Find shortest safe route in a path with landmines

Difficulty Level : Hard
Last Updated : 28 Jul, 2021

Given a path in the form of a rectangular matrix having few landmines arbitrarily placed (marked as 0), calculate length of the shortest safe route possible from any cell in the first column to any cell in the last column of the matrix. We have to avoid landmines and their four adjacent cells (left, right, above and below) as they are also unsafe. We are allowed to move to only adjacent cells which are not landmines. i.e. the route cannot contains any diagonal moves.

Examples:

Input: 
A 12 x 10 matrix with landmines marked as 0

[ 1  1  1  1  1  1  1  1  1  1 ]
[ 1  0  1  1  1  1  1  1  1  1 ]
[ 1  1  1  0  1  1  1  1  1  1 ]
[ 1  1  1  1  0  1  1  1  1  1 ]
[ 1  1  1  1  1  1  1  1  1  1 ]
[ 1  1  1  1  1  0  1  1  1  1 ]
[ 1  0  1  1  1  1  1  1  0  1 ]
[ 1  1  1  1  1  1  1  1  1  1 ]
[ 1  1  1  1  1  1  1  1  1  1 ]
[ 0  1  1  1  1  0  1  1  1  1 ]
[ 1  1  1  1  1  1  1  1  1  1 ]
[ 1  1  1  0  1  1  1  1  1  1 ]

Output:  
Length of shortest safe route is 13 (Highlighted in Bold)

The idea is to use Backtracking. We first mark all adjacent cells of the landmines as unsafe. Then for each safe cell of first column of the matrix, we move forward in all allowed directions and recursively checks if they leads to the destination or not. If destination is found, we update the value of shortest path else if none of the above solutions work we return false from our function.

Below is the implementation of above idea –

C++

// C++ program to find shortest safe Route in
// the matrix with landmines
#include <bits/stdc++.h>
using namespace std;
#define R 12
#define C 10
 
// These arrays are used to get row and column
// numbers of 4 neighbours of a given cell
int rowNum[] = { -1, 0, 0, 1 };
int colNum[] = { 0, -1, 1, 0 };
 
// A function to check if a given cell (x, y)
// can be visited or not
bool isSafe(int mat[R][C], int visited[R][C],
            int x, int y)
{
    if (mat[x][y] == 0 || visited[x][y])
        return false;
 
    return true;
}
 
// A function to check if a given cell (x, y) is
// a valid cell or not
bool isValid(int x, int y)
{
    if (x < R && y < C && x >= 0 && y >= 0)
        return true;
 
    return false;
}
 
// A function to mark all adjacent cells of
// landmines as unsafe. Landmines are shown with
// number 0
void markUnsafeCells(int mat[R][C])
{
    for (int i = 0; i < R; i++)
    {
        for (int j = 0; j < C; j++)
        {
            // if a landmines is found
            if (mat[i][j] == 0)
            {
              // mark all adjacent cells
              for (int k = 0; k < 4; k++)
                if (isValid(i + rowNum[k], j + colNum[k]))
                    mat[i + rowNum[k]][j + colNum[k]] = -1;
            }
        }
    }
 
    // mark all found adjacent cells as unsafe
    for (int i = 0; i < R; i++)
    {
        for (int j = 0; j < C; j++)
        {
            if (mat[i][j] == -1)
                mat[i][j] = 0;
        }
    }
 
    // Uncomment below lines to print the path
    /*for (int i = 0; i < R; i++)
    {
        for (int j = 0; j < C; j++)
        {
            cout << std::setw(3) << mat[i][j];
        }
        cout << endl;
    }*/
}
 
// Function to find shortest safe Route in the
// matrix with landmines
// mat[][] - binary input matrix with safe cells marked as 1
// visited[][] - store info about cells already visited in
// current route
// (i, j) are coordinates of the current cell
// min_dist --> stores minimum cost of shortest path so far
// dist --> stores current path cost
void findShortestPathUtil(int mat[R][C], int visited[R][C],
                          int i, int j, int &min_dist, int dist)
{
    // if destination is reached
    if (j == C-1)
    {
        // update shortest path found so far
        min_dist = min(dist, min_dist);
        return;
    }
 
    // if current path cost exceeds minimum so far
    if (dist > min_dist)
        return;
 
    // include (i, j) in current path
    visited[i][j] = 1;
 
    // Recurse for all safe adjacent neighbours
    for (int k = 0; k < 4; k++)
    {
        if (isValid(i + rowNum[k], j + colNum[k]) &&
            isSafe(mat, visited, i + rowNum[k], j + colNum[k]))
        {
            findShortestPathUtil(mat, visited, i + rowNum[k],
                           j + colNum[k], min_dist, dist + 1);
        }
    }
 
    // Backtrack
    visited[i][j] = 0;
}
 
// A wrapper function over findshortestPathUtil()
void findShortestPath(int mat[R][C])
{
    // stores minimum cost of shortest path so far
    int min_dist = INT_MAX;
 
    // create a boolean matrix to store info about
    // cells already visited in current route
    int visited[R][C];
 
    // mark adjacent cells of landmines as unsafe
    markUnsafeCells(mat);
 
    // start from first column and take minimum
    for (int i = 0; i < R; i++)
    {
        // if path is safe from current cell
        if (mat[i][0] == 1)
        {
            // initialize visited to false
            memset(visited, 0, sizeof visited);
 
            // find shortest route from (i, 0) to any
            // cell of last column (x, C - 1) where
            // 0 <= x < R
            findShortestPathUtil(mat, visited, i, 0,
                                 min_dist, 0);
 
            // if min distance is already found
            if(min_dist == C - 1)
                break;
        }
    }
 
    // if destination can be reached
    if (min_dist != INT_MAX)
        cout << "Length of shortest safe route is "
             << min_dist;
 
    else // if the destination is not reachable
        cout << "Destination not reachable from "
             << "given source";
}
 
// Driver code
int main()
{
    // input matrix with landmines shown with number 0
    int mat[R][C] =
    {
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 0, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 0, 1, 1, 1, 1 },
        { 1, 0, 1, 1, 1, 1, 1, 1, 0, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 0, 1, 1, 1, 1, 0, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 }
    };
 
    // find shortest path
    findShortestPath(mat);
 
    return 0;
}

Java

// Java program to find shortest safe Route
// in the matrix with landmines
import java.util.Arrays;
 
class GFG{
 
static final int R = 12;
static final int C = 10;
 
// These arrays are used to get row and column
// numbers of 4 neighbours of a given cell
static int rowNum[] = { -1, 0, 0, 1 };
static int colNum[] = { 0, -1, 1, 0 };
 
static int min_dist;
 
// A function to check if a given cell (x, y)
// can be visited or not
static boolean isSafe(int[][] mat, boolean[][] visited,
                      int x, int y)
{
    if (mat[x][y] == 0 || visited[x][y])
        return false;
 
    return true;
}
 
// A function to check if a given cell (x, y) is
// a valid cell or not
static boolean isValid(int x, int y)
{
    if (x < R && y < C && x >= 0 && y >= 0)
        return true;
 
    return false;
}
 
// A function to mark all adjacent cells of
// landmines as unsafe. Landmines are shown with
// number 0
static void markUnsafeCells(int[][] mat)
{
    for(int i = 0; i < R; i++)
    {
        for(int j = 0; j < C; j++)
        {
             
            // If a landmines is found
            if (mat[i][j] == 0)
            {
                 
                // Mark all adjacent cells
                for(int k = 0; k < 4; k++)
                    if (isValid(i + rowNum[k], j + colNum[k]))
                        mat[i + rowNum[k]][j + colNum[k]] = -1;
            }
        }
    }
 
    // Mark all found adjacent cells as unsafe
    for(int i = 0; i < R; i++)
    {
        for(int j = 0; j < C; j++)
        {
            if (mat[i][j] == -1)
                mat[i][j] = 0;
        }
    }
 
    // Uncomment below lines to print the path
    /*
     * for (int i = 0; i < R; i++) {
     * for (int j = 0; j < C; j++) { cout <<
     * std::setw(3) << mat[i][j]; } cout << endl; }
     */
}
 
// Function to find shortest safe Route in the
// matrix with landmines
// mat[][] - binary input matrix with safe cells marked as 1
// visited[][] - store info about cells already visited in
// current route
// (i, j) are coordinates of the current cell
// min_dist --> stores minimum cost of shortest path so far
// dist --> stores current path cost
static void findShortestPathUtil(int[][] mat,
                                 boolean[][] visited,
                                 int i, int j, int dist)
{
     
    // If destination is reached
    if (j == C - 1)
    {
         
        // Update shortest path found so far
        min_dist = Math.min(dist, min_dist);
        return;
    }
     
    // If current path cost exceeds minimum so far
    if (dist > min_dist)
        return;
 
    // include (i, j) in current path
    visited[i][j] = true;
 
    // Recurse for all safe adjacent neighbours
    for(int k = 0; k < 4; k++)
    {
        if (isValid(i + rowNum[k], j + colNum[k]) &&
            isSafe(mat, visited, i + rowNum[k],
                                 j + colNum[k]))
        {
            findShortestPathUtil(mat, visited, i + rowNum[k],
                             j + colNum[k], dist + 1);
        }
    }
 
    // Backtrack
    visited[i][j] = false;
}
 
// A wrapper function over findshortestPathUtil()
static void findShortestPath(int[][] mat)
{
     
    // Stores minimum cost of shortest path so far
    min_dist = Integer.MAX_VALUE;
     
    // Create a boolean matrix to store info about
    // cells already visited in current route
    boolean[][] visited = new boolean[R][C];
 
    // Mark adjacent cells of landmines as unsafe
    markUnsafeCells(mat);
 
    // Start from first column and take minimum
    for(int i = 0; i < R; i++)
    {
         
        // If path is safe from current cell
        if (mat[i][0] == 1)
        {
             
            // Initialize visited to false
            for(int k = 0; k < R; k++)
            {
                Arrays.fill(visited[k], false);
            }
 
            // Find shortest route from (i, 0) to any
            // cell of last column (x, C - 1) where
            // 0 <= x < R
            findShortestPathUtil(mat, visited, i, 0, 0);
 
            // If min distance is already found
            if (min_dist == C - 1)
                break;
        }
    }
     
    // If destination can be reached
    if (min_dist != Integer.MAX_VALUE)
        System.out.println("Length of shortest " +
                           "safe route is " + min_dist);
 
    else
     
        // If the destination is not reachable
        System.out.println("Destination not " +
                           "reachable from given source");
}
 
// Driver code
public static void main(String[] args)
{
     
    // Input matrix with landmines shown with number 0
    int[][] mat = {
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 0, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 0, 1, 1, 1, 1 },
        { 1, 0, 1, 1, 1, 1, 1, 1, 0, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 0, 1, 1, 1, 1, 0, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 } };
 
    // Find shortest path
    findShortestPath(mat);
}
}
 
// This code is contributed by sanjeev2552

Javascript

<script>
    // Javascript program to find shortest safe Route
    // in the matrix with landmines
     
    let R = 12;
    let C = 10;
 
    // These arrays are used to get row and column
    // numbers of 4 neighbours of a given cell
    let rowNum = [ -1, 0, 0, 1 ];
    let colNum = [ 0, -1, 1, 0 ];
 
    let min_dist;
 
    // A function to check if a given cell (x, y)
    // can be visited or not
    function isSafe(mat, visited, x, y)
    {
        if (mat[x][y] == 0 || visited[x][y])
            return false;
 
        return true;
    }
 
    // A function to check if a given cell (x, y) is
    // a valid cell or not
    function isValid(x, y)
    {
        if (x < R && y < C && x >= 0 && y >= 0)
            return true;
 
        return false;
    }
 
    // A function to mark all adjacent cells of
    // landmines as unsafe. Landmines are shown with
    // number 0
    function markUnsafeCells(mat)
    {
        for(let i = 0; i < R; i++)
        {
            for(let j = 0; j < C; j++)
            {
 
                // If a landmines is found
                if (mat[i][j] == 0)
                {
 
                    // Mark all adjacent cells
                    for(let k = 0; k < 4; k++)
                        if (isValid(i + rowNum[k], j + colNum[k]))
                            mat[i + rowNum[k]][j + colNum[k]] = -1;
                }
            }
        }
 
        // Mark all found adjacent cells as unsafe
        for(let i = 0; i < R; i++)
        {
            for(let j = 0; j < C; j++)
            {
                if (mat[i][j] == -1)
                    mat[i][j] = 0;
            }
        }
 
        // Uncomment below lines to print the path
        /*
         * for (int i = 0; i < R; i++) {
         * for (int j = 0; j < C; j++) { cout <<
         * std::setw(3) << mat[i][j]; } cout << endl; }
         */
    }
 
    // Function to find shortest safe Route in the
    // matrix with landmines
    // mat[][] - binary input matrix with safe cells marked as 1
    // visited[][] - store info about cells already visited in
    // current route
    // (i, j) are coordinates of the current cell
    // min_dist --> stores minimum cost of shortest path so far
    // dist --> stores current path cost
    function findShortestPathUtil(mat, visited, i, j, dist)
    {
 
        // If destination is reached
        if (j == C - 1)
        {
 
            // Update shortest path found so far
            min_dist = Math.min(dist, min_dist);
            return;
        }
 
        // If current path cost exceeds minimum so far
        if (dist > min_dist)
            return;
 
        // include (i, j) in current path
        visited[i][j] = true;
 
        // Recurse for all safe adjacent neighbours
        for(let k = 0; k < 4; k++)
        {
            if (isValid(i + rowNum[k], j + colNum[k]) &&
                isSafe(mat, visited, i + rowNum[k],
                                     j + colNum[k]))
            {
                findShortestPathUtil(mat, visited, i + rowNum[k],
                                 j + colNum[k], dist + 1);
            }
        }
 
        // Backtrack
        visited[i][j] = false;
    }
 
    // A wrapper function over findshortestPathUtil()
    function findShortestPath(mat)
    {
 
        // Stores minimum cost of shortest path so far
        min_dist = Number.MAX_VALUE;
 
        // Create a boolean matrix to store info about
        // cells already visited in current route
        let visited = new Array(R);
         
        for(let i = 0; i < R; i++)
        {
            visited[i] = new Array(C);
            for(let j = 0; j < C; j++)
            {
                visited[i][j] = false;
            }
        }
 
        // Mark adjacent cells of landmines as unsafe
        markUnsafeCells(mat);
 
        // Start from first column and take minimum
        for(let i = 0; i < R; i++)
        {
            // If path is safe from current cell
            if (mat[i][0] == 1)
            {
 
                // Find shortest route from (i, 0) to any
                // cell of last column (x, C - 1) where
                // 0 <= x < R
                findShortestPathUtil(mat, visited, i, 0, 0);
 
                // If min distance is already found
                if (min_dist == C - 1)
                    break;
            }
        }
 
        // If destination can be reached
        if (min_dist != Number.MAX_VALUE)
            document.write("Length of shortest " +
                               "safe route is " + min_dist + "</br>");
 
        else
 
            // If the destination is not reachable
            document.write("Destination not " +
                               "reachable from given source" + "</br>");
    }
     
    // Input matrix with landmines shown with number 0
    let mat = [
        [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
        [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 ],
        [ 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 ],
        [ 1, 1, 1, 1, 0, 1, 1, 1, 1, 1 ],
        [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
        [ 1, 1, 1, 1, 1, 0, 1, 1, 1, 1 ],
        [ 1, 0, 1, 1, 1, 1, 1, 1, 0, 1 ],
        [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
        [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
        [ 0, 1, 1, 1, 1, 0, 1, 1, 1, 1 ],
        [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
        [ 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 ] ];
  
    // Find shortest path
    findShortestPath(mat);
 
// This code is contributed by decode2207.
</script>

Output:

Length of shortest safe route is 13

Another method: It can be solved in polynomial time with the help of Breadth First Search. Enqueue the cells with 1 value in the queue with the distance as 0. As the BFS proceeds, shortest path to each cell from the first column is computed. Finally for the reachable cells in the last column, output the minimum distance.

The implementation in C++ is as follows:

C++

// C++ program to find shortest safe Route in
// the matrix with landmines
#include <bits/stdc++.h>
using namespace std;
 
#define R 12
#define C 10
 
struct Key{
    int x,y;
    Key(int i,int j){ x=i;y=j;};
};
 
// These arrays are used to get row and column
// numbers of 4 neighbours of a given cell
int rowNum[] = { -1, 0, 0, 1 };
int colNum[] = { 0, -1, 1, 0 };
 
// A function to check if a given cell (x, y) is
// a valid cell or not
bool isValid(int x, int y)
{
    if (x < R && y < C && x >= 0 && y >= 0)
        return true;
 
    return false;
}
 
// A function to mark all adjacent cells of
// landmines as unsafe. Landmines are shown with
// number 0
void findShortestPath(int mat[R][C]){
    int i,j;
 
    for (i = 0; i < R; i++)
    {
        for (j = 0; j < C; j++)
        {
            // if a landmines is found
            if (mat[i][j] == 0)
            {
            // mark all adjacent cells
            for (int k = 0; k < 4; k++)
                if (isValid(i + rowNum[k], j + colNum[k]))
                    mat[i + rowNum[k]][j + colNum[k]] = -1;
            }
        }
    }
// mark all found adjacent cells as unsafe
    for (i = 0; i < R; i++)
    {
        for (j = 0; j < C; j++)
        {
            if (mat[i][j] == -1)
                mat[i][j] = 0;
        }
    }
 
    int dist[R][C];
 
    for(i=0;i<R;i++){
        for(j=0;j<C;j++)
            dist[i][j] = -1;
    }
    queue<Key> q;
 
    for(i=0;i<R;i++){
        if(mat[i][0] == 1){
            q.push(Key(i,0));
            dist[i][0] = 0;
        }
    }
 
    while(!q.empty()){
        Key k = q.front();
        q.pop();
 
        int d = dist[k.x][k.y];
 
        int x = k.x;
        int y = k.y;
 
        for (int k = 0; k < 4; k++) {
            int xp = x + rowNum[k];
            int yp = y + colNum[k];
            if(isValid(xp,yp) && dist[xp][yp] == -1 && mat[xp][yp] == 1){
                dist[xp][yp] = d+1;
                q.push(Key(xp,yp));
            }
        }
    }
    // stores minimum cost of shortest path so far
    int ans = INT_MAX;
    // start from first column and take minimum
    for(i=0;i<R;i++){
        if(mat[i][C-1] == 1 && dist[i][C-1] != -1){
            ans = min(ans,dist[i][C-1]);
        }
    }
     
    // if destination can be reached
    if(ans == INT_MAX)
        cout << "NOT POSSIBLE\n";
         
    else// if the destination is not reachable
        cout << "Length of shortest safe route is " << ans << endl;
}
 
// Driver code
int main(){
     
    // input matrix with landmines shown with number 0
    int mat[R][C] =
    {
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 0, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 0, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 0, 1, 1, 1, 1 },
        { 1, 0, 1, 1, 1, 1, 1, 1, 0, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 0, 1, 1, 1, 1, 0, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 },
        { 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 }
    };
     
    // find shortest path
    findShortestPath(mat);
}

Output:

Length of shortest safe route is 13

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