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:
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++ 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;
}
Output:
Length of shortest safe route is 13