给定一组区间，如何找到它们之间的最大交点数， [英] Given a set of intervals, how to find the maximum number of intersections among them,

问题描述

假设您有一组区间 (1,5)、(6,10)、(3,8)、(7,9).我期望的输出是 3，因为最多有 3 个间隔 (3,8)、(6,10) 和 (7,9) 彼此相交.请注意，(1,5) 和 (3,8) 也彼此相交，但这只是其中的 2 个.所以这里的答案将是 3，因为 3 是彼此相交的最大间隔数.

Let's say you are given a set of intervals (1,5), (6,10), (3,8), (7,9). The output I expect is 3 since there are maximum 3 intervals (3,8), (6,10) and (7,9) that intersect with each other. Note that, (1,5) and (3,8) also intersect with each other but that's only 2 of them. So the answer here is going to be 3, since 3 is the maximum number of intervals that intersect with each other.

有哪些有效的方法可以找到它?我想第一步是根据起始值对间隔进行排序.之后有什么建议吗?

What are some efficient ways of finding it? I guess the first step would be to sort the intervals according to the starting value. Any suggestions after that?

推荐答案

标准的解决方案是将区间处理成一组(开始，结束)点.例如 (1,3) 生成 {1, begin}, {3, end}.然后对点进行排序并从左到右扫描，将 begin 计为 +1，将 end 计为 -1.计数器达到的最大值为最大重叠间隔数.

The standard solution is to process the intervals into a set of (begin,end) points. For example (1,3) generates {1, begin}, {3, end}. Then sort the points and scan left to right, counting begin as +1, end as -1. The max value reached by the counter is the maximum number of overlapping intervals.

这是从问题中的示例生成的中间数组:

This is the intermediate array generated from the example in the question:

[(1, 'begin'),
 (3, 'begin'),
 (5, 'end'),
 (6, 'begin'),
 (7, 'begin'),  # <--- counter reaches 3, its maximum value here.
 (8, 'end'),
 (9, 'end'), (10, 'end')]

这里有一个小问题.(1,end) 在 (1,begin) 之前还是之后?如果您将间隔视为开放，那么它应该在前面 - 这样 (0,1) &(1,2) 不会算作相交.否则它应该在后面，这些间隔将被视为相交间隔.

There is a minor tricky point here. Does (1,end) go before or after (1,begin)? If you treat intervals as open, then it should go before - this way (0,1) & (1,2) won't be counted as intersecting. Otherwise it should go after and these intervals will count as intersecting ones.

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