October 21, 2024
Problem Solving
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Trapping Rain Water: A Deep Dive into an Essential Coding Interview Problem


When it comes to coding interviews, especially those at prestigious tech companies like FAANG (Facebook, Amazon, Apple, Netflix, Google), certain problems have gained notoriety for their frequency and complexity. One such problem is the “Trapping Rain Water” question. This problem not only tests a candidate’s ability to think algorithmically but also challenges their problem-solving skills and understanding of data structures.

In this comprehensive guide, we’ll explore the Trapping Rain Water problem in depth, discussing its importance, various approaches to solve it, and how mastering this problem can help you in your coding journey and interview preparation.

Understanding the Trapping Rain Water Problem

Before we dive into the solutions, let’s first understand what the Trapping Rain Water problem is all about.

Problem Statement

Given an array of non-negative integers representing the heights of walls, where each integer represents the height of a wall, calculate how much water can be trapped between the walls after it rains.

For example, given the input array [0,1,0,2,1,0,1,3,2,1,2,1], the answer would be 6 units of water.

Visual Representation

To better understand the problem, let’s visualize it:


     |
 |   |
 | | |
_|_|_|_|_|_|_
0 1 0 2 1 0 1 3 2 1 2 1

In this representation, the ‘_’ symbols represent the ground, and the ‘|’ symbols represent the walls. The spaces between the walls where water can be trapped are left blank.

Why is the Trapping Rain Water Problem Important?

The Trapping Rain Water problem is significant for several reasons:

  1. Frequency in Interviews: This problem frequently appears in coding interviews, especially at top tech companies.
  2. Multiple Approaches: It can be solved using various techniques, allowing interviewers to assess a candidate’s problem-solving versatility.
  3. Algorithmic Thinking: It requires strong algorithmic thinking and the ability to optimize solutions.
  4. Data Structure Knowledge: Different solutions utilize various data structures, testing a candidate’s understanding of their applications.
  5. Time and Space Complexity Analysis: It provides an excellent opportunity to discuss trade-offs between time and space complexity.

Approaches to Solve the Trapping Rain Water Problem

Let’s explore different approaches to solve this problem, starting from the most intuitive to the most optimized solutions.

1. Brute Force Approach

The brute force approach is the most straightforward way to solve this problem. Here’s how it works:

  1. For each element in the array:
  2. Find the maximum height to its left
  3. Find the maximum height to its right
  4. The water trapped above this element is the minimum of these two heights minus the height of the current element
  5. Sum up the water trapped above each element

Here’s the implementation in Python:

def trap(height):
    if not height:
        return 0
    
    n = len(height)
    total_water = 0
    
    for i in range(1, n - 1):
        left_max = max(height[:i])
        right_max = max(height[i+1:])
        
        water = min(left_max, right_max) - height[i]
        if water > 0:
            total_water += water
    
    return total_water

Time Complexity: O(n^2), where n is the length of the input array.
Space Complexity: O(1), as we’re not using any extra space that scales with input size.

While this approach is intuitive and easy to understand, it’s not efficient for large inputs due to its quadratic time complexity.

2. Dynamic Programming Approach

We can optimize the brute force approach using dynamic programming. Instead of calculating the left and right max for each element every time, we can precompute these values:

def trap(height):
    if not height:
        return 0
    
    n = len(height)
    left_max = [0] * n
    right_max = [0] * n
    
    left_max[0] = height[0]
    for i in range(1, n):
        left_max[i] = max(left_max[i-1], height[i])
    
    right_max[n-1] = height[n-1]
    for i in range(n-2, -1, -1):
        right_max[i] = max(right_max[i+1], height[i])
    
    total_water = 0
    for i in range(n):
        water = min(left_max[i], right_max[i]) - height[i]
        if water > 0:
            total_water += water
    
    return total_water

Time Complexity: O(n), where n is the length of the input array.
Space Complexity: O(n), as we’re using two additional arrays of size n.

This approach significantly improves the time complexity at the cost of some additional space.

3. Two Pointer Approach

We can further optimize the solution using a two-pointer approach, which allows us to solve the problem in a single pass and with constant extra space:

def trap(height):
    if not height:
        return 0
    
    left, right = 0, len(height) - 1
    left_max = right_max = total_water = 0
    
    while left < right:
        if height[left] < height[right]:
            if height[left] >= left_max:
                left_max = height[left]
            else:
                total_water += left_max - height[left]
            left += 1
        else:
            if height[right] >= right_max:
                right_max = height[right]
            else:
                total_water += right_max - height[right]
            right -= 1
    
    return total_water

Time Complexity: O(n), where n is the length of the input array.
Space Complexity: O(1), as we’re only using a constant amount of extra space.

This approach is the most efficient in terms of both time and space complexity.

Understanding the Two Pointer Approach

The two-pointer approach might seem a bit tricky at first, so let’s break it down:

  1. We start with two pointers, one at the beginning (left) and one at the end (right) of the array.
  2. We also keep track of the maximum height seen from the left and right.
  3. At each step, we compare the heights at the left and right pointers:
  4. If the left height is smaller, we know that the amount of water that can be trapped at this position is determined by the left_max (because there’s a taller wall to the right).
  5. If the right height is smaller, we know that the amount of water that can be trapped at this position is determined by the right_max (because there’s a taller wall to the left).
  6. We add the trapped water (if any) to our total and move the pointer inward.
  7. We repeat this process until the pointers meet.

This approach works because at each step, we’re guaranteed that there’s a wall at least as tall as min(left_max, right_max) on the other side, allowing us to calculate the trapped water without needing to know the exact height of that wall.

Common Variations and Follow-up Questions

Interviewers often like to add twists to this problem or ask follow-up questions. Here are some common variations:

1. 2D Trapping Rain Water

In this variation, instead of a 1D array of heights, you’re given a 2D elevation map. The task is to calculate how much water can be trapped after it rains.

This problem is significantly more complex and usually requires a different approach, such as using a priority queue or applying the concept of “pouring water”.

2. Trapping Rain Water II

This is essentially the same as the 2D variation. It’s a good idea to be familiar with both names as they refer to the same problem.

3. Container With Most Water

While not exactly the same, this problem is related and often confused with Trapping Rain Water. In this problem, you need to find two lines that together with the x-axis forms a container that would hold the most water.

4. Largest Rectangle in Histogram

This problem shares some similarities in terms of the thinking process required. Understanding one can often help with understanding the other.

Tips for Solving the Trapping Rain Water Problem in Interviews

When faced with this problem in an interview setting, consider the following tips:

  1. Start with the Brute Force Approach: Even if you know the optimized solution, starting with the brute force approach shows your ability to break down the problem and come up with a working solution.
  2. Explain Your Thought Process: As you work through the problem, explain your thinking. This gives the interviewer insight into your problem-solving approach.
  3. Analyze Time and Space Complexity: For each approach you discuss, be prepared to analyze its time and space complexity.
  4. Consider Edge Cases: Discuss how your solution handles edge cases, such as an empty input array or an array with all equal heights.
  5. Optimize Incrementally: If you start with the brute force approach, discuss how you can optimize it step by step. This shows your ability to improve upon a solution.
  6. Code Clearly and Efficiently: When implementing your solution, write clean, well-commented code. This demonstrates your coding skills beyond just problem-solving.

How Mastering This Problem Helps in Your Coding Journey

Understanding and being able to solve the Trapping Rain Water problem is beneficial for several reasons:

  1. Problem-Solving Skills: This problem requires you to think creatively and approach the solution from different angles, enhancing your overall problem-solving abilities.
  2. Algorithmic Thinking: The various solutions to this problem showcase different algorithmic techniques, from brute force to dynamic programming to two-pointer approaches.
  3. Optimization Techniques: Learning how to optimize from a brute force solution to a more efficient one is a valuable skill applicable to many coding problems.
  4. Data Structure Knowledge: The different approaches utilize various data structures, reinforcing your understanding of when and how to use them effectively.
  5. Interview Preparation: Given its popularity in coding interviews, mastering this problem can boost your confidence and performance in technical interviews.
  6. Code Optimization: Understanding the trade-offs between different solutions in terms of time and space complexity is crucial for writing efficient code in real-world scenarios.

Conclusion

The Trapping Rain Water problem is a classic example of the type of algorithmic challenges you might face in coding interviews, especially at top tech companies. By understanding this problem deeply – from the brute force approach to the optimized two-pointer solution – you’re not just preparing for a specific interview question. You’re developing crucial skills in problem-solving, algorithmic thinking, and code optimization that will serve you well throughout your coding career.

Remember, the key to mastering such problems is practice and understanding. Don’t just memorize the solution; understand the logic behind each approach, the trade-offs involved, and how you can apply similar thinking to other problems. With this knowledge and practice, you’ll be well-equipped to tackle not just the Trapping Rain Water problem, but a wide range of coding challenges in your interviews and beyond.

Happy coding, and may your journey in mastering algorithms and data structures be as rewarding as it is challenging!