LWC 69: 774. Minimize Max Distance to Gas Station

传送门：774. Minimize Max Distance to Gas Station

Problem:

On a horizontal number line, we have gas stations at positions stations[0], stations1, …, stations[N-1], where N = stations.length.

Now, we add K more gas stations so that D, the maximum distance between adjacent gas stations, is minimized.

Return the smallest possible value of D.

Example:

Input: stations = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], K = 9
Output: 0.500000

Note:

stations.length will be an integer in range [10, 2000].

stations[i] will be an integer in range [0, 10^8].

K will be an integer in range [1, 10^6].

Answers within 10^-6 of the true value will be accepted as correct.

思路：
首先求出每个station之间的距离，考虑如下问题：两个station为[1, 9]，gap为8。要插入一个station使得最大的最小，显然插入后应该为[1, 5, 9]，最大间隔为4。举个反例，如果插入后为[1, 6, 9], [1, 3, 9]，它们的最大间隔分别为5， 6，明显不是最小。从这里可以看出，对于插入k个station使得最大的最小的唯一办法是均分。

一种贪心的做法是，找到最大的gap，插入1个station，依此类推，但很遗憾，这种贪心策略是错误的。问题的难点在于我们无法确定到底哪两个station之间需要插入station，插入几个station也无法得知。

换个思路，如果我们假设知道了答案会怎么样？因为知道了最大间隔，所以如果目前的两个station之间的gap没有符合最大间隔的约束，那么我们就必须添加新的station来让它们符合最大间隔的约束，这样一来，对于每个gap我们是能够求得需要添加station的个数。如果需求数<=K，说明我们还可以进一步减小最大间隔，直到需求数>K。

Java版本：

   public double minmaxGasDist(int[] stations, int K) {int n = stations.length;double[] gap = new double[n - 1];for (int i = 0; i < n - 1; ++i) {gap[i] = stations[i + 1] - stations[i];}double lf = 0;double rt = Integer.MAX_VALUE;double eps = 1e-7;while (Math.abs(rt - lf) > eps) {double mid = (lf + rt) /2;if (check(gap, mid, K)) {rt = mid;}else {lf = mid;}}return lf;}boolean check(double[] gap, double mid, int K) {int count = 0;for (int i = 0; i < gap.length; ++i) {count += (int)(gap[i] / mid);}return count <= K;}

Python版本：

class Solution(object):def minmaxGasDist(self, st, K):""":type stations: List[int]:type K: int:rtype: float"""lf = 1e-6rt = st[-1] - st[0]eps = 1e-7while rt - lf > eps:mid = (rt + lf) / 2cnt = 0for a, b in zip(st, st[1:]):cnt += (int)((b - a) / mid)if cnt <= K: rt = midelse: lf = midreturn rt

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LWC 69: 774. Minimize Max Distance to Gas Station

LWC 69: 774. Minimize Max Distance to Gas Station

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