Trapping Rain Water

This requires understanding how water can be trapped between bars of varying heights. We’ll break down the solution into simple steps and gradually increase the complexity.

Understand the Problem: The goal is to find out how much water could be trapped between the bars of varying heights. Water can only be trapped between two bars if there is a “valley” between them. The trapped water at a given index is determined by the minimum of the maximum height to the left and the maximum height to the right, minus the height at that index.
Initialize Variables: You’ll need to track the left and right maximums for each bar. Initialize two arrays leftMax and rightMax of the same length as the input array.
Find Left Maxima: Iterate from left to right and compute the maximum height from the start to the current index.
Find Right Maxima: Iterate from right to left and compute the maximum height from the end to the current index.
Calculate Trapped Water: Iterate through the height array and for each index, find the minimum of leftMax[i] and rightMax[i], and subtract height[i] from this minimum. Add this value to the result if it’s greater than zero.

Here’s the code:

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class Solution:
    def trap(self, height: List[int]) -> int:
        n = len(height)
        leftMax = [0] * n
        rightMax = [0] * n

        # Calculate left maxima
        leftMax[0] = height[0]
        for i in range(1, n):
            leftMax[i] = max(leftMax[i-1], height[i])

        # Calculate right maxima
        rightMax[n-1] = height[n-1]
        for i in range(n-2, -1, -1):
            rightMax[i] = max(rightMax[i+1], height[i])

        # Calculate trapped water
        result = 0
        for i in range(n):
            trapped = min(leftMax[i], rightMax[i]) - height[i]
            if trapped > 0:
                result += trapped

        return result

Insights and Key Takeaways:

This algorithm iterates through the height array three times, resulting in a time complexity of O(n).
It uses extra space for the left and right maxima arrays, leading to a space complexity of O(n).
Understanding the constraints and conditions for water trapping is crucial to solving this problem.
By keeping track of the maximum heights on both sides of each bar, we can efficiently calculate the amount of water trapped.

Identifying Problem Isomorphism

“Trapping Rain Water” can be mapped approximately to “Largest Rectangle in Histogram”.

Reasoning:

Both problems revolve around determining volumes or areas considering the height/length and the width of the structures. In “Trapping Rain Water”, we are given n non-negative integers representing an elevation map where the width of each bar is 1, and we have to calculate how much rainwater can be trapped.

In “Largest Rectangle in Histogram”, we are also given an array of integers representing the heights of bars in a histogram, and the width of each bar is 1. The task here is to find the largest rectangle that can be formed in the histogram.

Here’s the approximate mapping: Each bar in the histogram in the “Largest Rectangle in Histogram” problem corresponds to an elevation in the “Trapping Rain Water” problem. The height of the bar corresponds to the height of the elevation.

However, the conditions and constraints of these two problems differ significantly. “Trapping Rain Water” is concerned with the maximum amount of water that can be trapped between elevations, whereas “Largest Rectangle in Histogram” is interested in the maximum area of a rectangle that can be formed using the histogram bars.

Therefore, the mapping is approximate because while they share similarities in the problem structure, the actual problem conditions and objectives are different.

10 Prerequisite LeetCode Problems

This requires a understanding of arrays, two-pointer technique, dynamic programming, and stack data structure.

Here are 10 problems to build the necessary skills:

“Container With Most Water” (LeetCode problem #11): This problem also involves the use of two-pointer technique, which is a fundamental part of the “Trapping Rain Water” problem.
“Two Sum” (LeetCode problem #1): This is a classic problem to understand array manipulations and two-pointer technique.
“Longest Substring Without Repeating Characters” (LeetCode problem #3): This problem involves the use of two-pointer technique and is a good practice before tackling more complex problems.
“Max Area of Island” (LeetCode problem #695): This problem gives a good grounding in understanding grid-based problems, which can be helpful in picturing the “Trapping Rain Water” problem.
“Climbing Stairs” (LeetCode problem #70): This is a basic dynamic programming problem, which helps to understand the concept of storing and reusing previous computation results.
“Best Time to Buy and Sell Stock” (LeetCode problem #121): This problem helps in understanding how to scan arrays from left to right and how to keep track of intermediate computations.
“Product of Array Except Self” (LeetCode problem #238): This problem helps to understand the technique of scanning from both sides which is also used in “Trapping Rain Water”.
“Sliding Window Maximum” (LeetCode problem #239): This problem provides a good practice of using deque (double-ended queue) data structure and understanding the concept of sliding window.
“Valid Parentheses” (LeetCode problem #20): This problem is a basic one for understanding the stack data structure, which can be applied in the “Trapping Rain Water” problem.
“Largest Rectangle in Histogram” (LeetCode problem #84): This problem is similar to “Trapping Rain Water” in the sense that it requires calculating areas using heights. It also utilizes stack to keep track of the heights.

After practicing with these problems, one should have a solid foundation to tackle the “Trapping Rain Water” problem.

LeetCode 755, “Pour Water”, can be a good preparation problem before tackling “Trapping Rain Water”.

“Pour Water” is less complex than “Trapping Rain Water”. It still deals with the concept of water filling up spaces between elevations (similar to bars in the histogram), but it does not require the two-pointer technique or the use of stacks to solve.

In this problem, you simulate the process of pouring water on top of some bars and see how the water flows downwards to the left or the right, filling up the lowest points. This understanding of how water fills up spaces and how to navigate an array to find these spaces can be directly useful for understanding and solving “Trapping Rain Water”.

So yes, while it’s not necessary to solve “Pour Water” before “Trapping Rain Water”, doing so can help you grasp the underlying concepts better and improve your problem-solving skills.

Problem Classification

Problem Statement: Given n non-negative integers representing an elevation map where the width of each bar is 1, compute how much water it can trap after raining.

Input: height = [0,1,0,2,1,0,1,3,2,1,2,1] Output: 6 Explanation: The above elevation map (black section) is represented by array [0,1,0,2,1,0,1,3,2,1,2,1]. In this case, 6 units of rain water (blue section) are being trapped. Example 2:

Input: height = [4,2,0,3,2,5] Output: 9

Constraints:

n == height.length 1 <= n <= 2 * 104 0 <= height[i] <= 105

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?

Alternatively, if you’re working on a specific problem, you might ask something like:

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.

Q&A

Problem Classification

This problem can be classified as a Spatial Problem based on the problem domain. This category is chosen due to the following reasons:

The problem involves dealing with heights (elevations) and widths (distances between the elevations), which are spatial elements.
It requires an understanding of how water would be trapped between different elevations, a concept which requires an understanding of three-dimensional space.
The problem is about calculating the volume of water that can be trapped, which is a spatial concept.

Keep in mind that while this problem primarily fits into the Spatial Problems category, there might be elements of other categories involved as well, such as Sequential Problems (as we’re dealing with an array of heights) or Comparative Problems (as we’re comparing the heights of the bars to determine the amount of water that can be trapped). However, we’re focusing only on the most prominent domain, which is spatial in this case.

Language Agnostic Coding Drills

This code is a solution to the “Trapping Rain Water” problem, where the goal is to compute the maximum amount of water that can be trapped between the heights represented in the input array, height.

To break down the code into the smallest units of learning:

Understanding basic data types: integers, lists.
Understanding basic arithmetic operations like addition, subtraction, comparison.
Understanding conditionals and loops: if-else statements, while loop.
Understanding the len function to find the length of a list.
Understanding how to define a function and class in Python.
Understanding the use of return statement in a function.
Understanding array/list indexing and index-based operations.
Problem-specific: Maintaining two pointers in an array (left and right).
Problem-specific: Understanding and implementing the two-pointer technique to find the maximum trapped water.
Problem-specific: Updating a running total (ans in this code) based on certain conditions.

The problem-solving approach:

Initialize two pointers, i and j, to the start and end of the array respectively. Also, initialize lmax and rmax to the heights at these respective positions. lmax and rmax keep track of the maximum height of the bar from the start to the current index and from the end to the current index respectively.
Iterate the array from both ends using a while loop until i is not less than j.
Inside the loop, update lmax if the current height at i is more than lmax, and similarly, update rmax if the current height at j is more than rmax.
If lmax is less than or equal to rmax, add the difference between lmax and the height at i to ans (the total trapped water), and increment i. This is because if lmax is less than or equal to rmax, then lmax is the maximum height of the bar that can trap water at i.
If lmax is more than rmax, add the difference between rmax and the height at j to ans, and decrement j. This is because if lmax is more than rmax, then rmax is the maximum height of the bar that can trap water at j.
Continue this process until i and j meet, then return ans as the total trapped water.

Targeted Drills in Python

1. Understanding basic data types: integers, lists

Drill: Define an integer and a list, and print them.

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num = 10
lst = [1, 2, 3, 4, 5]

print("Integer: ", num)
print("List: ", lst)

2. Understanding basic arithmetic operations like addition, subtraction, comparison

Drill: Perform basic arithmetic operations.

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a = 6
b = 2

print("Addition: ", a + b)
print("Subtraction: ", a - b)
print("Is a > b?: ", a > b)

3. Understanding conditionals and loops: if-else statements, while loop

Drill: Write a simple while loop that prints numbers 1 to 5, and an if-else statement that checks if a number is even or odd.

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i = 1
while i <= 5:
    print(i)
    i += 1

num = 4
if num % 2 == 0:
    print(num, "is even.")
else:
    print(num, "is odd.")

4. Understanding the len function to find the length of a list

Drill: Create a list and use the len function to print its length.

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lst = [1, 2, 3, 4, 5]
print("Length of the list: ", len(lst))

5. Understanding how to define a function and class in Python

Drill: Define a simple class with a method that squares a number.

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class Squarer:
    def square(self, num):
        return num ** 2

s = Squarer()
print(s.square(5))

6. Understanding the use of return statement in a function

Drill: Write a function that returns the sum of two numbers.

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def add(a, b):
    return a + b

print(add(3, 7))

7. Understanding array/list indexing and index-based operations

Drill: Create a list and perform different indexing operations.

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lst = [1, 2, 3, 4, 5]
print("First element: ", lst[0])
print("Last element: ", lst[-1])
print("First 3 elements: ", lst[:3])

8. Problem-specific: Maintaining two pointers in an array

Drill: Implement a function to find the maximum and minimum elements in an array using the two pointer approach.

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def find_max_min(arr):
    start = 0
    end = len(arr) - 1
    max_val, min_val = float('-inf'), float('inf')
    
    while start <= end:
        max_val = max(max_val, arr[start], arr[end])
        min_val = min(min_val, arr[start], arr[end])
        start += 1
        end -= 1
    
    return max_val, min_val

print(find_max_min([1, 5, 3, 7, 4, 9, 2]))

9. Problem-specific: Implementing the two-pointer technique to find the maximum trapped water

Drill: Implement the function based on the problem-solving approach given.

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def trap(height):
    if len(height)<= 2:
        return 0

    ans = 0
    i = 1
    j = len(height) - 1
    lmax = height[0]
    rmax = height[-1]

    while i <=j:
        if height[i] > lmax:
            lmax = height[i]
        if height[j] > rmax:
            rmax = height[j]

        if lmax <= rmax:
            ans += lmax - height[i]
            i += 1

        else:
            ans += rmax - height[j]
            j -= 1

    return ans

print(trap([0,1,0,2,1,0,1,3,2,1,2,1]))

Each of these drills target a specific concept or technique required to implement the final solution. By practicing them separately, one can gradually build up the necessary skills and understanding to tackle the entire problem.

Trapping Rain Water

Identifying Problem Isomorphism

10 Prerequisite LeetCode Problems

Problem Classification

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Identification of Applicable Theoretical Concepts

Problem Breakdown and Solution Methodology

Inference of Problem-Solving Approach from the Problem Statement

Stepwise Refinement

Solution Approach and Analysis

Thought Process

From Brute Force to Optimal Solution

Coding Constructs

Language Agnostic Coding Drills

Targeted Drills in Python

Q&A

Similar Problems

Problem Classification

Language Agnostic Coding Drills

Targeted Drills in Python