ [-1 for col in range(MAX_K)]
            for col in range(C)]
            for row in range(R)]

mat = [[1, 2, 3],
       [4, 6, 5],
       [3, 2, 1]]

print(pathCountDP(mat, k))

# This code is contributed by ashutosh450

C#

// A Dynamic Programming based C# program
// to count paths with exactly 'k' coins
using System;

class GFG
{
    static readonly int R = 3;
    static readonly int C = 3;
    static readonly int MAX_K = 100;

    static int [,,]dp = new int[R, C, MAX_K];

    static int pathCountDPRecDP(int [,]mat, int m,
                                int n, int k)
    {
        // Base cases
        if (m < 0 || n < 0 || k < 0) return 0;
        if (m == 0 && n == 0)
            return (k == mat[m, n] ? 1 : 0);

        // If this subproblem is already solved
        if (dp[m, n, k] != -1) return dp[m, n, k];

        // (m, n) can be reached either through (m-1, n) or
        // through (m, n-1)
        dp[m, n, k] = pathCountDPRecDP(mat, m - 1, n, k - mat[m, n]) +
                      pathCountDPRecDP(mat, m, n - 1, k - mat[m, n]);

        return dp[m, n, k];
    }

    // This function mainly initializes [,]dp[] and
    // calls pathCountDPRecDP()
    static int pathCountDP(int [,]mat, int k)
    {
        for(int i = 0; i < R; i++)
            for(int j = 0; j < C; j++)
                for(int l = 0; l < MAX_K; l++)
                dp[i, j, l] = -1;

        return pathCountDPRecDP(mat, R - 1, C - 1, k);
    }

    // Driver Code
    public static void Main(String []args)
    {
        int k = 12;
        int[,] mat = { {1, 2, 3},
                       {4, 6, 5},
                       {3, 2, 1}};

        Console.WriteLine(pathCountDP(mat, k));
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>
// A Dynamic Programming based javascript program to count paths with
// exactly 'k' coins
     var R = 3;
     var C = 3;
     var MAX_K = 100;

     var dp = Array(R).fill().map(()=>Array(C).fill().map(()=>Array(MAX_K).fill(0)));

    function pathCountDPRecDP(mat , m , n , k) {
        // Base cases
        if (m < 0 || n < 0 || k < 0)
            return 0;
        if (m == 0 && n == 0)
            return (k == mat[m][n] ? 1 : 0);

        // If this subproblem is already solved
        if (dp[m][n][k] != -1)
            return dp[m][n][k];

        // (m, n) can be reached either through (m-1, n) or
        // through (m, n-1)
        dp[m][n][k] = pathCountDPRecDP(mat, m - 1, n, k - mat[m][n]) + pathCountDPRecDP(mat, m, n - 1, k - mat[m][n]);

        return dp[m][n][k];
    }

    // This function mainly initializes dp and calls
    // pathCountDPRecDP()
    function pathCountDP(mat , k) {
        for (i = 0; i < R; i++)
            for (j = 0; j < C; j++)
                for (l = 0; l < MAX_K; l++)
                    dp[i][j][l] = -1;

        return pathCountDPRecDP(mat, R - 1, C - 1, k);
    }

    // Driver Program to test above functions

        var k = 12;
        var mat = [[ 1, 2, 3 ],[ 4, 6, 5 ], [ 3, 2, 1 ] ];

        document.write(pathCountDP(mat, k));


// This code contributed by aashish1995
</script>

Output

2

Time complexity of this solution is O(m*n*k). 
Auxiliary Space: O(m*n*k)