Maximum Number of Fish in a Grid


Identifying Problem Isomorphism
“Maximum Number of Fish in a Grid” is based on the DepthFirst Search (DFS) algorithm where we traverse all cells, collecting as many fish as possible. We can map this problem to a similar problem “Gold Mine”.
In the “Path with Maximum Gold” problem, a grid is given representing a gold mine, and we need to get the maximum amount of gold from any cell in the first column to any cell in the last column. This problem is similar because it also involves a grid where each cell contains a certain amount (fish or gold), and we need to optimize our path to maximize the total amount collected.
The primary difference is that the “Gold Mine” problem involves traversing from the first column to the last, whereas in the “Maximum Number of Fish in a Grid” problem, the fisher can start at any cell. Nonetheless, the “Gold Mine” problem can be a useful reference for understanding how to implement a depthfirst search on a grid to maximize a cumulative value.
The constraints and conditions are not exactly the same, but the problemsolving strategies share similarities. Always adapt the algorithm to suit the specific conditions of your problem.


Problem Classification
The problem belongs to the graph traversal and dynamic programming.
‘What’ Components:
 Grid: The problem provides a 2D grid or matrix of size m x n. Each cell in the grid can either be a land cell (represented by 0) or a water cell (represented by a positive integer indicating the number of fish).
 Fisher: A fisher who can start at any water cell and either catch all the fish at that cell or move to any adjacent water cell.
 Operations: The fisher can perform two types of operations any number of times  Catch all fish at a cell or move to an adjacent water cell.
 Goal: The aim is to determine the maximum number of fish the fisher can catch, assuming the starting cell is chosen optimally.
The problem is a type of optimization problem in the field of graph theory. It involves traversing the grid (which can be seen as a graph) in a way that maximizes the total number of fish caught. It could be solved by a DepthFirst Search (DFS) or Dynamic Programming approach.
The problem also has elements of the Greedy algorithm as the fisher needs to make a locally optimal choice (catch fish or move to the next cell) at each step. But the overall solution is not purely greedy, as the fisher must choose the starting cell optimally, which requires considering the entire grid.
This problem is in the domain of Graph Theory and DepthFirst Search. It involves traversing a 2D grid (or a graph) and optimizing a quantity (the number of fish caught).
Here are the identified ‘What’ components:
 We are given a 2D matrix, each cell of which may be either a land cell or a water cell containing a certain number of fish.
 A fisher can start at any water cell and can perform certain operations an unlimited number of times.
 These operations are to catch all the fish at the current cell or to move to an adjacent water cell.
 We need to find the maximum number of fish that the fisher can catch if he starts at the optimal cell. If no water cell exists, we return 0.
This problem falls into the Optimization subclass of problems under the Graph Theory class. The problem involves searching for an optimal strategy (maximizing the number of fish caught), which needs to explore all potential paths (each possible sequence of adjacent water cells), hence graph traversal using DepthFirst Search is involved. Moreover, the representation of the land and water cells and their respective fish counts in a 2D grid hints at a need for effective grid manipulation. This is very typical in the landscape of Graph Theory problems.
Language Agnostic Coding Drills
Dissect the code and identify each distinct concept it contains:
a) Basic Python Syntax: Concepts like functions, loops, conditionals, and variables are fundamental to understanding the code.
b) Understanding and Manipulating 2D Lists (Matrices): The problem requires traversing and updating a 2D list, which is a crucial concept to grasp.
c) DepthFirst Search (DFS) on a Grid: The main algorithm implemented is DFS which involves moving to adjacent cells in a grid recursively until all viable paths are explored.
d) Grid Bounds Checking: As the algorithm involves moving across a 2D grid, checking whether a given cell lies within the grid boundaries is important.
e) Maximum Calculation: A running maximum calculation is required to track the best possible starting point.
Order the identified coding concepts by difficulty:
a) Basic Python Syntax: This is the easiest concept to understand, as it forms the foundation of all Python programs.
b) Understanding and Manipulating 2D Lists (Matrices): It requires understanding of lists and how to access and manipulate their elements.
c) Grid Bounds Checking: It involves conditional checking and understanding of how to work with indices in a 2D list. It’s harder than the previous concept because it requires knowledge about 2D lists and a bit of logical thinking.
d) Maximum Calculation: This involves keeping track of a variable as we loop through a data structure. The complexity comes from understanding how and when to update this variable.
e) DepthFirst Search (DFS) on a Grid: This is the most difficult concept as it requires a strong understanding of recursion and can be tricky to implement correctly.
Problemsolving approach leading to the final solution:
The problem can be solved by performing a depthfirst search from every water cell in the grid. The depthfirst search will explore all possible paths starting from the given cell, and add up the number of fish along the path.
a) Start by looping through each cell in the grid. If the cell is a water cell, perform a depthfirst search starting from this cell.
b) The depthfirst search function should start by recording the number of fish in the current cell and setting the cell to 0 (indicating that the fish have been caught).
c) Then, for each of the four possible directions (up, down, left, right), compute the new cell position and check if it is within the grid boundaries and if it is a water cell. If it is, recursively call the depthfirst search function for this new cell.
d) The depthfirst search function should return the total number of fish caught along the path it explored.
e) Keep track of the maximum number of fish caught across all paths explored, and return this as the solution to the problem. This involves understanding how to update a running maximum as we iterate through a data structure.
By integrating these concepts together, we can implement a solution that explores all possible paths in the grid and returns the maximum number of fish that can be caught.
Targeted Drills in Python
Pythonbased coding drills:
a) Basic Python Syntax:
1 2 3 4 5
def hello_world(): for i in range(5): if i % 2 == 0: print(f"Hello World {i}") hello_world()
b) Understanding and Manipulating 2D Lists (Matrices):
1 2 3 4 5
matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] for i in range(len(matrix)): for j in range(len(matrix[0])): matrix[i][j] *= 2 print(matrix)
c) Grid Bounds Checking:
1 2 3
def is_in_bounds(i, j, n, m): return 0 <= i < n and 0 <= j < m print(is_in_bounds(2, 2, 3, 3))
d) Maximum Calculation:
1 2 3 4 5 6
max_value = float('inf') numbers = [1, 3, 7, 2, 5] for num in numbers: if num > max_value: max_value = num print(max_value)
e) DepthFirst Search (DFS) on a Grid: Here’s a basic example of DFS, we will be visiting each cell in the grid exactly once.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
def dfs(grid, i, j): if i < 0 or i >= len(grid) or j < 0 or j >= len(grid[0]) or grid[i][j] == '0': return grid[i][j] = '0' dfs(grid, i+1, j) dfs(grid, i1, j) dfs(grid, i, j+1) dfs(grid, i, j1) grid = [['1', '1', '0', '0', '0'], ['1', '1', '0', '0', '0'], ['0', '0', '1', '0', '0'], ['0', '0', '0', '1', '1']] dfs(grid, 0, 0) print(grid)
Problemspecific concepts: The problemspecific concept here is the depthfirst search on a grid. We need this because the fisher can move to any adjacent cell to catch fish. Using depthfirst search, we can explore all possible paths starting from each water cell to find the maximum number of fish the fisher can catch.
Integration: To assemble the final solution, start by initializing the variable for the maximum number of fish and loop through each cell in the grid. If a cell is a water cell, perform depthfirst search from this cell to calculate the total number of fish that can be caught starting from this cell, update the maximum number of fish if needed. For the depthfirst search, record the number of fish in the current cell and set the cell to 0, then for each of the four possible directions, if the new cell is a valid water cell, recursively call the depthfirst search function for this cell. The depthfirst search function should return the total number of fish caught from the current cell.
10 Prerequisite LeetCode Problems
For the problem “2658. Maximum Number of Fish in a Grid”, the following problems are a good preparation:
“200. Number of Islands”: This problem helps you understand the concept of traversing a grid using depthfirst search (DFS) or breadthfirst search (BFS), which can be useful for the main problem.
“994. Rotting Oranges”: This problem involves a grid and using BFS to solve a problem of similar nature, which can help in the problem “Maximum Number of Fish in a Grid”.
“695. Max Area of Island”: This problem is about finding the maximum area in a 2D grid, which is similar to the task of finding the maximum number of fish in the main problem.
“127. Word Ladder”: Although not directly about a grid, this problem is about BFS in a graph which can be helpful in traversing through the grid in the main problem.
“542. 01 Matrix”: This problem involves manipulation of a 2D grid using BFS or DFS and could help understand grid manipulation techniques.
“286. Walls and Gates”: This problem can help you get comfortable with navigating a 2D grid and dealing with obstacles, a skill that is important in the main problem.
“1091. Shortest Path in Binary Matrix”: This problem involves finding paths in a grid which is similar to navigating through the grid in the main problem.
“130. Surrounded Regions”: This problem will give you a better understanding of how to traverse a grid using DFS, which is a crucial part in the main problem.
“417. Pacific Atlantic Water Flow”: This problem involves traversing a 2D grid similar to the main problem and could help understand the strategy to navigate through the grid.
“1293. Shortest Path in a Grid with Obstacles Elimination”: This problem can help you practice handling more complex constraints while traversing a grid.
These problems should give you a good grounding in 2D grid traversal problems and help you understand the methods and strategies used in solving such problems. This will be beneficial when attempting to solve “2658. Maximum Number of Fish in a Grid”.
Problem Classification
Problem Statement:You are given a 0indexed 2D matrix grid of size m x n, where (r, c) represents:
A land cell if grid[r][c] = 0, or A water cell containing grid[r][c] fish, if grid[r][c] > 0. A fisher can start at any water cell (r, c) and can do the following operations any number of times:
Catch all the fish at cell (r, c), or Move to any adjacent water cell. Return the maximum number of fish the fisher can catch if he chooses his starting cell optimally, or 0 if no water cell exists.
An adjacent cell of the cell (r, c), is one of the cells (r, c + 1), (r, c  1), (r + 1, c) or (r  1, c) if it exists.
Example 1:
Input: grid = [[0,2,1,0],[4,0,0,3],[1,0,0,4],[0,3,2,0]] Output: 7 Explanation: The fisher can start at cell (1,3) and collect 3 fish, then move to cell (2,3) and collect 4 fish.
Example 2:
Input: grid = [[1,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]] Output: 1 Explanation: The fisher can start at cells (0,0) or (3,3) and collect a single fish.
Constraints:
m == grid.length n == grid[i].length 1 <= m, n <= 10 0 <= grid[i][j] <= 10
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.
Clarification Questions
What are the clarification questions we can ask about this problem?
Problem Analysis and Key Insights
What are the key insights from analyzing the problem statement?
Problem Boundary
What is the scope of this problem?
How to establish the boundary of this problem?
Distilling the Problem to Its Core Elements
Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?
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 realworld 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.
Provide names by categorizing these cases
What are the edge cases?
How to visualize these cases?
What are the key insights from analyzing the different cases?
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?
Simple Explanation
Can you explain this problem in simple terms or like you would explain to a nontechnical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?
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 ProblemSolving Approach from the Problem Statement
Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?
How did you infer from the problem statement that this problem can be solved using ?
Simple Explanation of the Proof
I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel 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
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.
Identify Invariant
What is the invariant in this problem?
Identify Loop Invariant
What is the loop invariant in this problem?
Is invariant and loop invariant the same for this problem?
Thought Process
Can you explain the basic thought process and steps involved in solving this type of problem?
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.
Establishing Preconditions and Postconditions
Parameters:
 What are the inputs to the method?
 What types are these parameters?
 What do these parameters represent in the context of the problem?
Preconditions:
 Before this method is called, what must be true about the state of the program or the values of the parameters?
 Are there any constraints on the input parameters?
 Is there a specific state that the program or some part of it must be in?
Method Functionality:
 What is this method expected to do?
 How does it interact with the inputs and the current state of the program?
Postconditions:
 After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
 What does the return value represent or indicate?
 What side effects, if any, does the method have?
Error Handling:
 How does the method respond if the preconditions are not met?
 Does it throw an exception, return a special value, or do something else?
Problem Decomposition
Problem Understanding:
 Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
 Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
 For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
 Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
 For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
 Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
 How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?
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?
Code Explanation and Design Decisions
Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?
Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.
Coding Constructs
Consider the code for the solution of this problem.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem. The response text is of the following format:
Here are 10 problems that use similar underlying concepts: