Given a matrix of size **N X M** consisting of ‘#’, ‘.’ and ‘*’. ‘#’ means blocked path, ‘.’ means walkable path and ‘*’ means point you have to collect. Now consider you are at top left of the matrix. You have to reach bottom right of the matrix and come back to top left. When you are moving top left to bottom right, you are allowed to walk right or down. And when you are moving bottom right to top left, you are allowed to walk left or up. The task is find the maximum points you can grab during your whole journey. Points once taken will be converted into ‘.’ i.e walkable path.

Examples:

Input : N = 3, M = 4 **** .##. .##. **** Output : 8 Input : N = 9, M = 7 *........ .....**#. ..**...#* ..####*#. .*.#*.*#. ...#**... *........ Output : 7

If you consider two paths – one from (0, 0) to (N – 1, M – 1) and another from (N – 1, M – 1) to (0, 0) and collect maximum * in each path. You might end up with wrong answer. If you add the answer of both the paths independently, you will be calculating the intersecting points once again while backtracking. Essentially, ending up with the same path. Even if you keep a visited array, marking every * on the first path, you will still not end up with the correct answer.

Consider the following example:

If we consider as two independent path, then

total * collected is 7.

Whereas the maximum points collected can be 8 by following paths.

There are 4 * on each path. Hence, total = 8.

Thus we see that the optimal solution is not the sum of optimal solutions of both the path independently. The optimal answer for one does not ensure an optimal answer.

So, we will have to calculate both the paths simultaneously. This is needed because the answer for path 2 depends on the route chosen by path 1. Simultaneous calculation can be done by considering two path from (0, 0) to (N-1, M-1) and making four decisions at each position (two for each).

So instead of two traveling one path from top left to bottom right and other from bottom right to top left, we will travel two path from (0, 0) to (N-1, M-1) simultaneously, so at each step we take one step for both path. So our state will consist of (x1, y1, x2, y2) where (x1, y1) is position of first path and (x2, y2) is position of second tourist in grid.

Points to notice:

1. At each step either step can move right or down, so we have 4 choices for movement(2 choice for each path).

2. If both path are on the same cell (x1 == x2 and y1 == y2) then we can add only 1 to result if that cell have *.

3. We can reduce the complexity by reducing state dimesion from 4 to 3. If we know the position of first path (x1, y1) the x coordinate of second path x2, then we must have x1 + y1 = x2 + y2 since both path cover the same distance in same amount of time. So y2 = x1 + y1 – x2 and our state depends only on (x1, y1, x2).

Below is C++ implementation of this approach:

// CPP program to find maximum points that can // be collected in a journey from top to bottom // and then back from bottom to top, #include <bits/stdc++.h> #define MAX 5 #define N 5 #define M 5 #define inf 100000 using namespace std; // Calculating the points at a (row1, col1) && // (row2, col2) from path1 && path2 int cost(char grid[][M], int row1, int col1, int row2, int col2) { // If both path is at same cell if (row1 == row2 && col1 == col2) { // If the cell contain *, return 1 if (grid[row1][col1] == '*') return 1; // else return 0. return 0; } int ans = 0; // If path 1 contain *, add to answer. if (grid[row1][col1] == '*') ans++; // If path contain *, add to answer. if (grid[row2][col2] == '*') ans++; return ans; } // Calculate the maximum points that can be // collected. int solve(int n, int m, char grid[][M], int dp[MAX][MAX][MAX], int row1, int col1, int row2) { int col2 = (row1 + col1) - (row2); // If both path reach the bottom right cell if (row1 == n - 1 && col1 == m - 1 && row2 == n - 1 && col2 == m - 1) return 0; // If moving out of grid if (row1 >= n || col1 >= m || row2 >= n || col2 >= m) return -1 * inf; // If already calculated, return the value if (dp[row1][col1][row2] != -1) return dp[row1][col1][row2]; // Vaiable for 4 options. int ch1 = -1 * inf, ch2 = -1 * inf; int ch3 = -1 * inf, ch4 = -1 * inf; // If path 1 is moving right and path 2 // is moving down. if (grid[row1][col1 + 1] != '#' && grid[row2 + 1][col2] != '#') ch1 = cost(grid, row1, col1 + 1, row2 + 1, col2) + solve(n, m, grid, dp, row1, col1 + 1, row2 + 1); // If path 1 is moving right and path 2 is moving // right. if (grid[row1][col1 + 1] != '#' && grid[row2][col2 + 1] != '#') ch2 = cost(grid, row1, col1 + 1, row2, col2 + 1) + solve(n, m, grid, dp, row1, col1 + 1, row2); // If path 1 is moving down and path 2 is moving right. if (grid[row1 + 1][col1] != '#' && grid[row2][col2 + 1] != '#') ch3 = cost(grid, row1 + 1, col1, row2, col2 + 1) + solve(n, m, grid, dp, row1 + 1, col1, row2); // If path 1 is moving down and path 2 is moving down. if (grid[row1 + 1][col1] != '#' && grid[row2 + 1][col2] != '#') ch4 = cost(grid, row1 + 1, col1, row2 + 1, col2) + solve(n, m, grid, dp, row1 + 1, col1, row2 + 1); // Returning the maximum of 4 options. return dp[row1][col1][row2] = max({ch1, ch2, ch3, ch4}); } // Wrapper Function int wrapper(int n, int m, char grid[N][M]) { int ans = 0; int dp[MAX][MAX][MAX]; memset(dp, -1, sizeof dp); // If last bottom right cell is blcoked if (grid[n - 1][m - 1] == '#' || grid[0][0] == '#') ans = -1 * inf; // If top left cell contain * if (grid[0][0] == '*') ans++; grid[0][0] = '.'; // If bottom right cell contain * if (grid[n - 1][m - 1] == '*') ans++; grid[n - 1][m - 1] = '.'; ans += solve(n, m, grid, dp, 0, 0, 0); return max(ans, 0); } // Driven Program int main() { int n = 5, m = 5; char grid[N][M] = { { '.', '*', '.', '*', '.' }, { '*', '#', '#', '#', '.' }, { '*', '.', '*', '.', '*' }, { '.', '#', '#', '#', '*' }, { '.', '*', '.', '*', '.' } }; cout << wrapper(n, m, grid) << endl; return 0; }

Output:

8

**Time Complexity:** O(N^3)

This article is contributed by **Anuj Chauhan**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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