Check for possible path in 2D matrix

Given a 2D array(m x n). The task is to check if there is any path from top left to bottom right. In the matrix, -1 is considered as blockage (can’t go through this cell) and 0 is considered path cell (can go through it).

Note: Top left cell always contains 0

Examples:

Input : arr[][] = {{ 0, 0, 0, -1, 0},
{-1, 0, 0, -1, -1},
{ 0, 0, 0, -1, 0},
{-1, 0, 0, 0, 0},
{ 0, 0, -1, 0, 0}}
Output : Yes
Explanation:

The red cells are blocked, white cell denotes the path and the green cells are not blocked cells.

Input : arr[][] = {{ 0, 0, 0, -1, 0},
{-1, 0, 0, -1, -1},
{ 0, 0, 0, -1, 0},
{-1, 0, -1, 0, 0},
{ 0, 0, -1, 0, 0}}
Output : No
Explanation: There exists no path from start to end.

The red cells are blocked, white cell denotes the path and the green cells are not blocked cells.



Method 1

  • Approach: The solution is to perform BFS or DFS to find whether there is a path or not. The graph needs not to be created to perform the bfs, but the matrix itself will be used as a graph. Start the traversal from the top right corner and if there is a way to reach the bottom right corner then there is a path.
  • Algorithm:

    1. Create a queue that stores pairs (i,j) and insert the (0,0) in the queue.
    2. Run a loop till the queue is empty.
    3. In each iteration dequeue the queue (a,b), if the front element is the destination (row-1,col-1) then return 1, i,e there is a path and change the value of mat[a][b] to -1, i.e. visited.
    4. Else insert the adjacent indices where the value of matrix[i][j] is not -1.
  • Implementation:

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    // CPP program to find if there is path 
    // from top left to right bottom
    #include <bits/stdc++.h>
    using namespace std;
       
    #define row 5
    #define col 5
      
       
    // to find the path from
    // top left to bottom right
    bool isPath(int arr[row][col])
    {
        //directions
        int dir[4][2] = { {0,1}, {0,-1}, {1,0}, {-1,0}};
          
        //queue 
        queue<pair<int,int> > q;
          
        //insert the top right corner.
        q.push(make_pair(0,0));
          
        //until queue is empty
        while(q.size()>0)
        {
            pair<int,int> p = q.front();
            q.pop();
              
            //mark as visited
            arr[p.first][p.second] = -1;
              
            //destination is reached.
            if(p == make_pair(row-1,col-1))
                return true;
                  
            //check all four directions
            for(int i=0; i<4 ;i++)
            {
                //using the direction array
                int a = p.first + dir[i][0];
                int b = p.second + dir[i][1];
                  
                //not blocked and valid
                if(arr[a][b]!=-1&& a>=0 && b>=0
                        && a<row && b<col)
                {
                    q.push(make_pair(a,b));
                }
            }
        }
        return false;
    }
       
    // Driver Code
    int main()
    {
        // Given array
        int arr[row][col] = { { 0, 0, 0, -1, 0 },
                              { -1, 0, 0, -1, -1 },
                              { 0, 0, 0, -1, 0 },
                              { -1, 0, -1, 0, -1 },
                              { 0, 0, -1, 0, 0 } };
       
        // path from arr[0][0] to arr[row][col]
        if (isPath(arr))
          cout << "Yes";
        else
          cout << "No";
       
    return 0;
    }

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    Output:

    No
    
  • Complexity Analysis:

    • Time Complexity: O(R*C).
      Every element of the matrix can be inserted once in the queue, so time Complexity is O(R*C).
    • Space Complexity: O(R*C).
      To store all the elements in a queue O(R*C) space is needed.

Method 2

  • Approach: The only problem with the above solution is it uses extra space. This approach will eliminate the need for extra space. The basic idea is very similar. This algorithm will also perform BFS but the need for extra space will be eliminated by marking the array. So First run a loop and check which elements of the first column and the first row is accessible from 0,0 by using only the first row and column. mark them as 1. Now traverse the matrix from start to the end row-wise in increasing index of rows and columns. If the cell is not blocked then check that any of its adjacent cells is marked 1 or not. If marked 1 then mark the cell 1.
  • Algorithm:

    1. Mark the cell 0,0 as 1.
    2. Run a loop from 0 to row length and if the cell above is marked 1 and the current cell is not blocked then mark the current cell as 1.
    3. Run a loop from 0 to column length and if the left cell is marked 1 and the current cell is not blocked then mark the current cell as 1.
    4. Traverse the matrix from start to the end row-wise in increasing index of rows and columns.
    5. If the cell is not blocked then check that any of its adjacent cells (check only the cell above and the cell to the left). is marked 1 or not. If marked 1 then mark the cell 1.
    6. If the cell (row-1, col-1) is marked 1 return true else return false.
  • Implementation:

    C++

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    // CPP program to find if there is path 
    // from top left to right bottom
    #include <iostream>
    using namespace std;
      
    #define row 5
    #define col 5
      
    // to find the path from
    // top left to bottom right
    bool isPath(int arr[row][col])
    {
        // set arr[0][0] = 1
        arr[0][0] = 1;
      
        // Mark reachable (from top left) nodes 
        // in first row and first column.
        for (int i = 1; i < row; i++) 
            if (arr[i][0] != -1)
                arr[i][0] = arr[i - 1][0];   
      
        for (int j = 1; j < col; j++) 
            if (arr[0][j] != -1)
                arr[0][j] = arr[0][j - 1];    
      
        // Mark reachable nodes in remaining
        // matrix.
        for (int i = 1; i < row; i++) 
            for (int j = 1; j < col; j++) 
              if (arr[i][j] != -1)
                  arr[i][j] = max(arr[i][j - 1],
                                arr[i - 1][j]);       
          
        // return yes if right bottom 
        // index is 1
        return (arr[row - 1][col - 1] == 1);
    }
      
    // Driver Code
    int main()
    {
        // Given array
        int arr[row][col] = { { 0, 0, 0, -1, 0 },
                              { -1, 0, 0, -1, -1 },
                              { 0, 0, 0, -1, 0 },
                              { -1, 0, -1, 0, -1 },
                              { 0, 0, -1, 0, 0 } };
      
        // path from arr[0][0] to arr[row][col]
        if (isPath(arr))
          cout << "Yes";
        else
          cout << "No";
      
    return 0;
    }

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    Java

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    // Java program to find if there is path
    // from top left to right bottom
    class GFG
    {
        // to find the path from
        // top left to bottom right
        static boolean isPath(int arr[][])
        {
            // set arr[0][0] = 1
            arr[0][0] = 1;
      
            // Mark reachable (from top left) nodes
            // in first row and first column.
            for (int i = 1; i < 5; i++)
                if (arr[0][i] != -1)
                    arr[0][i] = arr[0][i - 1];
            for (int j = 1; j < 5; j++)
                if (arr[j][0] != -1)
                    arr[j][0] = arr[j - 1][0];
      
            // Mark reachable nodes in
            //  remaining matrix.
            for (int i = 1; i < 5; i++)
                for (int j = 1; j < 5; j++)
                    if (arr[i][j] != -1)
                        arr[i][j] = Math.max(arr[i][j - 1],
                                            arr[i - 1][j]);
      
            // return yes if right 
            // bottom index is 1
            return (arr[5 - 1][5 - 1] == 1);
        }
           
        //Driver code 
        public static void main(String[] args)
        {
            // Given array
            int arr[][] = { { 0, 0, 0, -1, 0 },
                            { -1, 0, 0, -1, -1 },
                            { 0, 0, 0, -1, 0 },
                            { -1, 0, -1, 0, -1 },
                            { 0, 0, -1, 0, 0 } };
      
            // path from arr[0][0] 
            // to arr[row][col]
            if (isPath(arr))
                System.out.println("Yes");
            else
                System.out.println("No");
        }
    }
    // This code is contributed 
    // by prerna saini 

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    Python3

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    # Python3 program to find if there 
    # is path from top left to right bottom 
    row = 5
    col = 5
      
    # to find the path from 
    # top left to bottom right 
    def isPath(arr):
          
        # set arr[0][0] = 1
        arr[0][0] = 1
      
        # Mark reachable (from top left) 
        # nodes in first row and first column. 
        for i in range(1, row):
            if (arr[i][0] != -1):
                arr[i][0] = arr[i-1][0]
      
        for j in range(1, col):
            if (arr[0][j] != -1):
                arr[0][j] = arr[0][j-1]
                  
        # Mark reachable nodes in 
        # remaining matrix. 
        for i in range(1, row):
            for j in range(1, col):
                if (arr[i][j] != -1):
                    arr[i][j] = max(arr[i][j - 1], 
                                    arr[i - 1][j])
                                      
        # return yes if right 
        # bottom index is 1
        return (arr[row - 1][col - 1] == 1)
      
    # Driver Code 
      
    # Given array 
    arr = [[ 0, 0, 0, -1, 0 ], 
           [-1, 0, 0, -1, -1], 
           [ 0, 0, 0, -1, 0 ], 
           [-1, 0, -1, 0, -1], 
           [ 0, 0, -1, 0, 0 ]] 
      
    # path from arr[0][0] to arr[row][col] 
    if (isPath(arr)):
        print("Yes"
    else:
        print("No")
      
    # This code is contributed 
    # by sahilshelangia

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    C#

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    // C# program to find if there is path
    // from top left to right bottom
    using System;
      
    class GFG
    {
        // to find the path from
        // top left to bottom right
        static bool isPath(int [,]arr)
        {
            // set arr[0][0] = 1
            arr[0, 0] = 1;
      
            // Mark reachable (from top left) nodes
            // in first row and first column.
            for (int i = 1; i < 5; i++)
                if (arr[i, 0] != -1)
                    arr[i, 0] = arr[i - 1, 0];
            for (int j = 1; j < 5; j++)
                if (arr[0,j] != -1)
                    arr[0,j] = arr[0, j - 1];
      
            // Mark reachable nodes in
            // remaining matrix.
            for (int i = 1; i < 5; i++)
                for (int j = 1; j < 5; j++)
                    if (arr[i, j] != -1)
                        arr[i, j] = Math.Max(arr[i, j - 1],
                                            arr[i - 1, j]);
      
            // return yes if right 
            // bottom index is 1
            return (arr[5 - 1, 5 - 1] == 1);
        }
          
        //Driver code 
        public static void Main()
        {
            // Given array
            int [,]arr = { { 0, 0, 0, -1, 0 },
                            { -1, 0, 0, -1, -1 },
                            { 0, 0, 0, -1, 0 },
                            { -1, 0, -1, 0, -1 },
                            { 0, 0, -1, 0, 0 } };
      
            // path from arr[0][0] 
            // to arr[row][col]
            if (isPath(arr))
                Console.WriteLine("Yes");
            else
                Console.WriteLine("No");
        }
    }
      
    // This code is contributed 
    // by vt_m

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    PHP

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    <?php
    // PHP program to find if 
    // there is path from top
    // left to right bottom
    $row = 5;
    $col = 5;
      
    // to find the path from
    // top left to bottom right
    function isPath($arr)
    {
        global $row, $col;
          
        $arr[0][0] = 1;
      
        // Mark reachable (from 
        // top left) nodes in 
        // first row and first column.
        for ($i = 1; $i < $row; $i++) 
            if ($arr[$i][0] != -1)
                $arr[$i][0] = $arr[$i - 1][0]; 
      
        for ($j = 1; $j < $col; $j++) 
            if ($arr[0][$j] != -1)
                $arr[0][$j] = $arr[0][$j - 1]; 
      
        // Mark reachable nodes 
        // in remaining matrix.
        for ($i = 1; $i < $row; $i++) 
            for ($j = 1; $j < $col; $j++) 
            if ($arr[$i][$j] != -1)
                $arr[$i][$j] = max($arr[$i][$j - 1],
                                   $arr[$i - 1][$j]); 
          
        // return yes if right 
        // bottom index is 1
        return ($arr[$row - 1][$col - 1] == 1);
    }
      
    // Driver Code
      
    // Given array
    $arr = array(array(0, 0, 0, 1, 0),
                 array(-1, 0, 0, -1, -1),
                 array(0, 0, 0, -1, 0),
                 array(-1, 0, -1, 0, -1),
                 array(0, 0, -1, 0, 0));
      
    // path from arr[0][0]
    // to arr[row][col]
    if (isPath($arr))
    echo "Yes";
    else
    echo "No";
          
    // This code is contributed by anuj_67.
    ?>

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    Output:

    No
    
  • Complexity Analysis:

    • Time Complexity: O(R*C).
      Every element of the matrix is traversed, so time Complexity is O(R*C).
    • Space Complexity: O(1).
      No extra space is needed.

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