The cherry pickup problem is a combinatorial optimization problem that involves finding the longest path through a grid of cells, where each cell may contain cherries that can be collected. The path must start and end at designated cells, and the path can only move in certain directions (usually up, down, left, or right).

In the standard version of the problem, there are two players who can move through the grid at the same time, and they can collect the cherries in each cell as they pass through it. The players are not allowed to move through the same cell at the same time, and they must return to the starting cell at the end of the game. The goal is to maximize the total number of cherries collected by both players.

Here is an example of the cherry pickup problem:

[ [0, 1, -1],

[1, 0, -1],

[-1, 1, 0]

]{codeBox}

- Dynamic programming: One way to solve the cherry pickup problem is to use dynamic programming. This involves defining a recursive function that takes as input the current positions of the players and the remaining time steps and returns the maximum number of cherries that can be collected in the remaining time. The function can be implemented using a 3D array to store the results of the subproblems, and the solutions to the subproblems can be combined to obtain the solution to the original problem.

- Breadth-first search: Another approach is to use breadth-first search (BFS) to explore the space of possible paths through the grid. This involves maintaining a queue of positions that have not yet been explored and iteratively expanding the search to new positions that are reachable from the current position. The BFS algorithm can be modified to keep track of the total number of cherries collected at each position and to return the maximum number of cherries collected when the search is complete.

- Depth-first search: A depth-first search (DFS) approach can also be used to solve the cherry pickup problem. This involves recursively exploring the grid, starting at the current position and moving in all possible directions until a solution is found or all paths have been exhausted. The DFS algorithm can be modified to keep track of the total number of cherries collected at each position and to return the maximum number of cherries collected when the search is complete.

Here is some example code that solves the cherry pickup problem using dynamic programming:

def cherryPickup(grid: List[List[int]]) -> int:

n = len(grid)

dp = [[[-1] * n for _ in range(n)] for _ in range(n)]

dp[0][0][0] = grid[0][0]

for t in range(1, 2 * n - 1):

for i in range(max(0, t - (n - 1)), min(n - 1, t) + 1):

for j in range(max(0, t - (n - 1)), min(n - 1, t) + 1):

for pi in (i - 1, i, i + 1):

for pj in (j - 1, j, j + 1):

if pi >= 0 and pj >= 0 and dp[pi][pj][i] != -1:

if grid[pi][pj] == -1 or grid[i][j] == -1:

continue

dp[i][j][pi] = max(dp[i][j][pi], dp[pi][pj][i] + grid[i][j] + (i != j) * grid[pi][pj])

return max(0, dp[n - 1][n - 1][n - 1]){codeBox}