Solution: Minimize Max Distance to Gas Station

Explore how to solve the problem of minimizing the largest gap between gas stations by strategically placing additional stations. Learn to apply modified binary search to efficiently narrow down the optimal maximum distance, understanding the importance of precision control. This lesson teaches key problem-solving techniques and complexity analysis relevant to binary search over continuous ranges.

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Statement
Solution
- Time complexity
- Space complexity

Statement

You are given an integer array, stations, representing the positions of existing gas stations along the x-axis. You are also given an integer k, indicating the number of additional gas stations you must add. These new gas stations can be placed at any position along the x-axis, including non-integer locations.

A penalty is the maximum distance between two adjacent gas stations after placing the k new stations. Your task is to return the smallest possible value of this penalty. An answer is correct if it is within $10^{-6}$ of the actual answer.

Constraints:

$10 \leq$ stations.length $\leq 2000$
$0 \leq$ stations[i] $\leq 10^8$
The stations array is sorted in a strictly increasing order.
$1 \leq$ k $\leq 10^{6}$ ...