Maximum Number of Moves in a Grid

This problem is a variant of the classic dynamic programming problem, longest increasing subsequence, with a twist that each cell can move to any of the three cells to its right (top-right, right, bottom-right) if the value in the next cell is greater. The task is to find the maximum number of moves.

We can solve this problem by using dynamic programming. We maintain a 2D array dp of size m x n, where dp[i][j] is the maximum number of moves we can make starting from cell (i, j). We fill this dp table in reverse column order (from right to left) and for each cell, we consider the three possible moves and take the maximum among them. If no valid move can be made, we don’t update the dp value for that cell. Finally, we return the maximum value in the first column of the dp table.

Python solution:

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class Solution:
    def maxMoves(self, grid: List[List[int]]) -> int:
        m, n = len(grid), len(grid[0])
        dp = [[0] * n for _ in range(m)]
        for j in range(n - 1, -1, -1):
            for i in range(m):
                if j == n - 1:
                    dp[i][j] = 0
                else:
                    for dx in [-1, 0, 1]:
                        nx = i + dx
                        if 0 <= nx < m and grid[nx][j + 1] > grid[i][j]:
                            dp[i][j] = max(dp[i][j], 1 + dp[nx][j + 1])
        return max(dp[i][0] for i in range(m))

In this code, we first initialize the dp table with 0’s. Then for each cell from the last column to the first column and for each row from top to bottom, if it’s not in the last column, we check the three possible moves and update the dp value for the current cell if the move is valid. Finally, we return the maximum value in the first column of the dp table, which is the maximum number of moves we can make starting from any cell in the first column.

Identifying Problem Isomorphism

“Maximum Number of Moves in a Grid” has an approximate isomorphic: “Shortest Path in Binary Matrix”

Both problems are based on finding the shortest path in a grid using Breadth-First Search (BFS). They require navigating a grid from a start point to an end point under specific movement rules, and the task is to either maximize or minimize a certain metric.

In “Shortest Path in Binary Matrix”, the grid is binary (consisting of 0s and 1s), and the objective is to find the shortest path from the top-left corner (0,0) to the bottom-right corner (n-1,n-1), where you can move in eight different directions, but cannot pass through a cell with a value of 1 (which represents an obstacle).

“Maximum Number of Moves in a Grid” includes more complex constraints and movement rules but is conceptually similar, involving grid traversal and path optimization.

The specific constraints and grid structures of the problems may vary, but the underlying problem-solving approach, involving BFS and pathfinding, is consistent. Therefore, understanding and solving one can provide insights and techniques that are helpful in addressing the other.

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def maxMoves(self, grid: List[List[int]]) -> int:
    m, n, dirs = len(grid), len(grid[0]), [(0, 1), (1, 1), (-1, 1)]
    @cache
    def dp(i, j):
        ans = 0
        for x, y in dirs:
            ni, nj = i + x, j + y
            if 0 <= ni < m and nj < n and grid[i][j] < grid[ni][nj]:
                ans = max(ans, 1 + dp(ni, nj))
        return ans
    return max(dp(i, 0) for i in range(m))

Problem Classification

The given problem falls into the category of Dynamic Programming and Grid Traversal problems in the Computer Science and Algorithm Design field.

What components:

  1. A 0-indexed m x n grid (matrix) of positive integers is given.
  2. You can start at any cell in the first column of the grid.
  3. From a cell at coordinates (row, col), you can move to any of these three cells: (row - 1, col + 1), (row, col + 1), or (row + 1, col + 1). However, the value of the cell you move to must be strictly bigger than the value of the current cell.
  4. You need to return the maximum number of moves that can be performed.

This is a classic example of a grid-based dynamic programming problem, where the optimal solution is sought for each cell in terms of the cells that can be reached from it. We need to find the maximum number of moves that can be performed following the given conditions, which involves making decisions at each step with the aim of maximizing the total outcome. This is why the problem is categorized as a dynamic programming problem.

Language Agnostic Coding Drills

  1. Dissect the code and identify each distinct concept it contains.

Concepts involved in the code:

a. List Manipulation and Comprehension: The code initializes the list of directions and utilizes list comprehension in the final return statement.

b. Looping Structures: For loop is used to iterate over possible directions and the rows of the grid.

c. Recursion: The code uses recursive function calls to explore all possible paths in the grid.

d. Memoization (Dynamic Programming): The code uses Python’s cache decorator to store the results of the recursive function, avoiding re-computation of previously calculated results.

e. Conditional Statements: The if statement is used to ensure valid moves within the grid.

f. Function Definition: The code includes a defined function dp inside the main function.

  1. List them out in order of increasing difficulty.

a. List Manipulation and Comprehension: Basic skill, necessary for handling data in Python.

b. Looping Structures: Slightly more complex, necessary for iterating through data.

c. Conditional Statements: Similar difficulty to looping, necessary for control flow.

d. Function Definition: Intermediate difficulty, necessary for organizing code and creating reusable components.

e. Recursion: Advanced skill, requiring understanding of the call stack and base/recursive case formulation.

f. Memoization (Dynamic Programming): Advanced skill, involves understanding how to store and retrieve intermediate results for efficient computation.

  1. Describe the problem-solving approach.

The problem can be solved through dynamic programming, specifically a top-down approach utilizing recursion and memoization.

a. Recursive Exploration: Starting from every cell in the first column, we explore all possible directions recursively.

b. Boundary Check & Valid Move: For each direction, we first check if the move is within the boundary of the grid and if the target cell’s value is larger than the current cell’s value. If both conditions are met, we proceed with the move.

c. Memoization: To avoid re-computing the number of moves for each cell, we cache the results. When a recursive call is made to a cell, we first check if the result has been previously computed and cached. If so, we retrieve the result from the cache instead of re-computing it.

d. Maximization: For each valid move, we calculate the total moves (1 plus the result of the recursive call) and keep track of the maximum value.

e. Result Computation: Finally, we return the maximum result among all starting cells in the first column.

Targeted Drills in Python

  1. Python Coding Drills

a. List Manipulation and Comprehension

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numbers = [1, 2, 3, 4, 5]
squares = [num**2 for num in numbers]
print(squares)  # Output: [1, 4, 9, 16, 25]

b. Looping Structures

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numbers = [1, 2, 3, 4, 5]
for num in numbers:
    print(num)

c. Conditional Statements

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numbers = [1, 2, 3, 4, 5]
for num in numbers:
    if num % 2 == 0:
        print(f"{num} is even")
    else:
        print(f"{num} is odd")

d. Function Definition

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def greet(name):
    print(f"Hello, {name}!")

greet("Alice")

e. Recursion

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def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n-1)

print(factorial(5))  # Output: 120

f. Memoization (Dynamic Programming)

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from functools import lru_cache

@lru_cache(maxsize=None)
def fibonacci(n):
    if n < 2:
        return n
    else:
        return fibonacci(n-1) + fibonacci(n-2)

print(fibonacci(10))  # Output: 55
  1. Problem-Specific Concepts

a. Matrix Navigation: The ability to navigate a matrix or 2D list is essential for this problem. For each cell in the grid, we need to evaluate three potential directions (top-right, right, bottom-right).

b. Caching Intermediate Results in a Grid: Given that the problem involves traversing a grid with possible overlapping paths, it is crucial to store intermediate results to avoid redundant calculations.

  1. Integrating the Drills

a. Start by defining a function dp(i, j) that calculates the maximum moves starting from cell (i, j).

b. Within dp(i, j), use a for loop to iterate over all possible directions. For each direction, check the boundary conditions and validity of the move. If valid, call dp(ni, nj) recursively, where ni and nj are the coordinates of the target cell.

c. Use memoization to store the results of dp(i, j) for all cells in the grid. If dp(i, j) is called again, retrieve the result from the cache instead of re-computing it.

d. In each recursive call of dp(i, j), compute the maximum moves by taking the maximum of the current maximum and 1 plus the result of the recursive call.

e. Finally, calculate the result by iterating over all cells in the first column (possible starting points) and returning the maximum result.

10 Prerequisite LeetCode Problems

The problem “2684. Maximum Number of Moves in a Grid” involves concepts of dynamic programming and grid traversal. Here are 10 related LeetCode problems of lesser complexity that can be solved as preparation:

  1. LeetCode 64: Minimum Path Sum: A classic dynamic programming problem which requires finding the minimum path sum from top left to bottom right of a grid.

  2. LeetCode 200: Number of Islands: A problem that involves grid traversal and could help understand more about grid manipulation.

  3. LeetCode 70: Climbing Stairs: This problem introduces the basic idea of dynamic programming, which is essential for the main problem.

  4. LeetCode 746: Min Cost Climbing Stairs: A more complex dynamic programming problem that requires maintaining state between different steps.

  5. LeetCode 55: Jump Game: This problem involves a similar concept of jumping to different indices based on certain rules.

  6. LeetCode 45: Jump Game II: An extension of the previous problem, this one requires finding the minimum number of jumps to reach the end.

  7. LeetCode 300: Longest Increasing Subsequence: This problem introduces the concept of increasing subsequences, which is a crucial part of the main problem.

  8. LeetCode 322: Coin Change: This dynamic programming problem requires finding the fewest number of coins that you need to make up a certain amount.

  9. LeetCode 72: Edit Distance: This problem requires understanding the concept of dynamic programming in a string context, which could be helpful.

  10. LeetCode 279: Perfect Squares: This problem requires understanding dynamic programming to find the least number of perfect square numbers that sum to a given number.

These problems will help you gain a better understanding of dynamic programming and grid traversal, which are key to solving “2684. Maximum Number of Moves in a Grid”.

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?

Problem Classification

Problem Statement:You are given a 0-indexed m x n matrix grid consisting of positive integers.

You can start at any cell in the first column of the matrix, and traverse the grid in the following way:

From a cell (row, col), you can move to any of the cells: (row - 1, col + 1), (row, col + 1) and (row + 1, col + 1) such that the value of the cell you move to, should be strictly bigger than the value of the current cell. Return the maximum number of moves that you can perform.

Example 1:

Input: grid = [[2,4,3,5],[5,4,9,3],[3,4,2,11],[10,9,13,15]] Output: 3 Explanation: We can start at the cell (0, 0) and make the following moves:

  • (0, 0) -> (0, 1).
  • (0, 1) -> (1, 2).
  • (1, 2) -> (2, 3). It can be shown that it is the maximum number of moves that can be made.

Example 2:

Input: grid = [[3,2,4],[2,1,9],[1,1,7]] Output: 0 Explanation: Starting from any cell in the first column we cannot perform any moves.

Constraints:

m == grid.length n == grid[i].length 2 <= m, n <= 1000 4 <= m * n <= 105 1 <= grid[i][j] <= 106

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.

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 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.

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 non-technical 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 Problem-Solving 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

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. 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?

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

  1. Parameters:

    • What are the inputs to the method?
    • What types are these parameters?
    • What do these parameters represent in the context of the problem?

  2. 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?

  3. Method Functionality:

    • What is this method expected to do?
    • How does it interact with the inputs and the current state of the program?

  4. 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?

  5. 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

  1. Problem Understanding:

    • Can you explain the problem in your own words? What are the key components and requirements?

  2. Initial Breakdown:

    • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

  3. 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?

  4. Task Identification:

    • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

  5. 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?

  6. Method Naming:

    • Can you give each task a simple, descriptive name that makes its purpose clear?

  7. 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

  1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

  2. 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?

  3. 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.

  4. 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?

  5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

  6. 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 following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. 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.

  1. 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.

  2. 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.

  3. 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.

  1. 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.

  2. 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.

  3. 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 problem-solving 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.