# Unique paths covering every non-obstacle block exactly once in a grid

Given a grid grid[][] with 4 types of blocks:

• 1 represents the starting block. There is exactly one starting block.
• 2 represents the ending block. There is exactly one ending block.
• 0 represents empty block we can walk over.
• -1 represents obstacles that we cannot walk over.

The task is to count the number of paths from the starting block to the ending block such that every non-obstacle block is covered exactly once.

Examples:

Input: grid[][] = {
{1, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 2, -1} }
Output:
Following are the only paths covering all the non-obstacle blocks:

Input: grid[][] = {
{1, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 2} }
Output:

Approach: We can use simple DFS here with backtracking. We can check that a particular path has covered all the non-obstacle blocks by counting all the blocks encountered in the way and finally comparing it with the total number of blocks available and if they match, then we add it as a valid solution.

Below is the implementation of the above approach:

 `// C++ implementation of the approach` `#include ` `using` `namespace` `std;`   `// Function for dfs.` `// i, j ==> Current cell indexes` `// vis ==> To mark visited cells` `// ans ==> Result` `// z ==> Current count 0s visited` `// z_count ==> Total 0s present` `void` `dfs(``int` `i, ``int` `j, vector >& grid,` `         ``vector >& vis, ``int``& ans,` `         ``int` `z, ``int` `z_count)` `{` `    ``int` `n = grid.size(), m = grid.size();`   `    ``// Mark the block as visited` `    ``vis[i][j] = 1;` `    ``if` `(grid[i][j] == 0)`   `        ``// update the count` `        ``z++;`   `    ``// If end block reached` `    ``if` `(grid[i][j] == 2) {`   `        ``// If path covered all the non-` `        ``// obstacle blocks` `        ``if` `(z == z_count)` `            ``ans++;` `        ``vis[i][j] = 0;` `        ``return``;` `    ``}`   `    ``// Up` `    ``if` `(i >= 1 && !vis[i - 1][j] && grid[i - 1][j] != -1)` `        ``dfs(i - 1, j, grid, vis, ans, z, z_count);`   `    ``// Down` `    ``if` `(i < n - 1 && !vis[i + 1][j] && grid[i + 1][j] != -1)` `        ``dfs(i + 1, j, grid, vis, ans, z, z_count);`   `    ``// Left` `    ``if` `(j >= 1 && !vis[i][j - 1] && grid[i][j - 1] != -1)` `        ``dfs(i, j - 1, grid, vis, ans, z, z_count);`   `    ``// Right` `    ``if` `(j < m - 1 && !vis[i][j + 1] && grid[i][j + 1] != -1)` `        ``dfs(i, j + 1, grid, vis, ans, z, z_count);`   `    ``// Unmark the block (unvisited)` `    ``vis[i][j] = 0;` `}`   `// Function to return the count of the unique paths` `int` `uniquePaths(vector >& grid)` `{` `    ``int` `z_count = 0; ``// Total 0s present` `    ``int` `n = grid.size(), m = grid.size();` `    ``int` `ans = 0;` `    ``vector > vis(n, vector<``bool``>(m, 0));` `    ``int` `x, y;` `    ``for` `(``int` `i = 0; i < n; ++i) {` `        ``for` `(``int` `j = 0; j < m; ++j) {`   `            ``// Count non-obstacle blocks` `            ``if` `(grid[i][j] == 0)` `                ``z_count++;` `            ``else` `if` `(grid[i][j] == 1) {`   `                ``// Starting position` `                ``x = i, y = j;` `            ``}` `        ``}` `    ``}` `    ``dfs(x, y, grid, vis, ans, 0, z_count);` `    ``return` `ans;` `}`   `// Driver code` `int` `main()` `{` `    ``vector > grid{ { 1, 0, 0, 0 },` `                               ``{ 0, 0, 0, 0 },` `                               ``{ 0, 0, 2, -1 } };`   `    ``cout << uniquePaths(grid);` `    ``return` `0;` `}`

 `# Python3 implementation of the approach `   `# Function for dfs. ` `# i, j ==> Current cell indexes ` `# vis ==> To mark visited cells ` `# ans ==> Result ` `# z ==> Current count 0s visited ` `# z_count ==> Total 0s present ` `def` `dfs(i, j, grid, vis, ans, z, z_count):`   `    ``n ``=` `len``(grid)` `    ``m ``=` `len``(grid[``0``])`   `    ``# Mark the block as visited ` `    ``vis[i][j] ``=` `1` `    `  `    ``if` `(grid[i][j] ``=``=` `0``):`   `        ``# Update the count ` `        ``z ``+``=` `1`   `    ``# If end block reached ` `    ``if` `(grid[i][j] ``=``=` `2``):`   `        ``# If path covered all the non- ` `        ``# obstacle blocks ` `        ``if` `(z ``=``=` `z_count):` `            ``ans ``+``=` `1` `            `  `        ``vis[i][j] ``=` `0` `        `  `        ``return` `grid, vis, ans ` `        `  `    ``# Up ` `    ``if` `(i >``=` `1` `and` `not` `vis[i ``-` `1``][j] ``and` `                      ``grid[i ``-` `1``][j] !``=` `-``1``): ` `        ``grid, vis, ans ``=` `dfs(i ``-` `1``, j, grid, ` `                             ``vis, ans, z, ` `                             ``z_count)`   `    ``# Down ` `    ``if` `(i < n ``-` `1` `and` `not` `vis[i ``+` `1``][j] ``and` `                         ``grid[i ``+` `1``][j] !``=` `-``1``): ` `        ``grid, vis, ans ``=` `dfs(i ``+` `1``, j, grid, ` `                             ``vis, ans, z, ` `                             ``z_count)`   `    ``# Left ` `    ``if` `(j >``=` `1` `and` `not` `vis[i][j ``-` `1``] ``and` `                      ``grid[i][j ``-` `1``] !``=` `-``1``): ` `        ``grid, vis, ans ``=` `dfs(i, j ``-` `1``, grid, ` `                             ``vis, ans, z, ` `                             ``z_count) `   `    ``# Right ` `    ``if` `(j < m ``-` `1` `and` `not` `vis[i][j ``+` `1``] ``and` `                         ``grid[i][j ``+` `1``] !``=` `-``1``): ` `        ``grid, vis, ans ``=` `dfs(i, j ``+` `1``, grid,` `                             ``vis, ans, z,` `                             ``z_count) `   `    ``# Unmark the block (unvisited) ` `    ``vis[i][j] ``=` `0` `    `  `    ``return` `grid, vis, ans`   `# Function to return the count ` `# of the unique paths ` `def` `uniquePaths(grid): ` `    `  `    ``# Total 0s present ` `    ``z_count ``=` `0` `    ``n ``=` `len``(grid)` `    ``m ``=` `len``(grid[``0``])` `    ``ans ``=` `0` `    `  `    ``vis ``=` `[[``0` `for` `j ``in` `range``(m)]` `              ``for` `i ``in` `range``(n)]` `    `  `    ``x ``=` `0` `    ``y ``=` `0` `    `  `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(m):` `            `  `            ``# Count non-obstacle blocks` `            ``if` `grid[i][j] ``=``=` `0``:` `                ``z_count ``+``=` `1` `                `  `            ``elif` `(grid[i][j] ``=``=` `1``):` `                `  `                ``# Starting position` `                ``x ``=` `i` `                ``y ``=` `j ` `        `  `    ``grid, vis, ans ``=` `dfs(x, y, grid, ` `                         ``vis, ans, ``0``,` `                         ``z_count)` `                         `  `    ``return` `ans`   `# Driver code ` `if` `__name__``=``=``'__main__'``:`   `    ``grid ``=` `[ [ ``1``, ``0``, ``0``, ``0` `], ` `             ``[ ``0``, ``0``, ``0``, ``0` `],` `             ``[ ``0``, ``0``, ``2``, ``-``1` `] ]` `             `  `    ``print``(uniquePaths(grid))` `    `  `# This code is contributed by rutvik_56`

Output:
`2`

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