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Code Matrix

Minimum sum path in a Matrix

The problem is to find the minimum sum path in a given matrix, where you can only move right or down. Starting from the top-left corner of the matrix, you need to reach the bottom-right corner while minimizing the sum of values along the path.

Example

Consider the following matrix:

1  3  4  6  8  2  4  15
2  10 1  4  5  7  7  2
13 3  1  3  1  7  4  12
5  8  1  7  2  7  4  2
7  9  3  13 5  10 3  2

The minimum sum path in this matrix would be: 1 -> 3 -> 1 -> 1 -> 3 -> 1 -> 2 -> 7 -> 4 -> 2 -> 2. The sum of the values along this path is 31.

Idea to Solve the Problem

To solve this problem, we can use a recursive approach that explores all possible paths from the top-left corner to the bottom-right corner of the matrix. At each step, we have two choices: move down or move right. We keep track of the current row, current column, and the sum of values along the path. We continue recursively exploring both choices until we reach the bottom-right corner.

Algorithm

  1. Start from the top-left corner of the matrix (0, 0) with initial sum as 0.
  2. If the current position is the bottom-right corner (R-1, C-1), update the output if the current sum is less than the current output.
  3. If the current position is within the matrix boundaries, explore the two choices: a. Move down: Call the function with updated row, column, count+1, k, and sum+current value. b. Move right: Call the function with updated row, column, count+1, k, and sum+current value.
  4. At the end of the recursive process, the output will contain the minimum sum path.

Pseudocode

path_sum(matrix, output, i, j, count, k, sum):
    if count == k and output > sum:
        output = sum
    else if i < R and j < C:
        path_sum(matrix, output, i+1, j, count+1, k, sum+matrix[i][j])
        path_sum(matrix, output, i, j+1, count+1, k, sum+matrix[i][j])

minimum_path_sum(matrix):
    output = INT_MAX
    path_sum(matrix, &output, 0, 0, 0, R+C-1, 0)
    print output

main:
    matrix = ...  // Define the matrix
    minimum_path_sum(matrix)

Code Solution

// C Program
// Minimum sum path in a Matrix
#include <stdio.h>
#include <limits.h>

#define R 5
#define C 8 

// Find the path sum of (0,0) to Matrix (R-1,C-1) 
void path_sum(int matrix[R][C],int *output,int i ,int j,int count,int k,int sum)
{

  if(count==k && *output > sum)
  {
    // When get a new result
    *output = sum;
  }
  else if(i < R && j < C)
  {
    //Recursive execute method with new parameters
    path_sum(matrix,output,i+1,j,count+1,k,sum+matrix[i][j]);
    path_sum(matrix,output,i,j+1,count+1,k,sum+matrix[i][j]);
  }
}

// Handles the request to find minimum path sum in a matrix
void minimum_path_sum(int matrix[R][C])
{
    // Set initially max value
    int output = INT_MAX;

    // Calculate max path sum
    path_sum(matrix,&output,0,0,0,R+C-1,0);

    // Display find result
    printf(" %d \n",output);
}

int main()
{
    // Define element of matrix
    int matrix[R][C] =
    {
        {1,  3,  4, 6,  8, 2,  4, 15 },
        {2,  10, 1, 4,  5, 7,  7, 2  },
        {13, 3,  1, 3,  1, 7,  4, 12 },
        {5,  8,  1, 7,  2, 7,  4, 2  },
        {7,  9,  3, 13, 5, 10, 3, 2  }
    };
    // 1 -> 3 -> 4 -> 1 -> 1 -> 3 -> 1 -> 2 -> 7 -> 4 -> 2 -> 2
    minimum_path_sum(matrix);

  return 0;
}

Output

 31
/*
    Java program 
    Minimum sum path in a Matrix
*/
// Define TreeNode
public class MyMatrix
{
    public int output;
    // Find the path sum of (0,0) to Matrix (R-1,C-1) 
    public void path_sum(int[][] matrix, 
    int i, int j, 
    int count, int k, 
    int sum, 
    int rows, int cols)
    {
        if (count == k && this.output > sum)
        {
            // When get a new result
            this.output = sum;
        }
        else if (i < rows && j < cols)
        {
            //Recursive execute method with new parameters
            path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j],rows,cols);
            path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j],rows,cols);
        }
    }
    // Handles the request to find max path sum in a matrix
    public void minimum_path_sum(int[][] matrix)
    {
        int r = matrix.length;
        int c = matrix[0].length;
        // Set initially max value
        this.output = Integer.MAX_VALUE;
        // Calculate max path sum
        path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c);
        // Display find result
        System.out.print("  " + this.output + " \n");
    }
    public static void main(String[] args)
    {
        MyMatrix obj = new MyMatrix();
        int[][] matrix =
        {
            {1,  3,  4, 6,  8, 2,  4, 15 },
            {2,  10, 1, 4,  5, 7,  7, 2  },
            {13, 3,  1, 3,  1, 7,  4, 12 },
            {5,  8,  1, 7,  2, 7,  4, 2  },
            {7,  9,  3, 13, 5, 10, 3, 2  }
        };
         // 1 -> 3 -> 4 -> 1 -> 1 -> 3 -> 1 -> 2 -> 7 -> 4 -> 2 -> 2
        obj.minimum_path_sum(matrix);
    }
}

Output

  31
// Include header file
#include <iostream>
#include<limits.h>
#define R 5
#define C 8 

using namespace std;
/*
    C++ program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
class MyMatrix
{
	public: int output;
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	void path_sum(int matrix[R][C], int i, int j, int count, int k, int sum)
	{
		if (count == k && this->output > sum)
		{
			//  When get a new result
			this->output = sum;
		}
		else if (i < R && j < C)
		{
			// Recursive execute method with new parameters
			this->path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j]);
			this->path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j]);
		}
	}
	//  Handles the request to find max path sum in a matrix
	void minimum_path_sum(int matrix[R][C])
	{
		
		//  Set initially max value
		this->output = INT_MAX;
		//  Calculate max path sum
		this->path_sum(matrix, 0, 0, 0, R + C - 1, 0);
		//  Display find result
		cout << "  " << this->output << " \n";
	}
};
int main()
{
	MyMatrix obj = MyMatrix();
    int matrix[R][C] =
    {
        {1,  3,  4, 6,  8, 2,  4, 15 },
        {2,  10, 1, 4,  5, 7,  7, 2  },
        {13, 3,  1, 3,  1, 7,  4, 12 },
        {5,  8,  1, 7,  2, 7,  4, 2  },
        {7,  9,  3, 13, 5, 10, 3, 2  }
    };
	//  1->3->4->1->1->3->1->2->7->4->2->2
	obj.minimum_path_sum(matrix);
	return 0;
}

Output

  31
// Include namespace system
using System;
/*
    C# program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
public class MyMatrix
{
	public int output;
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	public void path_sum(int[,] matrix, int i, int j, int count, int k, int sum, int rows, int cols)
	{
		if (count == k && this.output > sum)
		{
			//  When get a new result
			this.output = sum;
		}
		else if (i < rows && j < cols)
		{
			// Recursive execute method with new parameters
			path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i,j], rows, cols);
			path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i,j], rows, cols);
		}
	}
	//  Handles the request to find max path sum in a matrix
	public void minimum_path_sum(int[,] matrix)
	{
		int r = matrix.GetLength(0);
		int c = matrix.GetLength(1);
		//  Set initially max value
		this.output = int.MaxValue;
		//  Calculate max path sum
		path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c);
		//  Display find result
		Console.Write("  " + this.output + " \n");
	}
	public static void Main(String[] args)
	{
		MyMatrix obj = new MyMatrix();
		int[,] matrix = 
        {
            {1,  3,  4, 6,  8, 2,  4, 15 },
            {2,  10, 1, 4,  5, 7,  7, 2  },
            {13, 3,  1, 3,  1, 7,  4, 12 },
            {5,  8,  1, 7,  2, 7,  4, 2  },
            {7,  9,  3, 13, 5, 10, 3, 2  }
        };
		//  1->3->4->1->1->3->1->2->7->4->2->2
		obj.minimum_path_sum(matrix);
	}
}

Output

  31
<?php
/*
    Php program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
class MyMatrix
{
	public $output;
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	public	function path_sum( & $matrix, $i, $j, $count, $k, $sum, $rows, $cols)
	{
		if ($count == $k && $this->output > $sum)
		{
			//  When get a new result
			$this->output = $sum;
		}
		else if ($i < $rows && $j < $cols)
		{
			// Recursive execute method with new parameters
			$this->path_sum($matrix, $i + 1, $j, $count + 1, $k, $sum + $matrix[$i][$j], $rows, $cols);
			$this->path_sum($matrix, $i, $j + 1, $count + 1, $k, $sum + $matrix[$i][$j], $rows, $cols);
		}
	}
	//  Handles the request to find max path sum in a matrix
	public	function minimum_path_sum( & $matrix)
	{
		$r = count($matrix);
		$c = count($matrix[0]);
		//  Set initially max value
		$this->output = PHP_INT_MAX;
		//  Calculate max path sum
		$this->path_sum($matrix, 0, 0, 0, $r + $c - 1, 0, $r, $c);
		//  Display find result
		echo "  ". $this->output ." \n";
	}
}

function main()
{
	$obj = new MyMatrix();
	$matrix = array(
        array(1, 3, 4, 6, 8, 2, 4, 15), 
        array(2, 10, 1, 4, 5, 7, 7, 2), 
        array(13, 3, 1, 3, 1, 7, 4, 12), 
        array(5, 8, 1, 7, 2, 7, 4, 2), 
        array(7, 9, 3, 13, 5, 10, 3, 2)
    );
	//  1->3->4->1->1->3->1->2->7->4->2->2
	$obj->minimum_path_sum($matrix);
}
main();

Output

  31
/*
    Node Js program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
class MyMatrix
{
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	path_sum(matrix, i, j, count, k, sum, rows, cols)
	{
		if (count == k && this.output > sum)
		{
			//  When get a new result
			this.output = sum;
		}
		else if (i < rows && j < cols)
		{
			// Recursive execute method with new parameters
			this.path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j], rows, cols);
			this.path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j], rows, cols);
		}
	}
	//  Handles the request to find max path sum in a matrix
	minimum_path_sum(matrix)
	{
		var r = matrix.length;
		var c = matrix[0].length;
		//  Set initially max value
		this.output = Number.MAX_VALUE;
		//  Calculate max path sum
		this.path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c);
		//  Display find result
		process.stdout.write("  " + this.output + " \n");
	}
}

function main()
{
	var obj = new MyMatrix();
	var matrix = 
    [
        [1, 3, 4, 6, 8, 2, 4, 15] , 
        [2, 10, 1, 4, 5, 7, 7, 2] , 
        [13, 3, 1, 3, 1, 7, 4, 12] , 
        [5, 8, 1, 7, 2, 7, 4, 2] , 
        [7, 9, 3, 13, 5, 10, 3, 2]
	];
	//  1->3->4->1->1->3->1->2->7->4->2->2
	obj.minimum_path_sum(matrix);
}
main();

Output

  31
import sys

#  Python 3 program 
#  Minimum sum path in a Matrix

#  Define TreeNode
class MyMatrix :
	
	#  Find the path sum of (0,0) to Matrix (R-1,C-1) 
	def path_sum(self, matrix, i, j, count, k, sum, rows, cols) :
		if (count == k and self.output > sum) :
			#  When get a new result
			self.output = sum
		
		elif(i < rows and j < cols) :
			# Recursive execute method with new parameters
			self.path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j], rows, cols)
			self.path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j], rows, cols)
		
	
	#  Handles the request to find max path sum in a matrix
	def minimum_path_sum(self, matrix) :
		r = len(matrix)
		c = len(matrix[0])
		#  Set initially max value
		self.output = sys.maxsize
		#  Calculate max path sum
		self.path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c)
		#  Display find result
		print("  ", self.output ," ")
	

def main() :
	obj = MyMatrix()
	matrix = [
		[1, 3, 4, 6, 8, 2, 4, 15] , 
        [2, 10, 1, 4, 5, 7, 7, 2] , 
        [13, 3, 1, 3, 1, 7, 4, 12] , 
        [5, 8, 1, 7, 2, 7, 4, 2] , 
        [7, 9, 3, 13, 5, 10, 3, 2]
	]
	#  1->3->4->1->1->3->1->2->7->4->2->2
	obj.minimum_path_sum(matrix)

if __name__ == "__main__": main()

Output

   31
#  Ruby program 
#  Minimum sum path in a Matrix

#  Define TreeNode
class MyMatrix  
	# Define the accessor and reader of class MyMatrix  
	attr_reader :output
	attr_accessor :output
 
	
	#  Find the path sum of (0,0) to Matrix (R-1,C-1) 
	def path_sum(matrix, i, j, count, k, sum, rows, cols) 
		if (count == k && self.output > sum) 
			#  When get a new result
			self.output = sum
		elsif(i < rows && j < cols) 
			# Recursive execute method with new parameters
			self.path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j], rows, cols)
			self.path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j], rows, cols)
		end

	end

	#  Handles the request to find max path sum in a matrix
	def minimum_path_sum(matrix) 
		r = matrix.length
		c = matrix[0].length
		#  Set initially max value
		self.output = (2 ** (0. size * 8 - 2))
		#  Calculate max path sum
		self.path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c)
		#  Display find result
		print("  ", self.output ," \n")
	end

end

def main() 
	obj = MyMatrix.new()
	matrix = [
		[1, 3, 4, 6, 8, 2, 4, 15] , 
        [2, 10, 1, 4, 5, 7, 7, 2] , 
        [13, 3, 1, 3, 1, 7, 4, 12] , 
        [5, 8, 1, 7, 2, 7, 4, 2] , 
        [7, 9, 3, 13, 5, 10, 3, 2]
	]
	#  1->3->4->1->1->3->1->2->7->4->2->2
	obj.minimum_path_sum(matrix)
end

main()

Output

  31 

/*
    Scala program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
class MyMatrix(var output: Int)
{
  	def this()
    {
      	this(Int.MaxValue)
    }
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	def path_sum(matrix: Array[Array[Int]], i: Int, j: Int, count: Int, k: Int, sum: Int, rows: Int, cols: Int): Unit = {
		if (count == k && this.output > sum)
		{
			//  When get a new result
			this.output = sum;
		}
		else if (i < rows && j < cols)
		{
			// Recursive execute method with new parameters
			this.path_sum(matrix, i + 1, j, count + 1, k, sum + matrix(i)(j), rows, cols);
			this.path_sum(matrix, i, j + 1, count + 1, k, sum + matrix(i)(j), rows, cols);
		}
	}
	//  Handles the request to find max path sum in a matrix
	def minimum_path_sum(matrix: Array[Array[Int]]): Unit = {
		var r: Int = matrix.length;
		var c: Int = matrix(0).length;
		//  Set initially max value
		this.output = Int.MaxValue;
		//  Calculate max path sum
		this.path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c);
		//  Display find result
		print("  " + this.output + " \n");
	}
}
object Main
{
	def main(args: Array[String]): Unit = {
		var obj: MyMatrix = new MyMatrix();
		var matrix: Array[Array[Int]] = Array(
            Array(1, 3, 4, 6, 8, 2, 4, 15), 
            Array(2, 10, 1, 4, 5, 7, 7, 2), 
            Array(13, 3, 1, 3, 1, 7, 4, 12), 
            Array(5, 8, 1, 7, 2, 7, 4, 2), 
            Array(7, 9, 3, 13, 5, 10, 3, 2)
        );
		//  1->3->4->1->1->3->1->2->7->4->2->2
		obj.minimum_path_sum(matrix);
	}
}

Output

  31
/*
    Swift 4 program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
class MyMatrix
{
	var output: Int;
	init()
	{
		self.output = Int.max;
	}
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	func path_sum(_ matrix: [[Int]], _ i: Int, _ j: Int, _ count: Int, _ k: Int, _ sum: Int, _ rows: Int, _ cols: Int)
	{
		if (count == k && self.output > sum)
		{
			//  When get a new result
			self.output = sum;
		}
		else if (i < rows && j < cols)
		{
			// Recursive execute method with new parameters
			self.path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j], rows, cols);
			self.path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j], rows, cols);
		}
	}
	//  Handles the request to find max path sum in a matrix
	func minimum_path_sum(_ matrix: [[Int]])
	{
		let r: Int = matrix.count;
		let c: Int = matrix[0].count;
		//  Set initially max value
		self.output = Int.max;
		//  Calculate max path sum
		self.path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c);
		//  Display find result
		print("  ", self.output ," ");
	}
}
func main()
{
	let obj: MyMatrix = MyMatrix();
	let matrix: [[Int]] = 
    [
      [1, 3, 4, 6, 8, 2, 4, 15] , 
      [2, 10, 1, 4, 5, 7, 7, 2] , 
      [13, 3, 1, 3, 1, 7, 4, 12] , 
      [5, 8, 1, 7, 2, 7, 4, 2] , 
      [7, 9, 3, 13, 5, 10, 3, 2]
    ];
	//  1->3->4->1->1->3->1->2->7->4->2->2
	obj.minimum_path_sum(matrix);
}
main();

Output

   31
/*
    Kotlin program 
    Minimum sum path in a Matrix
*/
//  Define TreeNode
class MyMatrix
{
	var output: Int;
	constructor()
	{
		this.output = Int.MAX_VALUE;
	}
	//  Find the path sum of (0,0) to Matrix (R-1,C-1)
	fun path_sum(matrix: Array<Array<Int>> , i: Int, j: Int, count: Int, k: Int, sum: Int, rows: Int, cols: Int): Unit
	{
		if (count == k && this.output>sum)
		{
			//  When get a new result
			this.output = sum;
		}
		else
		if (i<rows && j<cols)
		{
			// Recursive execute method with new parameters
			this.path_sum(matrix, i + 1, j, count + 1, k, sum + matrix[i][j], rows, cols);
			this.path_sum(matrix, i, j + 1, count + 1, k, sum + matrix[i][j], rows, cols);
		}
	}
	//  Handles the request to find max path sum in a matrix
	fun minimum_path_sum(matrix: Array<Array<Int>> ): Unit
	{
		var r: Int = matrix.count();
		var c: Int = matrix[0].count();
		//  Set initially max value
		this.output = Int.MAX_VALUE;
		//  Calculate max path sum
		this.path_sum(matrix, 0, 0, 0, r + c - 1, 0, r, c);
		//  Display find result
		print("  " + this.output + " \n");
	}
}
fun main(args: Array<String>): Unit
{
	var obj: MyMatrix = MyMatrix();
	var matrix: Array<Array<Int>> = arrayOf(
      arrayOf(1, 3, 4, 6, 8, 2, 4, 15), 
      arrayOf(2, 10, 1, 4, 5, 7, 7, 2), 
      arrayOf(13, 3, 1, 3, 1, 7, 4, 12), 
      arrayOf(5, 8, 1, 7, 2, 7, 4, 2), 
      arrayOf(7, 9, 3, 13, 5, 10, 3, 2)
    );
	//  1->3->4->1->1->3->1->2->7->4->2->2
	obj.minimum_path_sum(matrix);
}

Output

  31

Output Explanation

The mentioned C code implements the above recursive approach to find the minimum sum path in the matrix. It explores all possible paths and updates the output with the minimum sum. The output matches the expected minimum sum path in the matrix, which is 1 -> 3 -> 1 -> 1 -> 3 -> 1 -> 2 -> 7 -> 4 -> 2 -> 2, and the sum is 31.

Time Complexity

The time complexity of the mentioned solution is exponential, as it explores all possible paths from the top-left corner to the bottom-right corner. In the worst case, the time complexity is O(2^(R+C)), where R is the number of rows and C is the number of columns in the matrix.

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