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# Path to reach border cells from a given cell in a 2D Grid without crossing specially marked cells

• Last Updated : 28 Jun, 2021

Given a matrix of dimensions N*M consisting of characters ‘M’, ‘#’, ‘.’ and only a single instance of ‘A’. The task is to print any one path from the cell having value A to any border cell of the matrix according to the following rules:

• Every second the path from cell ‘A’ can move in all four adjacent cells having ‘.’ characters only. The characters L, R, U, and D represent the directions for the cell (X – 1, Y), (X + 1, Y), (X, Y – 1), and (X, Y + 1) respectively moving from the cell (X, Y).
• The cells having characters ‘#’ and ‘M’ are the block cells.
• Every second, the cell having characters ‘M’ spread itself in all four directions time simultaneously with the character ‘A’.

Note: If there doesn’t exist any such path from A to any border cell of the matrix, then print “-1”.

Examples:

Input: mat[][] = {{‘#’, ‘M’, ‘.’}, {‘#’, ‘A’, ‘M’}, {‘#’, ‘.’, ‘#’}}
Output: D
Explanation:
The matrix changes as follows: Thus, by going 1 cell down, the border cells can be reached.

Input: mat[][] = {{‘#’, ‘#’, ‘#’, ‘#’, ‘#’, ‘#’, ‘#’, ‘#’}, {‘#’, ‘M’, ‘.’, ‘.’, ‘A’, ‘.’, ‘.’, ‘#’}, {‘#’, ‘#’, ‘#’, ‘.’, ‘M’, ‘#’, ‘.’, ‘#’}, {‘#’, ‘.’, ‘#’, ‘.’, ‘.’, ‘#’, ‘.’, ‘.’}, {‘5’, ‘#’, ‘.’, ‘#’, ‘#’, ‘#’, ‘#’, ‘#’, ‘#’}}
Output: RRDDR

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The given problem can be solved by first simulating the multisource BFS on the grid for all the block cells ‘M’ and then perform BFS from the cell ‘A’ to check if any border cell can be reached or not. Follow the steps below to solve the problem:

• Initialize a matrix, say G[][] that stores the minimum to reach the cell having values ‘.’ from all the cells having the value ‘M’.
• Perform the Multisource BFS from all the cells having values ‘M’ for finding the minimum time to reach every cell from its nearest cell having ‘M’ and store it in the matrix G[][].
• Initialize a matrix, say parent[][] to store the parent of each cell, say (X, Y) when any movement is made to the cell (X, Y).
• Perform the BFS Traversal on the grid from the position where ‘A’ occurs, say (SX, SY) using the following steps:
• Push the current cell (SX, SY) with the distance as 0 in the queue.
• Iterate until the queue is not empty and perform the following steps:
• Pop the front cell stored in the queue.
• Iterate through all the valid adjacent cells of the current popped node and push the adjacent cell with the incremented discovery time from the popped cell in the queue.
• Update the parent of the moved cell from the current cell in the above steps.
• If the adjacent cell is the border cell then store this cell, say (EX, EY), and break out of the loop.
• If the border of the matrix is not reached, then print “-1”. Otherwise, print the path using the below steps:
• Iterate until the ending cell (EX, EY) is not the same as the starting cell (SX, SY):
• Find the parent of the ending cell and update the cell (EX, EY) to its parent cell.
• Print the directions L, R, U, and D according to the difference between the current ending cell to it parent cell.

Below is an implementation of the above approach:

## C++

 `// C++ program for tha above approach``#include ``using` `namespace` `std;`` ` `#define largest 10000000`` ` `// store the grid (2-d grid)``vector > g;`` ` `// store the coordinates of the 'M' cells``vector > M;`` ` `// record the parent of a index``vector, ``int``> > > parent;`` ` `// start indices of A``int` `sx, sy;`` ` `// For traversing to adjacent cells``int` `dx[] = { 1, 0, -1, 0 };``int` `dy[] = { 0, -1, 0, 1 };`` ` `// rows, columns, end-x, end-y``int` `n, m, ex = -1, ey = -1;`` ` `// function to check if the index``// to be processed is valid or not``bool` `isvalid(``int` `x, ``int` `y)``{``    ``// should not exceed any of the boundary walls``    ``if` `(x < 1 || x > n || y < 0 || y > m)``        ``return` `false``;`` ` `    ``// if current cell has '#'``    ``if` `(g[x][y] == largest + 1)``        ``return` `false``;`` ` `    ``return` `true``;``}`` ` `// function to check if the current``// index is at border ornot``bool` `isborder(``int` `x, ``int` `y)``{``    ``if` `(x == 1 || y == 1 || x == n || y == m)``        ``return` `true``;`` ` `    ``return` `false``;``}`` ` `// function to get the direction when``// movement is made from (x --> ex) and (y --> ey)``char` `cal(``int` `x, ``int` `y, ``int` `ex, ``int` `ey)``{``    ``if` `(x + 1 == ex && y == ey)``        ``return` `'D'``;`` ` `    ``if` `(x - 1 == ex && y == ey)``        ``return` `'U'``;`` ` `    ``if` `(x == ex && y + 1 == ey)``        ``return` `'R'``;`` ` `    ``return` `'L'``;``}`` ` `// Function for the multisource``// bfs on M's to store the timers``void` `fillMatrix()``{``    ``// queue to store index``    ``// for bfs and shortest time``    ``// for each (i, j)``    ``queue, ``int``> > q;``    ``for` `(``auto` `m : M) {`` ` `        ``// time at this moment is passed as zero``        ``q.push({ m, 0 });`` ` `        ``// insert time 0 for this (x, y)``        ``g[m.first][m.second] = 0;``    ``}`` ` `    ``// process till the queue becomes empty``    ``while` `(!q.empty()) {`` ` `        ``// get the index and time of``        ``// the element at front of queue``        ``int` `x = q.front().first.first;``        ``int` `y = q.front().first.second;``        ``int` `time` `= q.front().second;`` ` `        ``// remove it``        ``q.pop();`` ` `        ``// iterate to all the positions``        ``// from the given position i.e.``        ``// (x+1, y), (x-1, y), (x, y+1), (x, y-1)``        ``for` `(``auto` `i : { 0, 1, 2, 3 }) {`` ` `            ``int` `newx = x + dx[i];``            ``int` `newy = y + dy[i];`` ` `            ``// check for validity of current index``            ``if` `(!isvalid(newx, newy))``                ``continue``;`` ` `            ``// not visited before``            ``if` `(g[newx][newy] == -1) {`` ` `                ``// update time``                ``g[newx][newy] = (``time` `+ 1);`` ` `                ``// push it into the queue``                ``q.push({ { newx, newy }, ``time` `+ 1 });``            ``}``        ``}``    ``}`` ` `    ``// in the end if there are some places on grid``    ``// that were blocked and BFS couldn't reach there``    ``// then just manually iterate over them and``    ``// change them to largest +1 i.e. treat them as '#'`` ` `    ``for` `(``int` `i = 1; i <= n; i++) {``        ``for` `(``int` `j = 1; j <= m; j++) {``            ``if` `(g[i][j] == -1) {``                ``g[i][j] = largest;``            ``}``        ``}``    ``}``}`` ` `// A's BFS. It will return the time``// when A reaches boundary``// if it is not possible, it will return -1``int` `bfs()``{``    ``queue, ``int``> > q;`` ` `    ``// push the starting (x, y)``    ``// and it's time as 0``    ``q.push({ { sx, sy }, 0 });`` ` `    ``while` `(!q.empty()) {``        ``int` `x = q.front().first.first;``        ``int` `y = q.front().first.second;``        ``int` `time` `= q.front().second;`` ` `        ``q.pop();`` ` `        ``for` `(``auto` `i : { 0, 1, 2, 3 }) {``            ``int` `newx = x + dx[i];``            ``int` `newy = y + dy[i];`` ` `            ``if` `(!isvalid(newx, newy))``                ``continue``;`` ` `            ``// Moving to this index is not possible``            ``if` `((``time` `+ 1) >= (g[newx][newy]))``                ``continue``;`` ` `            ``// index to move on already has``            ``// a parent i.e. already visited``            ``if` `(parent[newx][newy].first.first != -1)``                ``continue``;`` ` `            ``// Move to this index and mark the parents``            ``parent[newx][newy].first.first = x;``            ``parent[newx][newy].first.second = y;``            ``parent[newx][newy].second = ``time` `+ 1;`` ` `            ``q.push({ { newx, newy }, ``time` `+ 1 });`` ` `            ``// if this index is a border``            ``if` `(isborder(newx, newy)) {`` ` `                ``// update ex and ey``                ``ex = newx;``                ``ey = newy;``                ``return` `time` `+ 1;``            ``}``        ``}``    ``}`` ` `    ``// if not possible``    ``return` `-1;``}`` ` `// Function to solve the above problem``void` `isItPossible(vector > Mat)``{``    ``// Resize the global vectors``    ``g.resize(n + 1, vector<``int``>(m + 1));``    ``parent.resize(``        ``n + 1, vector, ``int``> >(m + 1));`` ` `    ``for` `(``int` `i = 1; i <= n; i++) {``        ``for` `(``int` `j = 1; j <= m; j++) {`` ` `            ``// initialize that no one is currently the``            ``// parent of (i, j)``            ``parent[i][j].first.first = -1;``            ``parent[i][j].first.second = -1;``            ``parent[i][j].second = -1;`` ` `            ``char` `x = Mat[i - 1][j - 1];``            ``if` `(x == ``'M'``) {``                ``// if the input character is 'M',``                ``// push the coordinates in M and``                ``// in the grid take 0 as input``                ``M.push_back({ i, j });``                ``g[i][j] = 0;``            ``}`` ` `            ``else` `if` `(x == ``'A'``) {``                ``// this is the start x and start y`` ` `                ``sx = i, sy = j;``                ``g[i][j] = -1;``            ``}`` ` `            ``else` `if` `(x == ``'.'``)``                ``g[i][j] = -1;`` ` `            ``// denote '#' with largest+1``            ``else``                ``g[i][j] = largest + 1;``        ``}``    ``}`` ` `    ``// if already at the border``    ``if` `(isborder(sx, sy)) {``        ``cout << ``"Already a boundary cell\n"``;``        ``return``;``    ``}`` ` `    ``// Multisource bfs``    ``fillMatrix();`` ` `    ``// bfs of A``    ``int` `time` `= bfs();`` ` `    ``// if (end x) index is -1 and``    ``// boundary has not been``    ``if` `(ex == -1) {``        ``cout << ex;``    ``}``    ``else` `{`` ` `        ``vector<``char``> ans; ``// record the path`` ` `        ``while` `(!(ex == sx && ey == sy)) {``            ``int` `x = parent[ex][ey].first.first;``            ``int` `y = parent[ex][ey].first.second;`` ` `            ``// get the direction of movement``            ``char` `dir = cal(x, y, ex, ey);`` ` `            ``ans.push_back(dir);``            ``ex = x;``            ``ey = y;``        ``}`` ` `        ``reverse(ans.begin(), ans.end());`` ` `        ``for` `(``auto` `x : ans) {``            ``cout << x;``        ``}``    ``}``}``// Driver code``int` `main()``{``    ``// Input``    ``vector > Mat = { { ``'#'``, ``'M'``, ``'.'` `},``                                  ``{ ``'#'``, ``'A'``, ``'M'` `},``                                  ``{ ``'#'``, ``'.'``, ``'#'` `} };``    ``n = Mat.size();``    ``m = Mat.size();`` ` `    ``// Function call``    ``isItPossible(Mat);``    ``return` `0;``}`
Output
`D`

Time Complexity: O(n*m)
Auxiliary Space: O(n*m)

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