Non-overlapping Intervals
Given intervals [start, end], return the minimum number of intervals to remove to make the rest non-overlapping.
Input: intervals as start:end comma-separated.
Example:
1:2,2:3,3:4,1:3
1
- The given intervals are
[1,2],[2,3],[3,4], and[1,3], which can be sorted by their end values as[1,2],[2,3],[3,4], and[1,3]. - After sorting, we can see that
[1,3]overlaps with[1,2]and[2,3], but if we remove[1,3], the remaining intervals are non-overlapping. - The number of intervals to remove is 1, as removing
[1,3]makes the rest non-overlapping. - The final output is 1, which is the minimum number of intervals to remove.
Constraints:
- 1 <= len(intervals) <= 10^5
- -5 * 10^4 <= start < end <= 5 * 10^4
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Background Knowledge
The "Non-overlapping Intervals" problem falls under the category of interval scheduling problems, which are a classic topic in computer science and operations research. Interval scheduling involves scheduling a set of tasks or events, each represented by a start and end time, to minimize conflicts or overlaps. In this problem, we're given a set of intervals and need to find the minimum number of intervals to remove to make the rest non-overlapping. This requires understanding the concept of overlapping intervals and how to efficiently compare and select intervals to remove.
The key concept here is to recognize that two intervals overlap if the start time of one interval is less than the end time of the other. This can be represented mathematically as start1​<end2​ and start2​<end1​. To solve this problem, we'll need to use a greedy algorithm or a dynamic programming approach, both of which are common techniques for solving interval scheduling problems. Greedy algorithms make locally optimal choices at each step, hoping to find a global optimum solution, while dynamic programming breaks down the problem into smaller sub-problems and solves each sub-problem only once.
Understanding the time complexity and space complexity of the algorithm is also crucial. The time complexity refers to the amount of time the algorithm takes to complete, usually measured in terms of the input size. The space complexity refers to the amount of memory the algorithm uses. For interval scheduling problems, the time complexity can range from O(n) to O(n2), depending on the approach used, where n is the number of intervals. The space complexity is usually O(n), as we need to store the intervals in memory.
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