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This approach utilizes backtracking and depth-first search (DFS) to explore all possible paths to collect the maximum gold. Start from each cell containing gold and perform DFS to explore in all four possible directions (up, down, left, right). During the search, keep track of the total gold collected. Use backtracking to unvisit a cell after exploring all potential paths from it.
Time complexity: O((m*n)^2) - In the worst-case scenario, each cell containing gold gets visited and backtracked.
Space complexity: O(m*n) - This results from recursive stack space and the grid modifications.
1public class Solution {
2 private int DFS(int[][] grid, int x, int y) {
3 if (x < 0 || y < 0 || x >= grid.Length || y >= grid[0].Length || grid[x][y] == 0) return 0;
4
5 int originalGold = grid[x][y];
6 grid[x][y] = 0; // Mark as visited
7 int maxGold = 0;
8 int[][] directions = new int[][] { new int[] {1, 0}, new int[] {-1, 0}, new int[] {0, 1}, new int[] {0, -1} };
9 foreach(var dir in directions) {
10 maxGold = Math.Max(maxGold, DFS(grid, x + dir[0], y + dir[1]));
11 }
12 grid[x][y] = originalGold; // Backtrack
13 return originalGold + maxGold;
14 }
15
16 public int GetMaximumGold(int[][] grid) {
17 int maxGold = 0;
18 for (int i = 0; i < grid.Length; i++) {
19 for (int j = 0; j < grid[0].Length; j++) {
20 if (grid[i][j] > 0) {
21 maxGold = Math.Max(maxGold, DFS(grid, i, j));
22 }
23 }
24 }
25 return maxGold;
26 }
27}The C# solution models its logic on recursive depth-first search, optimizing direction handling within a loop for potential path traversal. It accomplishes thorough evaluation via state-managed grid manipulations and backtracking, ensuring all feasible paths are scrutinized to ascertain maximum achievable gold.
This approach involves using dynamic programming with bit masking to track visited states and optimize the path of maximum gold collection. It systematically calculates the maximum gold that can be gathered by starting from each gold cell and considering valid moves while ensuring no cell is revisited.
Time complexity: O(2^(m*n) * m * n) given bitmask state examination.
Space complexity: O(2^(m*n) * m * n) due to DP storage needs for various path states.
1#include <algorithm>
#include <cstring>
using namespace std;
class Solution {
int dp[1 << 15][15][15];
int dirs[5] = {0, 1, 0, -1, 0};
public:
int solve(vector<vector<int>>& grid, int mask, int x, int y) {
if (dp[mask][x][y] != -1) return dp[mask][x][y];
int maxGold = 0;
for (int d = 0; d < 4; ++d) {
int nx = x + dirs[d], ny = y + dirs[d + 1];
if (nx >= 0 && ny >= 0 && nx < grid.size() && ny < grid[0].size() && grid[nx][ny] > 0
&& !(mask & (1 << (nx * grid[0].size() + ny)))) {
int newMask = mask | (1 << (nx * grid[0].size() + ny));
maxGold = max(maxGold, solve(grid, newMask, nx, ny));
}
}
return dp[mask][x][y] = grid[x][y] + maxGold;
}
int getMaximumGold(vector<vector<int>>& grid) {
memset(dp, -1, sizeof(dp));
int maxGold = 0, m = grid.size(), n = grid[0].size();
for (int i = 0; i < m; ++i) {
for (int j = 0; j < n; ++j) {
if (grid[i][j] > 0) {
int mask = 1 << (i * n + j);
maxGold = max(maxGold, solve(grid, mask, i, j));
}
}
}
return maxGold;
}
};The C++ strategy employs dynamic programming and bit-masking to address cell visted states, similar to the C-underpinning logic. This coverage aims to derive optimal routes leveraging memory efficiently through DP arrays. Referential paths are stored as bitmask iterations with recursive resolution aiding rapid path enumeration.