Solution Explanation

This problem asks to find the minimum maximum distance between adjacent gas stations after adding k new stations. The key insight is to use binary search.

Core Idea:

The solution uses binary search to find the minimum maximum distance. We define a check function that determines if a given maximum distance x is feasible, meaning we can add at most k stations to achieve this distance.

Binary Search:

The binary search space is from 0 (minimum possible distance if we place stations everywhere) to the maximum distance between any two adjacent stations in the input. The check function helps us determine if a given mid value (the midpoint in the binary search) is a feasible maximum distance. If it is, we try a smaller distance; otherwise, we try a larger distance.

The check function:

For a given maximum distance x, the check function calculates how many new stations are needed to ensure the distance between any two adjacent stations is at most x. It iterates through pairs of adjacent existing stations and computes how many new stations must be added between them. If the total number of added stations needed is less than or equal to k, then x is a feasible maximum distance.

Time Complexity Analysis:

• Binary Search: The binary search iterates log(R - L) times, where R is the upper bound (maximum distance between stations) and L is the lower bound (0). The log is base 2.

• check Function: The check function iterates through the stations array once, which takes O(n) time, where n is the number of gas stations.

• Overall: The overall time complexity is dominated by the binary search and the check function calls, making it O(n * log(R - L)). Since R - L is not very large compared to n in many cases, we can consider it approximately O(n log n) in practice. But the formal complexity is O(n log R), where R is the maximum distance between two consecutive stations.

Space Complexity Analysis:

The space complexity is O(1) because the algorithm uses only a constant amount of extra space regardless of input size. The check function doesn't use any significant extra space.

Code Explanation (Python3):

class Solution:
    def minmaxGasDist(self, stations: List[int], k: int) -> float:
        def check(x):
            return sum(int((b - a) / x) for a, b in pairwise(stations)) <= k
 
        left, right = 0, max(stations[i+1] - stations[i] for i in range(len(stations)-1))
        while right - left > 1e-6: #1e-6 is the tolerance for precision
            mid = (left + right) / 2
            if check(mid):
                right = mid
            else:
                left = mid
        return left
 
from itertools import pairwise

The pairwise function from itertools is used to efficiently iterate through pairs of adjacent elements in stations. The rest of the code directly implements the binary search and the check function as described above. Note the use of 1e-6 for the termination condition to accommodate for floating-point precision issues. Other language implementations are very similar in structure, differing mostly in syntax.