Tallest Billboard

You can solve this problem using Dynamic Programming (DP). The goal is to find two disjoint subsets of rods that have the same sum, and that sum will be the height of the billboard.

Let’s define a DP array dp, where dp[d] represents the maximum total length we can achieve using some rods, given that one side of the steel support is d units longer than the other side.

The base case is dp[0] = 0, as we can always make both sides equal length with no rods.

Now, let’s iterate through the given rods and update the dp array. For each rod, we have three choices:

Add the rod to the longer side (the side that is d units longer).
Add the rod to the shorter side.
Do not use the rod at all.

Here’s the code implementing the logic:

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class Solution:
    def tallestBillboard(self, rods: List[int]) -> int:
        # Initialize the DP array with negative infinity except for dp[0]
        dp = {0: 0}

        # Iterate through the rods
        for rod in rods:
            # Create a temporary copy to store the updated values
            tmp = dict(dp)
            for d in dp:
                # Add the rod to the longer side
                tmp[d + rod] = max(tmp.get(d + rod, -float('inf')), dp[d])
                # Add the rod to the shorter side
                tmp[abs(d - rod)] = max(tmp.get(abs(d - rod), -float('inf')), dp[d] + min(rod, d))
            # Update the DP array
            dp = tmp

        # The maximum height of the billboard is dp[0]
        return dp[0]

The time complexity of this solution is O(N * S), where N is the number of rods, and S is the sum of the rods. The space complexity is O(S).

Identifying Problem Isomorphism

“Tallest Billboard” can be approximately mapped to “Partition Equal Subset Sum”.

In “Tallest Billboard”, you are given rods of different lengths and you need to use them to make two sets of rods such that their total lengths are equal. The aim is to maximize the total length.

In “Partition Equal Subset Sum”, you are given a set of numbers and you have to partition them into two subsets such that the sum of elements in both subsets is equal.

The reason is that both require partitioning a set into two subsets of equal sum. In “Tallest Billboard”, the rods represent the set and the total length of rods represents the sum. In “Partition Equal Subset Sum”, the set is explicitly provided and you need to find the subset partition.

“Partition Equal Subset Sum” is simpler because it is a standard subset partition problem, while “Tallest Billboard” is a more specific application with an additional requirement of maximizing the total length.

10 Prerequisite LeetCode Problems

“Tallest Billboard” is a dynamic programming problem. Here are 10 problems to prepare for this problem:

70. Climbing Stairs: A very basic introduction to the concepts behind dynamic programming.
198. House Robber: Another beginner-friendly dynamic programming problem.
322. Coin Change: This problem introduces the concept of optimizing a certain value, which is often necessary in dynamic programming problems.
300. Longest Increasing Subsequence: A step up in complexity, this problem requires dynamic programming to keep track of multiple possible answers.
416. Partition Equal Subset Sum: This problem is a good practice for working with dynamic programming problems that involve arrays.
120. Triangle: This problem introduces the concept of dynamic programming with a grid or a 2D array.
494. Target Sum: This problem has similarities to the billboard problem in that they both involve finding specific sums.
139. Word Break: This problem shows a different use of dynamic programming, where the solution involves partitioning a string rather than working with numbers.
518. Coin Change 2: As an extension of the Coin Change problem, this problem asks for the number of combinations that make up a given amount.
1043. Partition Array for Maximum Sum: This problem also involves dividing up an array and optimizing the sum, similar to the billboard problem.

Problem Classification

Problem Statement: You are installing a billboard and want it to have the largest height. The billboard will have two steel supports, one on each side. Each steel support must be an equal height.

You are given a collection of rods that can be welded together. For example, if you have rods of lengths 1, 2, and 3, you can weld them together to make a support of length 6.

Return the largest possible height of your billboard installation. If you cannot support the billboard, return 0.

Example 1:

Input: rods = [1,2,3,6] Output: 6 Explanation: We have two disjoint subsets {1,2,3} and {6}, which have the same sum = 6.

Example 2:

Input: rods = [1,2,3,4,5,6] Output: 10 Explanation: We have two disjoint subsets {2,3,5} and {4,6}, which have the same sum = 10.

Example 3:

Input: rods = [1,2] Output: 0 Explanation: The billboard cannot be supported, so we return 0.

Constraints:

1 <= rods.length <= 20 1 <= rods[i] <= 1000 sum(rods[i]) <= 5000

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.