Count unique paths is a matrix whose product of elements contains odd number of divisors

Given a matrix mat[][] of dimension NxM, the task is to count the number of unique paths from the top-left cell, i.e. mat[0][0], to the bottom-right cell, i.e. mat[N – 1][M – 1] of the given matrix such that product of elements in that path contains an odd number of divisors. Possible moves from any cell (i, j) are (i, j + 1) or (i + 1, j).

Examples:

Input: mat[][] = {{1, 1}, {3, 1}, {3, 1}}
Output: 2
Explanation: Two possible paths satisfying the condition: 

• 1->3->3->1, Product = 9, Number of Divisors of 9 are 1, 3, 9 which is odd.
• 1->1->1->1, Product = 1, Number of Divisors of 1 is 1 only which is odd.

Input: mat[][] = {{4, 1}, {4, 4}}
Output: 2
Explanation: Two possible paths satisfying the condition: 

• 4->4->4, Product = 64, Number of Divisors of 9 are 1, 2, 4, 8, 16, 32, 64 which is odd.
• 4->1->4, Product = 16, Number of Divisors of 16 are 1, 2, 4, 8, 16 which is odd.

Naive Approach: The simplest approach is to generate all possible path from the top-left cell to the bottom-right cell for the given matrix and check if the product of all the elements for all such path has an odd number of divisors by finding all the divisors using the approach discussed in this article.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

int countPaths = 0;

// Store the product of all paths
vector<int> v;

// Function to calculate and store
// all the paths product in vector
void CountUniquePaths(int a[][105], int i,
                      int j, int m,
                      int n, int ans)
{
    // Base Case
    if (i >= m || j >= n)
        return;

    // If reaches the bottom right
    // corner and product is a
    // perfect square
    if (i == m - 1 && j == n - 1) {

        // Find square root
        long double sr = sqrt(ans * a[i][j]);

        // If square root is an integer
        if ((sr - floor(sr)) == 0)
            countPaths++;
    }

    // Move towards down in the matrix
    CountUniquePaths(a, i + 1, j, m,
                     n, ans * a[i][j]);

    // Move towards right in the matrix
    CountUniquePaths(a, i, j + 1, m,
                     n, ans * a[i][j]);
}

// Driver Code
int main()
{
    int M = 3, N = 2;

    // Given matrix mat[][]
    int mat[M][105] = { { 1, 1 },
                        { 3, 1 },
                        { 3, 1 } };

    // Function Call
    CountUniquePaths(mat, 0, 0, M, N, 1);

    // Print the result
    cout << countPaths;

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;
import java.lang.*;

class GFG{

static int countPaths = 0;

// Function to calculate and store
// all the paths product in vector
static void CountUniquePaths(int[][] a, int i,
                             int j, int m,
                             int n, int ans)
{

    // Base Case
    if (i >= m || j >= n)
        return;

    // If reaches the bottom right
    // corner and product is a
    // perfect square
    if (i == m - 1 && j == n - 1)
    {

        // Find square root
        double sr = Math.sqrt(ans * a[i][j]);

        // If square root is an integer
        if ((sr - Math.floor(sr)) == 0)
            countPaths++;
    }

    // Move towards down in the matrix
     CountUniquePaths(a, i + 1, j, m,
                      n, ans * a[i][j]);

    // Move towards right in the matrix
     CountUniquePaths(a, i, j + 1, m,
                      n, ans * a[i][j]);
}

// Driver Code
public static void main (String[] args)
{
    int M = 3, N = 2;

    // Given matrix mat[][]
    int[][] mat = { { 1, 1 },
                    { 3, 1 },
                    { 3, 1 } };

    // Function call
    CountUniquePaths(mat, 0, 0, M, N, 1);

    System.out.println(countPaths);
}
}

// This code is contributed by sallagondaavinashreddy7

Python3

# Python3 program for
# the above approach
import math
countPaths = 0;

# Function to calculate
# and store all the paths
# product in vector
def CountUniquePaths(a, i, j,
                     m, n, ans):

    # Base Case
    if (i >= m or j >= n):
        return;

    # If reaches the bottom
    # right corner and product
    # is a perfect square
    global countPaths;

    if (i == m - 1 and
        j == n - 1):

        # Find square root
        sr = math.sqrt(ans * a[i][j]);

        # If square root is an integer
        if ((sr - math.floor(sr)) == 0):
            countPaths += 1;

    # Move towards down
    # in the matrix
    CountUniquePaths(a, i + 1, j,
                     m, n, ans * a[i][j]);

    # Move towards right
    # in the matrix
    CountUniquePaths(a, i, j + 1,
                     m, n, ans * a[i][j]);

# Driver Code
if __name__ == '__main__':

    M = 3; N = 2;

    # Given matrix mat
    mat = [[1, 1],
           [3, 1],
           [3, 1]];

    # Function call
    CountUniquePaths(mat, 0,
                     0, M, N, 1);

    print(countPaths);

# This code is contributed by Princi Singh

C#

// C# program for the
// above approach
using System;
class GFG{

static int countPaths = 0;

// Function to calculate and store
// all the paths product in vector
static void CountUniquePaths(int[,] a, int i,
                             int j, int m,
                             int n, int ans)
{
  // Base Case
  if (i >= m || j >= n)
    return;

  // If reaches the bottom right
  // corner and product is a
  // perfect square
  if (i == m - 1 && j == n - 1)
  {

    // Find square root
    double sr = Math.Sqrt(ans *
                          a[i, j]);

    // If square root is an integer
    if ((sr - Math.Floor(sr)) == 0)
      countPaths++;
  }

  // Move towards down in the matrix
  CountUniquePaths(a, i + 1, j, m,
                   n, ans * a[i, j]);

  // Move towards right in the matrix
  CountUniquePaths(a, i, j + 1, m,
                   n, ans * a[i, j]);
}

// Driver Code
public static void Main (String[] args)
{
  int M = 3, N = 2;

  // Given matrix mat[][]
  int[,] mat = {{1, 1},
                {3, 1},
                {3, 1}};

  // Function call
  CountUniquePaths(mat, 0, 0,
                   M, N, 1);

  Console.Write(countPaths);
}
}

// This code is contributed by Chitranayal

Javascript

<script>

// Javascript program for the
// above approach
let countPaths = 0;

// Function to calculate and store
// all the paths product in vector
function CountUniquePaths(a, i, j, m, n, ans)
{

    // Base Case
    if (i >= m || j >= n)
        return;

    // If reaches the bottom right
    // corner and product is a
    // perfect square
    if (i == m - 1 && j == n - 1)
    {

        // Find square root
        let sr = Math.sqrt(ans * a[i][j]);

        // If square root is an integer
        if ((sr - Math.floor(sr)) == 0)
            countPaths++;
    }

    // Move towards down in the matrix
     CountUniquePaths(a, i + 1, j, m,
                      n, ans * a[i][j]);

    // Move towards right in the matrix
     CountUniquePaths(a, i, j + 1, m,
                      n, ans * a[i][j]);
}

// Driver Code
let M = 3, N = 2;

// Given matrix mat[][]
let mat = [ [ 1, 1 ],
            [ 3, 1 ],
            [ 3, 1 ] ];

// Function call
CountUniquePaths(mat, 0, 0, M, N, 1);

document.write(countPaths);

// This code is contributed by souravghosh0416

</script>

Output:

2

Time Complexity: O((2^N)*sqrt(N))
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to use Dynamic Programming. Initialize an auxiliary space dp[][][] that will store the numbers having a product of all the path from top-left to end of each row for each row of the given matrix. Below are the steps:

• Initialize an 3D auxiliary space dp[][][] where dp[i][j] will store the product of all the path from cell (0, 0) to (i, j).
• To compute each state values calculate dp[i][j] by using all the values from dp(i – 1, j) and dp(i, j – 1).
• For the first row of the given matrix mat[][] store the prefix product of the first row at dp[i][0].
• For the first column of the given matrix mat[][] store the prefix product of the first column at dp[0][i].
• Now iterate over (1, 1) to (N, M) using two nested loops i and j and do the following:
  • Store the vector top at index dp[i – 1][j] and left at index dp[i][j – 1].
  • Store the product of current element mat[i][j] with the elements stored in top[] and left[] in another auxiliary array curr[].
  • Update the current dp state as dp[i][j] = curr.
• Now traverse the array stored at dp(N – 1, M – 1) and count all the numbers which is a perfect square.
• Print the final count after the above steps.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Stores the results
vector<vector<vector<int> > >
    dp(105, vector<vector<int> >(105));

// Count of unique product paths
int countPaths = 0;

// Function to check whether number
// is perfect square or not
bool isPerfectSquare(int n)
{
    long double sr = sqrt(n);

    // If square root is an integer
    return ((sr - floor(sr)) == 0);
}

// Function to calculate and store
// all the paths product in vector
void countUniquePaths(int a[][105],
                      int m, int n,
                      int ans)
{
    // Store the value a[0][0]
    dp[0][0].push_back(a[0][0]);

    // Initialize first row of dp
    for (int i = 1; i < m; i++) {

        // Find prefix product
        a[i][0] *= a[i - 1][0];
        dp[i][0].push_back(a[i][0]);
    }

    // Initialize first column of dp
    for (int i = 1; i < n; i++) {

        // Find the prefix product
        a[0][i] *= a[0][i - 1];
        dp[0][i].push_back(a[0][i]);
    }

    // Iterate over range (1, 1) to (N, M)
    for (int i = 1; i < m; i++) {
        for (int j = 1; j < n; j++) {

            // Copy  dp[i-1][j] in top[]
            vector<int> top = dp[i - 1][j];

            // Copy dp[i][j-1] into left[]
            vector<int> left = dp[i][j - 1];

            // Compute the values of current
            // state and store it in curr[]
            vector<int> curr;

            // Find the product of a[i][j]
            // with elements at top[]
            for (int k = 0;
                 k < top.size(); k++) {
                curr.push_back(top[k] * a[i][j]);
            }

            // Find the product of a[i][j]
            // with elements at left[]
            for (int k = 0;
                 k < left.size(); k++) {
                curr.push_back(left[k] * a[i][j]);
            }

            // Update the current state
            dp[i][j] = curr;
        }
    }

    // Traverse dp[m - 1][n - 1]
    for (auto i : dp[m - 1][n - 1]) {

        // Check if perfect square
        if (isPerfectSquare(i)) {
            countPaths++;
        }
    }
}

// Driver Code
int main()
{
    int M = 3, N = 4;

    // Given matrix mat[][]
    int mat[M][105] = { { 1, 2, 3, 1 },
                        { 3, 1, 2, 4 },
                        { 2, 3, 1, 1 } };

    // Function Call
    countUniquePaths(mat, M, N, 1);

    // Print the final count
    cout << countPaths;

    return 0;
}

Java

// Java program for the above approach
import java.util.*;
class GFG
{

  // Stores the results
  static ArrayList<ArrayList<ArrayList<Integer>>> dp;

  // Count of unique product paths
  static int countPaths = 0;

  // Function to check whether number
  // is perfect square or not
  static boolean isPerfectSquare(int n)
  {
    double  sr = Math.sqrt(n);

    // If square root is an integer
    return ((sr - Math.floor(sr)) == 0);
  }

  // Function to calculate and store
  // all the paths product in vector
  static void countUniquePaths(int a[][],
                               int m, int n,
                               int ans)
  {

    // Store the value a[0][0]
    dp.get(0).get(0).add(a[0][0]);

    // Initialize first row of dp
    for (int i = 1; i < m; i++)
    {

      // Find prefix product
      a[i][0] *= a[i - 1][0];
      dp.get(i).get(0).add(a[i][0]);
    }

    // Initialize first column of dp
    for (int i = 1; i < n; i++)
    {

      // Find the prefix product
      a[0][i] *= a[0][i - 1];
      dp.get(0).get(i).add(a[0][i]);
    }

    // Iterate over range (1, 1) to (N, M)
    for (int i = 1; i < m; i++)
    {
      for (int j = 1; j < n; j++)
      {

        // Copy  dp[i-1][j] in top[]
        ArrayList<Integer> top =
          dp.get(i - 1).get(j);

        // Copy dp[i][j-1] into left[]
        ArrayList<Integer> left =
          dp.get(i).get(j - 1);

        // Compute the values of current
        // state and store it in curr[]
        ArrayList<Integer> curr = new ArrayList<>();

        // Find the product of a[i][j]
        // with elements at top[]
        for (int k = 0;
             k < top.size(); k++)
        {
          curr.add(top.get(k) * a[i][j]);
        }

        // Find the product of a[i][j]
        // with elements at left[]
        for (int k = 0;
             k < left.size(); k++)
        {
          curr.add(left.get(k) * a[i][j]);
        }

        // Update the current state
        dp.get(i).set(j, curr);
      }
    }

    // Traverse dp[m - 1][n - 1]
    for (Integer i : dp.get(m - 1).get(n - 1))
    {

      // Check if perfect square
      if (isPerfectSquare(i))
      {
        countPaths++;
      }
    }
  }

  // Driver code
  public static void main (String[] args)
  {
    int M = 3, N = 4;

    // Given matrix mat[][]
    int mat[][] = { { 1, 2, 3, 1 },
                   { 3, 1, 2, 4 },
                   { 2, 3, 1, 1 } };

    dp = new ArrayList<>();

    for(int i = 0; i < 105; i++)
    {
      dp.add(new ArrayList<>());
      for(int j = 0; j < 105; j++)
        dp.get(i).add(new ArrayList<Integer>());
    }

    // Function Call
    countUniquePaths(mat, M, N, 1);

    // Print the final count
    System.out.println(countPaths);
  }
}

// This code is contributed by offbeat

Python3

# Python3 program for the above approach
from math import sqrt, floor

# Stores the results
dp = [[[] for j in range(105)]
          for k in range(105)]

# Count of unique product paths
countPaths = 0

# Function to check whether number
# is perfect square or not
def isPerfectSquare(n):

    sr = sqrt(n)

    # If square root is an integer
    return ((sr - floor(sr)) == 0)

# Function to calculate and store
# all the paths product in vector
def countUniquePaths(a, m, n, ans):

    global dp
    global countPaths

    # Store the value a[0][0]
    dp[0][0].append(a[0][0])

    # Initialize first row of dp
    for i in range(1, m):

        # Find prefix product
        a[i][0] *= a[i - 1][0]
        dp[i][0].append(a[i][0])

    # Initialize first column of dp
    for i in range(1, n):

        # Find the prefix product
        a[0][i] *= a[0][i - 1]
        dp[0][i].append(a[0][i])

    # Iterate over range (1, 1) to (N, M)
    for i in range(1, m):
        for j in range(1, n):

            # Copy  dp[i-1][j] in top[]
            top = dp[i - 1][j]

            # Copy dp[i][j-1] into left[]
            left = dp[i][j - 1]

            # Compute the values of current
            # state and store it in curr[]
            curr = []

            # Find the product of a[i][j]
            # with elements at top[]
            for k in range(len(top)):
                curr.append(top[k] * a[i][j])

            # Find the product of a[i][j]
            # with elements at left[]
            for k in range(len(left)):
                curr.append(left[k] * a[i][j])

            # Update the current state
            dp[i][j] = curr

    # Traverse dp[m - 1][n - 1]
    for i in dp[m - 1][n - 1]:

        # Check if perfect square
        if (isPerfectSquare(i)):
            countPaths += 1

# Driver Code
if __name__ == '__main__':

    M = 3
    N = 4

    # Given matrix mat[][]
    mat = [ [ 1, 2, 3, 1 ],
            [ 3, 1, 2, 4 ],
            [ 2, 3, 1, 1 ] ]

    # Function Call
    countUniquePaths(mat, M, N, 1)

    # Print the final count
    print(countPaths)

# This code is contributed by ipg2016107

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG
{

    // Stores the results
    static List<List<List<int>>> dp;

    // Count of unique product paths
    static int countPaths = 0;

    // Function to check whether number
    // is perfect square or not
    static bool isPerfectSquare(int n)
    {
        double sr = Math.Sqrt(n);

        // If square root is an integer
        return ((sr - Math.Floor(sr)) == 0);
    }

    // Function to calculate and store
    // all the paths product in vector
    static void countUniquePaths(int[,] a,
                                 int m, int n,
                                 int ans)
    {

        // Store the value a[0][0]
        dp[0][0].Add(a[0, 0]);

        // Initialize first row of dp
        for (int i = 1; i < m; i++)
        {

            // Find prefix product
            a[i, 0] *= a[i - 1, 0];
            dp[i][0].Add(a[i, 0]);
        }

        // Initialize first column of dp
        for (int i = 1; i < n; i++)
        {

            // Find the prefix product
            a[0, i] *= a[0, i - 1];
            dp[0][i].Add(a[0, i]);
        }

        // Iterate over range (1, 1) to (N, M)
        for (int i = 1; i < m; i++)
        {
            for (int j = 1; j < n; j++)
            {

                // Copy  dp[i-1][j] in top[]
                List<int> top = dp[i - 1][j];

                // Copy dp[i][j-1] into left[]
                List<int> left = dp[i][j - 1];

                // Compute the values of current
                // state and store it in curr[]
                List<int> curr = new List<int>();

                // Find the product of a[i][j]
                // with elements at top[]
                for (int k = 0;
                     k < top.Count; k++)
                {
                    curr.Add(top[k] * a[i, j]);
                }

                // Find the product of a[i][j]
                // with elements at left[]
                for (int k = 0;
                     k < left.Count; k++)
                {
                    curr.Add(left[k] * a[i, j]);
                }

                // Update the current state
                dp[i][j] = curr;
            }
        }

        // Traverse dp[m - 1][n - 1]
        foreach (int i in dp[m - 1][n - 1])
        {

            // Check if perfect square
            if (isPerfectSquare(i))
            {
                countPaths++;
            }
        }
    }

    // Driver code
    public static void Main()
    {
        int M = 3, N = 4;

        // Given matrix mat[][]
        int[,] mat = { { 1, 2, 3, 1 },
                   { 3, 1, 2, 4 },
                   { 2, 3, 1, 1 } };

        dp = new List<List<List<int>>>();

        for (int i = 0; i < 105; i++)
        {
            dp.Add(new List<List<int>>());
            for (int j = 0; j < 105; j++)
                dp[i].Add(new List<int>());
        }

        // Function Call
        countUniquePaths(mat, M, N, 1);

        // Print the final count
        Console.Write(countPaths);
    }
}

// This code is contributed by Saurabh Jaiswal

Javascript

<script>

// Javascript program for the above approach

// Stores the results
let dp = [];

// Count of unique product paths
let countPaths = 0;

// Function to check whether number
// is perfect square or not
function isPerfectSquare(n)
{
    let sr = Math.sqrt(n);

    // If square root is an integer
    return((sr - Math.floor(sr)) == 0);
}

// Function to calculate and store
// all the paths product in vector
function countUniquePaths(a, m, n, ans)
{

    // Store the value a[0][0]
    dp[0][0].push(a[0][0]);

    // Initialize first row of dp
    for(let i = 1; i < m; i++)
    {

        // Find prefix product
        a[i][0] *= a[i - 1][0];
        dp[i][0].push(a[i][0]);
    }

    // Initialize first column of dp
    for(let i = 1; i < n; i++)
    {

        // Find the prefix product
        a[0][i] *= a[0][i - 1];
        dp[0][i].push(a[0][i]);
    }

    // Iterate over range (1, 1) to (N, M)
    for(let i = 1; i < m; i++)
    {
        for(let j = 1; j < n; j++)
        {

            // Copy  dp[i-1][j] in top[]
            let top = dp[i - 1][j];

            // Copy dp[i][j-1] into left[]
            let left = dp[i][j - 1];

            // Compute the values of current
            // state and store it in curr[]
            let curr = [];

            // Find the product of a[i][j]
            // with elements at top[]
            for(let k = 0; k < top.length; k++)
            {
                curr.push(top[k] * a[i][j]);
            }

            // Find the product of a[i][j]
            // with elements at left[]
            for(let k = 0; k < left.length; k++)
            {
                curr.push(left[k] * a[i][j]);
            }

            // Update the current state
            dp[i][j] = curr;
        }
    }

    // Traverse dp[m - 1][n - 1]
    for(let i = 0; i < dp[m - 1][n - 1].length; i++)
    {

        // Check if perfect square
        if (isPerfectSquare(dp[m - 1][n - 1][i]))
        {
            countPaths++;
        }
    }
}

// Driver code
let M = 3, N = 4;
let mat = [ [ 1, 2, 3, 1 ],
            [ 3, 1, 2, 4 ],
            [ 2, 3, 1, 1 ] ];
for(let i = 0; i < 105; i++)
{
    dp.push([]);
    for(let j = 0; j < 105; j++)
        dp[i].push([]);
}

// Function Call
countUniquePaths(mat, M, N, 1);

// Print the final count
document.write(countPaths);

// This code is contributed by avanitrachhadiya2155

</script>

Output:

3

Time Complexity: O(N³)
Auxiliary Space: O(N³)