Maximize the Minimum Powered City

Problem Description

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. 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, and you need to return the maximum possible minimum power of a city, if the additional power stations are built optimally.

Key Insights

• The power of each city is influenced by the power stations in its range.
• We want to maximize the minimum power across all cities after optimally placing k additional power stations.
• A binary search approach can be applied on the minimum power value to determine feasibility.
• Greedy placement of power stations within the range can help achieve the desired minimum power.

Space and Time Complexity

Time Complexity: O(n log(max_p)), where max_p is the maximum possible power. Space Complexity: O(n), for storing the power values during calculations.

Solution

To solve this problem, we will employ a binary search strategy to find the maximum possible minimum power. The basic steps are:

1. Binary Search: We will search for the maximum possible minimum power x in the range from 0 to the maximum possible power after adding all k stations.
2. Feasibility Check: For each potential minimum power x, we will check if it is possible to ensure that all cities have at least x power by distributing the k additional power stations optimally.
3. Greedy Distribution: For each city, calculate how many additional power stations are needed to reach at least x power. Distribute the stations within the allowed range r.

This approach effectively reduces the problem space and allows us to determine the maximum minimum power efficiently.

Code Solutions