Cherry Pickup II

This can be solved using dynamic programming. We can use a 3D dp array to keep track of the maximum cherries that both robots can collect.

Here’s a step-by-step explanation:

Initialize Variables: We’ll initialize a 3D array dp with dimensions (rows, cols, cols) and fill it with -1 as a marker for unvisited states.
Define Base Case: The base case is when both robots are at the last row, and their cherries are simply the sum of the cherries in their current cells (considering that they can’t be at the same cell).
Recursive Function: We’ll define a recursive function that takes the current row and the columns of both robots. It calculates the cherries picked by trying all possible combinations of moving down or diagonally down left or right.
Return the Answer: Finally, we’ll return the result from the recursive function for the starting positions of the robots.

Python solution:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
class Solution:
    def cherryPickup(self, grid: List[List[int]]) -> int:
        rows, cols = len(grid), len(grid[0])
        dp = [[[None] * cols for _ in range(cols)] for _ in range(rows)]

        # Recursive function to calculate cherries
        def helper(r, c1, c2):
            if r == rows:
                return 0
            if c1 < 0 or c1 >= cols or c2 < 0 or c2 >= cols:
                return float('-inf')
            if dp[r][c1][c2] != None:
                return dp[r][c1][c2]

            # Picking cherries for current cell
            cherries = grid[r][c1] + (grid[r][c2] if c1 != c2 else 0)

            # Trying all combinations of moves for both robots
            maxCherries = 0
            for d1 in range(-1, 2):
                for d2 in range(-1, 2):
                    nextCherries = helper(r + 1, c1 + d1, c2 + d2)
                    maxCherries = max(maxCherries, nextCherries)

            dp[r][c1][c2] = cherries + maxCherries
            return dp[r][c1][c2]

        return helper(0, 0, cols - 1)

This solution is efficient enough for the given constraints, with a time complexity of (O(rows \cdot cols^3)).

This involves dynamic programming and multi-dimensional array manipulation. Here are some simpler problems to understand these concepts:

LeetCode 62. Unique Paths
- This problem is a classic dynamic programming problem that can help you understand how to count paths in a grid.
LeetCode 63. Unique Paths II
- This is a follow-up to the previous problem, introducing obstacles in the grid and showing you how to adjust your dynamic programming approach to handle them.
LeetCode 64. Minimum Path Sum
- This problem asks you to find a path from top left to bottom right which minimizes the sum of all numbers along its path, another grid-based dynamic programming problem.
LeetCode 221. Maximal Square
- This problem will help you learn how to work with 2D arrays and apply dynamic programming techniques to find a solution.
LeetCode 72. Edit Distance
- This problem is a classical dynamic programming problem which requires considering several possibilities for each state.
LeetCode 198. House Robber
- This problem is a simpler dynamic programming problem which will help you understand how to handle dynamic programming problems.
LeetCode 121. Best Time to Buy and Sell Stock
- This problem involves dynamic programming and will help you understand the concept of maintaining a dynamic ‘maximum’ or ‘minimum’ value during traversal.
LeetCode 518. Coin Change 2
- This problem is a good practice for understanding dynamic programming involving counting combinations that add up to a specific value.
LeetCode 1143. Longest Common Subsequence
- This problem is a classical dynamic programming problem which requires considering several possibilities for each state.
LeetCode 741. Cherry Pickup
- This problem is a precursor to Cherry Pickup II, and understanding how to solve this problem would be directly beneficial to solving Cherry Pickup II.

You’ll understand how to set up a dynamic programming table, how to handle dynamic programming transitions, and how to traverse grids.

Problem Classification

Problem Statement:You are given a rows x cols matrix grid representing a field of cherries where grid[i][j] represents the number of cherries that you can collect from the (i, j) cell.

You have two robots that can collect cherries for you:

Robot #1 is located at the top-left corner (0, 0), and Robot #2 is located at the top-right corner (0, cols - 1). Return the maximum number of cherries collection using both robots by following the rules below:

From a cell (i, j), robots can move to cell (i + 1, j - 1), (i + 1, j), or (i + 1, j + 1). When any robot passes through a cell, It picks up all cherries, and the cell becomes an empty cell. When both robots stay in the same cell, only one takes the cherries. Both robots cannot move outside of the grid at any moment. Both robots should reach the bottom row in grid.

Example 1:

Input: grid = [[3,1,1],[2,5,1],[1,5,5],[2,1,1]] Output: 24 Explanation: Path of robot #1 and #2 are described in color green and blue respectively. Cherries taken by Robot #1, (3 + 2 + 5 + 2) = 12. Cherries taken by Robot #2, (1 + 5 + 5 + 1) = 12. Total of cherries: 12 + 12 = 24. Example 2:

Input: grid = [[1,0,0,0,0,0,1],[2,0,0,0,0,3,0],[2,0,9,0,0,0,0],[0,3,0,5,4,0,0],[1,0,2,3,0,0,6]] Output: 28 Explanation: Path of robot #1 and #2 are described in color green and blue respectively. Cherries taken by Robot #1, (1 + 9 + 5 + 2) = 17. Cherries taken by Robot #2, (1 + 3 + 4 + 3) = 11. Total of cherries: 17 + 11 = 28.

Constraints:

rows == grid.length cols == grid[i].length 2 <= rows, cols <= 70 0 <= grid[i][j] <= 100

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.