Maximize the Minimum Powered City

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class Solution:
    def maxPower(self, stations: List[int], r: int, k: int) -> int:
        n = len(stations)

        def isGood(minPowerRequired, additionalStations):
            windowPower = sum(stations[:r])  # init windowPower to store power of 0th city (minus stations[r])
            additions = [0] * n
            for i in range(n):
                if i + r < n:  # now, windowPower stores sum of power stations from [i-r..i+r], it also means it's the power of city `ith`
                    windowPower += stations[i + r]

                if windowPower < minPowerRequired:
                    needed = minPowerRequired - windowPower
                    if needed > additionalStations:  # Not enough additional stations to plant
                        return False
                    # Plant the additional stations on the farthest city in the range to cover as many cities as possible
                    additions[min(n - 1, i + r)] += needed
                    windowPower = minPowerRequired
                    additionalStations -= needed

                if i - r >= 0:  # out of window range
                    windowPower -= stations[i - r] + additions[i - r]

            return True

        left = 0
        right = sum(stations) + k  # The answer = `right`, when `r = n`, all value of stations are the same!
        ans = 0
        while left <= right:
            mid = (left + right) // 2
            if isGood(mid, k):
                ans = mid  # This is the maximum possible minimum power so far
                left = mid + 1  # Search for a larger value in the right side
            else:
                right = mid - 1  # Decrease minPowerRequired to need fewer additional power stations
        return ans

Identifying Problem Isomorphism

“Maximize the Minimum Powered City” can be seen as an application of the “Minimum Spanning Tree” problem.

Reasoning: Both problems involve the concept of connecting different nodes (cities) in a network (graph) to achieve some optimization goal. In the “Minimum Spanning Tree” problem, the aim is to connect all nodes in a graph such that the total weight of the edges is minimized. Similarly, in “Maximize the Minimum Powered City”, the aim is to connect all cities while maximizing the minimum power of any city.

“Minimum Spanning Tree” is simpler as it involves finding a tree that connects all nodes with the minimum total weight. “Maximize the Minimum Powered City” has an additional constraint of maximizing the minimum power, which makes it more complex. Also, the nature of the problem implies that a variation of the algorithm used to solve the “Minimum Spanning Tree” problem would be required, possibly something similar to Kruskal’s or Prim’s algorithm but with a modification to account for the additional constraint.

10 Prerequisite LeetCode Problems

“2528. Maximize the Minimum Powered City” is about graph theory, spanning trees, and binary search. Here are some simpler problems to prepare for it:

Prim’s Minimum Spanning Tree (MST): This problem introduces the concept of Prim’s algorithm which is used to find a minimum spanning tree for a weighted undirected graph.
1135. Connecting Cities With Minimum Cost: Similar to Prim’s MST, this problem uses Kruskal’s algorithm to find the minimum cost to connect all points.
743. Network Delay Time: This problem requires understanding of Dijkstra’s algorithm, a common algorithm for finding shortest paths in a weighted graph.
1202. Smallest String With Swaps: This problem is about union-find and can be helpful to solve problems that involve connecting components in a graph.
787. Cheapest Flights Within K Stops: It’s about finding shortest paths in a weighted graph with an additional condition.
378. Kth Smallest Element in a Sorted Matrix: This problem will help with understanding binary search in the context of multidimensional arrays, which is a concept used in the printer problem.
1631. Path With Minimum Effort: This problem is about finding the minimum total effort required to travel from the top-left cell to the bottom-right cell of a matrix.
778. Swim in Rising Water: This problem requires you to use binary search and Depth-First Search to find the minimum possible maximum value over a path in a grid.
721. Accounts Merge: This problem helps in understanding how to construct and traverse graphs and solve problems that involve connecting components in a graph.
200. Number of Islands: This problem helps you understand how to traverse a grid and identify connected components.

These cover minimum spanning trees, binary search, and handling weighted graphs.

Problem Classification

Problem Statement:You are given a 0-indexed integer array stations of length n, where stations[i] represents the number of power stations in the ith city. Each power station can provide power to every city in a fixed range. In other words, if the range is denoted by r, then a power station at city i can provide power to all cities j such that |i - j| <= r and 0 <= i, j <= n - 1. Note that |x| denotes absolute value. For example, |7 - 5| = 2 and |3 - 10| = 7. The power of a city is the total number of power stations it is being provided power from. The government has sanctioned building k more power stations, each of which can be built in any city, and have the same range as the pre-existing ones. Given the two integers r and k, return the maximum possible minimum power of a city, if the additional power stations are built optimally. Note that you can build the k power stations in multiple cities.

Example 1:

Input: stations = [1,2,4,5,0], r = 1, k = 2 Output: 5 Explanation: One of the optimal ways is to install both the power stations at city 1. So stations will become [1,4,4,5,0].

City 0 is provided by 1 + 4 = 5 power stations.
City 1 is provided by 1 + 4 + 4 = 9 power stations.
City 2 is provided by 4 + 4 + 5 = 13 power stations.
City 3 is provided by 5 + 4 = 9 power stations.
City 4 is provided by 5 + 0 = 5 power stations. So the minimum power of a city is 5. Since it is not possible to obtain a larger power, we return 5.

Example 2:

Input: stations = [4,4,4,4], r = 0, k = 3 Output: 4 Explanation: It can be proved that we cannot make the minimum power of a city greater than 4.

Constraints:

n == stations.length 1 <= n <= 105 0 <= stations[i] <= 105 0 <= r <= n - 1 0 <= k <= 109

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.