Max Increase to Keep City Skyline

10 Prerequisite LeetCode Problems

“Max Increase to Keep City Skyline” has an isomorph: “Reshape the Matrix”.

In both problems, we have a matrix and need to make modifications without changing some aspect of its structure. In “Max Increase to Keep City Skyline”, we are increasing the height of buildings (elements in the matrix) without changing the skyline. In “Reshape the Matrix”, we change the shape of the matrix while maintaining the row reading order.

While these problems have a similar thematic, their specific goals and methods are different. “Max Increase to Keep City Skyline” focuses on finding maximum possible increase in the height of buildings while “Reshape the Matrix” focuses on reshaping while maintaining order of elements. “Reshape the Matrix” is simpler as it does not require finding an optimal solution.

An advanced problem would be “Walls and Gates”, where we deal with a matrix of rooms and gates, and have to update each room’s distance to its nearest gate. This involves not only updating the matrix elements, but also considering their relative positions to each other, making it more complex than both of the aforementioned problems.

“Max Increase to Keep City Skyline” is about matrix manipulation and maximum value extraction. Unfortunately, I don’t have an exact isomorphic problem for this, but I can suggest a couple of problems that share some similarities.

“Set Matrix Zeroes” is simpler and shares the concept of matrix manipulation, where you traverse through the matrix and make changes based on certain conditions.

A more complex problem is “Rotate Image”, it involves more sophisticated matrix manipulation, which requires a rotation operation.

In these problems, the focus lies on how you handle matrix operations, which is a key element in “Max Increase to Keep City Skyline”.

The problem statement for “Max Increase to Keep City Skyline” is: “In a 2D grid of size m x n, each cell is a building of certain height. The skyline viewed from top or bottom is the maximum height of each column. The skyline viewed from left or right is the maximum height of each row. Suppose you increase the height of each non-zero building to the maximum possible height without altering the skyline from any side, how much total increase in the height of the buildings can be done?”

The main steps to solve it are:

  1. First, find the maximum height in each row and each column. This gives us the skyline when viewed from left/right and top/bottom respectively.

  2. Then, traverse through each cell in the grid again. For each cell, we determine the lesser of the maximum height in its row and column. This represents the maximum height this building can be increased to without affecting the skyline.

  3. We calculate the difference between the new height and the original height, and add it to a running total.

  4. At the end, the total represents the maximum increase in height that can be achieved.

Python code:

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class Solution:
    def maxIncreaseKeepingSkyline(self, grid: List[List[int]]) -> int:
        # Find the maximum height in each row
        maxRow = [max(row) for row in grid]

        # Find the maximum height in each column
        maxCol = [max(col) for col in zip(*grid)]

        # Initialize a variable to store total increase
        total_increase = 0

        # Traverse through each cell in the grid
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                # Find the lesser of the maximum height in its row and column
                new_height = min(maxRow[i], maxCol[j])

                # Calculate the difference and add to total increase
                total_increase += new_height - grid[i][j]

        # Return the total increase
        return total_increase

This approach works because to maintain the skylines from both views, a building’s new height cannot exceed the maximum height in its row or column. Among these two maximums, we have to pick the lesser one to ensure both skylines are not affected. By adding up the increases for each building, we get the total maximum increase.

10 Prerequisite LeetCode Problems

For “Max Increase to Keep City Skyline”, the following problems are a good preparation:

  1. “121. Best Time to Buy and Sell Stock” - This problem helps you understand how to track the maximum and minimum values in an array, which is a skill that can be applied to the row and column maximums in the city skyline problem.

  2. “238. Product of Array Except Self” - This problem involves calculations with multiple arrays, similar to handling both the row and column arrays in the city skyline problem.

  3. “73. Set Matrix Zeroes” - This problem involves manipulating a 2D matrix based on conditions of its rows and columns.

  4. “695. Max Area of Island” - This problem helps you understand how to traverse a 2D matrix and calculate area, similar to calculating the skyline.

  5. “994. Rotting Oranges” - This problem deals with grid traversal and time, which can help with understanding and tracking transformations on a grid.

  6. “200. Number of Islands” - This problem also involves traversing a 2D grid and could be useful for visualizing and working with the buildings in the skyline problem.

  7. “64. Minimum Path Sum” - This problem involves calculating paths in a 2D grid, a technique that could be useful for calculating the maximum increase in the skyline problem.

  8. “48. Rotate Image” - This problem gives practice in manipulating 2D grids in place, which could be useful in the skyline problem when determining the maximum possible heights for buildings.

  9. “88. Merge Sorted Array” - While this problem is about one-dimensional arrays, it helps practice combining two different arrays into one, a technique used in the skyline problem to calculate the maximum increases.

  10. “42. Trapping Rain Water” - This problem requires an understanding of how to calculate volumes in a grid-like structure, a skill that is crucial to solving the skyline problem.

These cover traversing and manipulating 2D arrays, understanding and applying conditions to elements in a grid, and performing calculations with multiple arrays.

Problem Classification

Problem Statement:

There is a city composed of n x n blocks, where each block contains a single building shaped like a vertical square prism. You are given a 0-indexed n x n integer matrix grid where grid[r][c] represents the height of the building located in the block at row r and column c.

A city’s skyline is the outer contour formed by all the building when viewing the side of the city from a distance. The skyline from each cardinal direction north, east, south, and west may be different.

We are allowed to increase the height of any number of buildings by any amount (the amount can be different per building). The height of a 0-height building can also be increased. However, increasing the height of a building should not affect the city’s skyline from any cardinal direction.

Return the maximum total sum that the height of the buildings can be increased by without changing the city’s skyline from any cardinal direction.

Example 1:

Input: grid = [[3,0,8,4],[2,4,5,7],[9,2,6,3],[0,3,1,0]] Output: 35 Explanation: The building heights are shown in the center of the above image. The skylines when viewed from each cardinal direction are drawn in red. The grid after increasing the height of buildings without affecting skylines is: gridNew = [ [8, 4, 8, 7], [7, 4, 7, 7], [9, 4, 8, 7], [3, 3, 3, 3] ]

Example 2:

Input: grid = [[0,0,0],[0,0,0],[0,0,0]] Output: 0 Explanation: Increasing the height of any building will result in the skyline changing.

Constraints:

n == grid.length n == grid[r].length 2 <= n <= 50 0 <= grid[r][c] <= 100

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 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?

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

  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 code for the solution of this problem.

  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. The response text is of the following format:

Here are 10 problems that use similar underlying concepts: