Solution: Minimize Max Distance to Gas Station

Explore how to apply modified binary search to minimize the largest distance between gas stations by optimally placing additional stations along an axis. Understand the use of precision-based binary search over a continuous range and the method to verify feasibility of station placement, enabling you to solve complex gap minimization problems efficiently.

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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 $\leq 10^{6}$ ...