Solution: Cherry Pickup

Explore a dynamic programming approach to solve the Cherry Pickup problem where you find the maximum cherries collected on a round trip path across an n by n grid. Understand the use of memoization and how synchronized dual-path traversal simplifies the solution and optimizes performance.

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Statement
Solution
- Time complexity
- Space complexity

Statement

You are given an $n \times n$ grid representing a field of cherries. Each cell in the grid can have one of three possible values:

$0$ : An empty cell that can be walked through
$1$ : A cell containing a cherry that can be picked and then passed through
$-1$ : A cell containing a thorn, which cannot be crossed

Your task is to find the maximum number of cherries that can be collected while following these rules:

Start at the top-left corner $(0,0)$ and reach the bottom-right corner $(n-1,~n-1)$ by only moving right or down through valid cells ( $0$ or $1$ ).
After reaching the bottom-right corner, return to the starting point $(0, 0)$ ...