Count the number of ways to traverse a Matrix

The problem involves counting the number of ways to traverse a matrix from the top-left corner to the bottom-right corner. The matrix has rows and columns, and we can only move down or right at each step. The goal is to find the total number of unique paths that can be taken to reach the bottom-right corner.

Problem Statement

Given a matrix with rows and columns, the task is to count the number of ways to traverse the matrix from the top-left corner to the bottom-right corner, only moving down or right.

Example Scenario

Consider a matrix with 9 rows and 7 columns. We want to count the number of unique paths to traverse this matrix. The result is 3003.

Idea to Solve the Problem

To solve this problem, we can use a recursive approach. At each step, we have two choices: either move down or move right. We recursively calculate the number of paths by summing up the number of paths from the cell below and the cell to the right.

Pseudocode

int countPath(int i, int j, int row, int col)
{
    if (i == row && col == j)
        return 1;
    else if (i <= row && j <= col)
        return countPath(i + 1, j, row, col) + countPath(i, j + 1, row, col);
    else
        return 0;
}

Algorithm Explanation

1. Implement a function countPath that takes four arguments: i (current row), j (current column), row (total number of rows), and col (total number of columns).
2. Implement base cases:
   • If i reaches the last row and j reaches the last column, return 1 (we reached the bottom-right corner).
   • If both i and j are within the matrix bounds, recursively calculate the number of paths from the cell below (i + 1) and the cell to the right (j + 1).
   • If we go out of bounds, return 0.
3. The recursive cases involve adding the number of paths from the cell below and the cell to the right.

Code Solution

• C
• Java
• C++
• C#
• Php
• Node Js
• Python
• Ruby
• Scala
• Swift 4
• Kotlin

// C Program
// Count the number of ways to traverse a Matrix
#include <stdio.h>

// Count matrix path
int countPath(int i, int j, int row, int col)
{
	if (i == row && col == j)
	{
		return 1;
	}
	else if (i <= row && j <= col)
	{
		//  Count matrix path recursively
		return countPath(i + 1, j, row, col) + countPath(i, j + 1, row, col);
	}
	else
	{
		return 0;
	}
}
// Handles the request to count path of the matrix
void paths(int row, int col)
{
	if (row <= 0 || col <= 0)
	{
		return;
	}
	// Display given rows and columns
	printf(" Row : %d, Col : %d", row, col);
	// Display calculated result
	printf("\n Paths : %d\n\n", countPath(0, 0, row - 1, col - 1));
}
int main()
{
	// Test Cases
	paths(9, 7);
	paths(3, 7);
	paths(4, 4);
	return 0;
}

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

/*
    Java Program
    Count the number of ways to traverse a Matrix
*/

public class Counting
{

    // Count matrix path
    public int countPath(int i, int j, int row, int col)
    {
        if (i == row && col == j)
        {
            return 1;
        }
        else if (i <= row && j <= col)
        {
            //  Count matrix path recursively
            return countPath(i + 1, j, row, col) + 
                   countPath(i, j + 1, row, col);
        }
        else
        {
            return 0;
        }
    }
    // Handles the request to count path of the matrix
    public void paths(int row, int col)
    {
        if (row <= 0 || col <= 0)
        {
            return;
        }
        // Display given rows and columns
        System.out.print(" Row : " + row + ", Col : " + col );
        // Display calculated result
        System.out.print("\n Paths : " + 
                         countPath(0, 0, row - 1, col - 1) + "\n\n");
    }
    public static void main(String[] args)
    {

        Counting task = new Counting();
        // Test Cases
        task.paths(9, 7);
        task.paths(3, 7);
        task.paths(4, 4);
    }
}

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

// Include header file
#include <iostream>
using namespace std;

/*
    C++ Program
    Count the number of ways to traverse a Matrix
*/

class Counting
{
	public:
		// Count matrix path
		int countPath(int i, int j, int row, int col)
		{
			if (i == row && col == j)
			{
				return 1;
			}
			else if (i <= row && j <= col)
			{
				//  Count matrix path recursively
				return this->countPath(i + 1, j, row, col) 
                  + this->countPath(i, j + 1, row, col);
			}
			else
			{
				return 0;
			}
		}
	// Handles the request to count path of the matrix
	void paths(int row, int col)
	{
		if (row <= 0 || col <= 0)
		{
			return;
		}
		// Display given rows and columns
		cout << " Row : " << row << ", Col : " << col;
		// Display calculated result
		cout << "\n Paths : " << this->countPath(0, 0, row - 1, col - 1) 
      		 << "\n\n";
	}
};
int main()
{
	Counting *task = new Counting();
	// Test Cases
	task->paths(9, 7);
	task->paths(3, 7);
	task->paths(4, 4);
	return 0;
}

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

// Include namespace system
using System;
/*
    Csharp Program
    Count the number of ways to traverse a Matrix
*/
public class Counting
{
	// Count matrix path
	public int countPath(int i, int j, int row, int col)
	{
		if (i == row && col == j)
		{
			return 1;
		}
		else if (i <= row && j <= col)
		{
			//  Count matrix path recursively
			return this.countPath(i + 1, j, row, col) + 
              	   this.countPath(i, j + 1, row, col);
		}
		else
		{
			return 0;
		}
	}
	// Handles the request to count path of the matrix
	public void paths(int row, int col)
	{
		if (row <= 0 || col <= 0)
		{
			return;
		}
		// Display given rows and columns
		Console.Write(" Row : " + row + ", Col : " + col);
		// Display calculated result
		Console.WriteLine("\n Paths : " + this.countPath(0, 0, row - 1, col - 1) + "\n");
	}
	public static void Main(String[] args)
	{
		Counting task = new Counting();
		// Test Cases
		task.paths(9, 7);
		task.paths(3, 7);
		task.paths(4, 4);
	}
}

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

<?php
/*
    Php Program
    Count the number of ways to traverse a Matrix
*/
class Counting
{
	// Count matrix path
	public	function countPath($i, $j, $row, $col)
	{
		if ($i == $row && $col == $j)
		{
			return 1;
		}
		else if ($i <= $row && $j <= $col)
		{
			//  Count matrix path recursively
			return $this->countPath($i + 1, $j, $row, $col) 
              	  + $this->countPath($i, $j + 1, $row, $col);
		}
		else
		{
			return 0;
		}
	}
	// Handles the request to count path of the matrix
	public	function paths($row, $col)
	{
		if ($row <= 0 || $col <= 0)
		{
			return;
		}
		// Display given rows and columns
		echo(" Row : ".$row.
			", Col : ".$col);
		// Display calculated result
		echo("\n Paths : ".$this->countPath(0, 0, $row - 1, $col - 1).
			"\n\n");
	}
}

function main()
{
	$task = new Counting();
	// Test Cases
	$task->paths(9, 7);
	$task->paths(3, 7);
	$task->paths(4, 4);
}
main();

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

/*
    Node JS Program
    Count the number of ways to traverse a Matrix
*/
class Counting
{
	// Count matrix path
	countPath(i, j, row, col)
	{
		if (i == row && col == j)
		{
			return 1;
		}
		else if (i <= row && j <= col)
		{
			//  Count matrix path recursively
			return this.countPath(i + 1, j, row, col) 
              + this.countPath(i, j + 1, row, col);
		}
		else
		{
			return 0;
		}
	}
	// Handles the request to count path of the matrix
	paths(row, col)
	{
		if (row <= 0 || col <= 0)
		{
			return;
		}
		// Display given rows and columns
		process.stdout.write(" Row : " + row + ", Col : " + col);
		// Display calculated result
		console.log("\n Paths : " 
                             + this.countPath(0, 0, row - 1, col - 1) 
                             + "\n");
	}
}

function main()
{
	var task = new Counting();
	// Test Cases
	task.paths(9, 7);
	task.paths(3, 7);
	task.paths(4, 4);
}
main();

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

#    Python 3 Program
#    Count the number of ways to traverse a Matrix
class Counting :
	#  Count matrix path
	def countPath(self, i, j, row, col) :
		if (i == row and col == j) :
			return 1
		elif (i <= row and j <= col) :
			#   Count matrix path recursively
			return self.countPath(
              i + 1, j, row, col
            ) + self.countPath(
              i, j + 1, row, col
            )
		else :
			return 0
		
	
	#  Handles the request to count path of the matrix
	def paths(self, row, col) :
		if (row <= 0 or col <= 0) :
			return
		
		#  Display given rows and columns
		print(" Row : ", row ,", Col : ", col, end = "")
		#  Display calculated result
		print("\n Paths : ", self.countPath(0, 0, row - 1, col - 1) ,"\n")
	

def main() :
	task = Counting()
	#  Test Cases
	task.paths(9, 7)
	task.paths(3, 7)
	task.paths(4, 4)

if __name__ == "__main__": main()

 Row :  9 , Col :  7
 Paths :  3003

 Row :  3 , Col :  7
 Paths :  28

 Row :  4 , Col :  4
 Paths :  20

#    Ruby Program
#    Count the number of ways to traverse a Matrix
class Counting 
	#  Count matrix path
	def countPath(i, j, row, col) 
		if (i == row && col == j) 
			return 1
		elsif (i <= row && j <= col) 
			#   Count matrix path recursively
			return self.countPath(i + 1, j, row, col) + 
              self.countPath(i, j + 1, row, col)
		else
 
			return 0
		end

	end

	#  Handles the request to count path of the matrix
	def paths(row, col) 
		if (row <= 0 || col <= 0) 
			return
		end

		#  Display given rows and columns
		print(" Row : ", row ,", Col : ", col)
		#  Display calculated result
		print("\n Paths : ", 
              self.countPath(0, 0, row - 1, col - 1) ,
              "\n\n")
	end

end

def main() 
	task = Counting.new()
	#  Test Cases
	task.paths(9, 7)
	task.paths(3, 7)
	task.paths(4, 4)
end

main()

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

/*
    Scala Program
    Count the number of ways to traverse a Matrix
*/
class Counting()
{
	// Count matrix path
	def countPath(i: Int, j: Int, row: Int, col: Int): Int = {
		if (i == row && col == j)
		{
			return 1;
		}
		else if (i <= row && j <= col)
		{
			//  Count matrix path recursively
			return countPath(i + 1, j, row, col) + 
                   countPath(i, j + 1, row, col);
		}
		else
		{
			return 0;
		}
	}
	// Handles the request to count path of the matrix
	def paths(row: Int, col: Int): Unit = {
		if (row <= 0 || col <= 0)
		{
			return;
		}
		// Display given rows and columns
		print(" Row : " + row + ", Col : " + col);
		// Display calculated result
		print("\n Paths : " + countPath(0, 0, row - 1, col - 1) + "\n\n");
	}
}
object Main
{
	def main(args: Array[String]): Unit = {
		var task: Counting = new Counting();
		// Test Cases
		task.paths(9, 7);
		task.paths(3, 7);
		task.paths(4, 4);
	}
}

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

/*
    Swift 4 Program
    Count the number of ways to traverse a Matrix
*/
class Counting
{
	// Count matrix path
	func countPath(_ i: Int, _ j: Int, _ row: Int, _ col: Int) -> Int
	{
		if (i == row && col == j)
		{
			return 1;
		}
		else if (i <= row && j <= col)
		{
			//  Count matrix path recursively
			return self.countPath(i + 1, j, row, col) + 
              self.countPath(i, j + 1, row, col);
		}
		else
		{
			return 0;
		}
	}
	// Handles the request to count path of the matrix
	func paths(_ row: Int, _ col: Int)
	{
		if (row <= 0 || col <= 0)
		{
			return;
		}
		// Display given rows and columns
		print(" Row : ", row ,", Col : ", col, terminator: "");
		// Display calculated result
		print("\n Paths : ", self.countPath(0, 0, row - 1, col - 1) ,"\n");
	}
}
func main()
{
	let task = Counting();
	// Test Cases
	task.paths(9, 7);
	task.paths(3, 7);
	task.paths(4, 4);
}
main();

 Row :  9 , Col :  7
 Paths :  3003

 Row :  3 , Col :  7
 Paths :  28

 Row :  4 , Col :  4
 Paths :  20

/*
    Kotlin Program
    Count the number of ways to traverse a Matrix
*/
class Counting
{
	// Count matrix path
	fun countPath(i: Int, j: Int, row: Int, col: Int): Int
	{
		if (i == row && col == j)
		{
			return 1;
		}
		else if (i <= row && j <= col)
		{
			//  Count matrix path recursively
			return this.countPath(i + 1, j, row, col) + 
              this.countPath(i, j + 1, row, col);
		}
		else
		{
			return 0;
		}
	}
	// Handles the request to count path of the matrix
	fun paths(row: Int, col: Int): Unit
	{
		if (row <= 0 || col <= 0)
		{
			return;
		}
		// Display given rows and columns
		print(" Row : " + row + ", Col : " + col);
		// Display calculated result
		print("\n Paths : " + this.countPath(0, 0, row - 1, col - 1) + "\n\n");
	}
}
fun main(args: Array < String > ): Unit
{
	val task: Counting = Counting();
	// Test Cases
	task.paths(9, 7);
	task.paths(3, 7);
	task.paths(4, 4);
}

 Row : 9, Col : 7
 Paths : 3003

 Row : 3, Col : 7
 Paths : 28

 Row : 4, Col : 4
 Paths : 20

Output Explanation

The code implements the algorithm and finds the number of unique paths to traverse a matrix based on the provided input. It demonstrates three test cases and displays the resulting number of paths.

Time Complexity

The time complexity of this algorithm is O(2^(m + n)), where m is the number of rows and n is the number of columns in the matrix. This is because for each cell, we consider two choices: moving down or moving right. The recursion tree can have up to 2^(m + n) nodes. While this approach works for small matrices, it becomes inefficient for larger ones. Dynamic programming can be used to optimize the solution.