Find shortest safe route in a path with landmines
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 cellint 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 notbool 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 notbool 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 0void 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 cordinates of the current cell// min_dist --> stores minimum cost of shortest path so far// dist --> stores current path costvoid 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) { // initailize 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 codeint 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;} |
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Java
// Java program to find shortest safe Route// in the matrix with landminesimport 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 cellstatic 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 notstatic 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 notstatic 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 0static 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 cordinates of the current cell// min_dist --> stores minimum cost of shortest path so far// dist --> stores current path coststatic 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) { // Initailize 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 codepublic 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 |
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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 cellint 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 notbool 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 0void 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 codeint 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); } |
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Output:
Length of shortest safe route is 13
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