Geek in a Maze

Geek is in a maze of size N * M. Each cell in the maze is made of either ‘.’ or ‘#’. An empty cell is represented by ‘.’ and an obstacle is represented by ‘#’. The task is to find out how many different empty cells he can pass through If Geek starts at cell (R, C) and avoids the obstacles and he can move in any of the four directions but he can move up at most U times and he can move down at most D times. 

Examples:

Input: N = 3, M = 3, 
R = 1, C = 0
U = 1, D = 1
mat = {{. . .}, {. # .}, {# . .}}
Output: 5
Explanation: Geek can reach 
(1, 0), (0, 0), (0, 1), (0, 2), (1, 2) 

Input: N = 3, M = 4, R = 1, C = 0, U = 1, D = 2 
mat = {{. . .},  {. # .}, {. . .}, {# . .}} 
Output: 10
Explanation: Geek can reach all the 
cells except for the obstacles.

Approach: The idea is to solve this problem is based on the following idea:

Keep moving radially in all four directions (up, down, left, right) and keep counting the number of turning taken in moving up and down. If the number of turns left for given upward and downward movements are not 0 then move to up and down and keep counting the empty cells.

Follow the steps mentioned below to implement the idea:

• Check if the starting point is blocked by an obstacle (#)
  •  If true, return 0.
• Keep a queue of arrays to store rows, columns, ups, and downs for any cell.
• Do a BFS traversal:
  • Check if the cell is empty then increment the count variable (say cnt).
  • Check if any up move is left or not.
    • If moves for up is left then move to up and decrement the move count for up and push the current status of the cell in the queue.
  • Check if any down move is left or not.
    • If moves for down is left then move to down and decrement the move count for down and push the current status of the cell in the queue.
• Finally, return the cnt.

Below is the implementation of the above approach:

C++

// C++ code to implement the approach

#include <bits/stdc++.h>
using namespace std;

// Function to count different empty cells
// he can pass through while avoiding
// the obstacles
int numberOfCells(int n, int m, int r,
                  int c, int u, int d,
                  vector<vector<char> >& mat)
{
    // If cell having Obstacle
    if (mat[r] == '#')
        return 0;

    queue<vector<int> > que;
    int cnt = 0;
    int i = 0;
    int j = 0;

    mat[r] = '#';
    que.push({ r, c, u, d });

    // BFS traversal of the matrix
    while (que.size()) {
        auto& f = que.front();
        int rr = f[0];
        int cc = f[1];
        int uu = f[2];
        int dd = f[3];
        que.pop();

        ++cnt;

        // Move left
        i = rr;
        j = cc - 1;
        if (0 <= j && mat[i][j] == '.') {

            // Mark the cell visited
            mat[i][j] = '#';
            que.push({ i, j, uu, dd });
        }

        // Move right
        i = rr;
        j = cc + 1;
        if (j < m && mat[i][j] == '.') {

            // Mark the cell visited
            mat[i][j] = '#';
            que.push({ i, j, uu, dd });
        }

        // Move up
        i = rr - 1;
        j = cc;
        if (0 <= i && mat[i][j] == '.' && uu) {

            // Mark the cell visited
            mat[i][j] = '#';
            que.push({ i, j, uu - 1, dd });
        }

        // Move down
        i = rr + 1;
        j = cc;
        if (i < n && mat[i][j] == '.' && dd) {

            // Mark the cell visited
            mat[i][j] = '#';
            que.push({ i, j, uu, dd - 1 });
        }
    }

    // Return the count
    return cnt;
}

// Driver code
int main()
{
    int N = 3, M = 3, R = 1, C = 0;
    int U = 1, D = 1;
    vector<vector<char> > mat = { { '.', '.', '.' },
                                  { '.', '#', '.' },
                                  { '#', '.', '.' } };

    // Function call
    cout << numberOfCells(N, M, R, C, U, D, mat);
    return 0;
}

Java

import java.util.*;
import java.io.*;

// Java program for the above approach
class GFG{

    // Function to count different empty cells
    // he can pass through while avoiding
    // the obstacles
    public static int numberOfCells(int n, int m, int r, int c, int u, int d, ArrayList<ArrayList<Character>> mat)
    {
        // If cell having Obstacle
        if (mat.get(r).get(c) == '#')
            return 0;

        Queue<ArrayList<Integer>> que = new ArrayDeque<ArrayList<Integer>>();
        int cnt = 0;
        int i = 0;
        int j = 0;

        mat.get(r).set(c, '#');
        que.add(new ArrayList<Integer>(List.of(r, c, u, d)));

        // BFS traversal of the matrix
        while (!que.isEmpty()) {
            ArrayList<Integer> f = que.peek();
            int rr = f.get(0);
            int cc = f.get(1);
            int uu = f.get(2);
            int dd = f.get(3);
            que.remove();

            ++cnt;

            // Move left
            i = rr;
            j = cc - 1;
            if (0 <= j && mat.get(i).get(j) == '.') {

                // Mark the cell visited
                mat.get(i).set(j, '#');
                que.add(new ArrayList<Integer>(List.of(i, j, uu, dd)));
            }

            // Move right
            i = rr;
            j = cc + 1;
            if (j < m && mat.get(i).get(j) == '.') {

                // Mark the cell visited
                mat.get(i).set(j, '#');
                que.add(new ArrayList<Integer>(List.of(i, j, uu, dd)));
            }

            // Move up
            i = rr - 1;
            j = cc;
            if (0 <= i && mat.get(i).get(j) == '.' && uu > 0) {

                // Mark the cell visited
                mat.get(i).set(j, '#');
                que.add(new ArrayList<Integer>(List.of(i, j, uu-1, dd)));
            }

            // Move down
            i = rr + 1;
            j = cc;
            if (i < n && mat.get(i).get(j) == '.' && dd > 0) {

                // Mark the cell visited
                mat.get(i).set(j, '#');
                que.add(new ArrayList<Integer>(List.of(i, j, uu, dd-1)));
            }
        }

        // Return the count
        return cnt;
    }

    // Driver code
    public static void main(String args[])
    {
        int N = 3, M = 3, R = 1, C = 0;
        int U = 1, D = 1;
        ArrayList<ArrayList<Character>> mat = new ArrayList<ArrayList<Character>>(
            List.of(new ArrayList<Character>(
                List.of('.', '.', '.')
            ),
            new ArrayList<Character>(
                List.of('.', '#', '.')
            ),
            new ArrayList<Character>(
                List.of('#', '.', '.')
            ))
        );

        // Function call
        System.out.println(numberOfCells(N, M, R, C, U, D, mat));
    }
}

// This code is contributed by subhamgoyal2014.

Output

5

Time Complexity: O(N * M)
Auxiliary Space: O(N * M)