Problem Description

The problem asks us to find the minimum number of intervals that need to be removed from a given list of time intervals to ensure that no two intervals overlap with each other. Intervals overlap when one interval starts before the other one ends. For example, if one interval is [1, 2] and another is [2, 3], they do not overlap, but if the second interval is [1, 3], they do overlap because the first interval has not ended before the second one starts.

Our input is an array of intervals, where each interval is represented as a list with two elements, signifying the start and the end times. The output should be an integer that represents the minimum number of intervals that must be removed to eliminate all overlaps.

Intuition

The key intuition behind the solution lies in the greedy algorithm approach. A greedy algorithm makes the locally optimal choice at each stage with the hope of finding the global optimum. In the context of this problem, the greedy choice would be to always pick the interval that ends the earliest, because this would leave the most room for subsequent intervals.

Here is the thinking process for arriving at the solution:

1. Sort the intervals based on their end times. This way, we encounter the intervals that finish earliest first and can thus make the greedy choice.
2. Start with the first interval, considering it as non-overlapping by default, and make a note of its end time.
3. Iterate through the subsequent intervals:
   • If the start time of the current interval is not less than the end time of the last non-overlapping interval, it means this interval does not overlap with the previously considered intervals. We can then update our last known end time to be the end time of the current interval.
   • If the start time is less than the last known end time, an overlap occurs, and we must choose to remove an interval. Following the greedy approach, we keep the interval with the earlier end time and remove the other by incrementing our answer (the number of intervals to remove).

By following these steps and always choosing the interval that finishes earliest, we ensure that we take up the least possible space on the timeline for each interval, and therefore maximize the number of intervals we can include without overlapping.

Learn more about Greedy, Dynamic Programming and Sorting patterns.

Solution Approach

The provided solution follows the greedy strategy mentioned in the reference solution approach. Let's discuss how the solution is implemented in more detail:

1. Sorting Intervals: The input list of intervals is sorted based on the end times of the intervals. This is done using Python's sort function with a lambda function as the key that retrieves the end time x[1] from each interval x.

   intervals.sort(key=lambda x: x[1])

   Sorting the intervals by their end times allows us to apply the greedy algorithm effectively.

2. Initializing Variables: Two variables are initialized:

   • ans: set to 0, it keeps count of the number of intervals we need to remove.
   • t: set to the end time of the first interval, it represents the latest end time of the last interval that we decided to keep.

3. Iterating Over Intervals: The code iterates through the rest of the intervals starting from the second interval (since the first interval's end time is stored in t).

   for s, e in intervals[1:]:

   Each interval [s, e] consists of a start time s and an end time e.

4. Checking for Overlapping: In each iteration, the code checks if the current interval's start time s is greater than or equal to the variable t:

   • If s >= t, there is no overlap with the last chosen interval, so the current interval can be kept. We then update t to the current interval's end time e.

   • If s < t, there is an overlap with the last chosen interval, so the current interval needs to be removed, and we increment ans by 1.

   if s >= t:
       t = e
   else:
       ans += 1

5. Returning the Result: After the loop finishes, ans holds the minimum number of intervals that need to be removed to make all remaining intervals non-overlapping, so it is returned as the final result.

   return ans

By following the greedy approach, the algorithm ensures that the intervals with the earliest end times are considered first, minimizing potential overlap with future intervals and thus minimizing the number of intervals that need to be removed.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we are given the following list of intervals:

intervals = [[1, 2], [2, 3], [3, 4], [1, 3]]

We want to remove the minimum number of intervals so that no two intervals overlap. Here is how we apply the solution approach:

1. Sorting Intervals: First, we sort the intervals by their end times:

   sorted_intervals = [[1, 2], [2, 3], [1, 3], [3, 4]]

2. Initializing Variables: We initialize ans = 0 and t = 2, where t is the end time of the first interval after sorting.

3. Iterating Over Intervals: We iterate through the intervals starting from the second one.

4. Checking for Overlapping:

   • Comparing the 2nd interval [2, 3] with t: since 2 >= 2, we update t to 3, and no interval is removed.
   • Comparing the 3rd interval [1, 3] with t: since 1 < 3, this interval overlaps with the previously chosen intervals. We increment ans to 1.
   • Comparing the 4th interval [3, 4] with t: since 3 >= 3, we update t to 4, and no interval is removed.

After checking all intervals, the number of intervals to be removed is ans = 1. This is because the interval [1, 3] was overlapping, and by removing it, we ensured that no intervals overlap.

The output of our algorithm for this example would be 1, indicating we need to remove a single interval to eliminate all overlaps.

Solution Implementation

Time and Space Complexity

The given Python code aims to find the minimum number of intervals that can be removed from a set of intervals so that none of them overlap. Here's an analysis of its time complexity and space complexity:

Time Complexity

The time complexity of the code is primarily determined by the sorting operation and the single pass through the sorted interval list:

1. intervals.sort(key=lambda x: x[1]) sorts the intervals based on the end time. Sorting in Python uses the Timsort algorithm, which has a time complexity of O(n log n), where n is the number of intervals.
2. The for loop iterates through the list of intervals once. The loop runs in O(n) as each interval is visited exactly once.

Combining these steps, the overall time complexity is dominated by the sorting step, leading to a total time complexity of O(n log n).

Space Complexity

The space complexity of the code is mainly the space required to hold the input and the variables used for the function:

1. In-place sorting of the intervals doesn't require additional space proportional to the input size, so the space complexity of the sorting operation is O(1).
2. The only extra variables that are used include ans, and t. These are constant-size variables and do not scale with the input size.

Hence, the overall space complexity of the code is O(1), indicating it requires constant additional space.

Learn more about how to find time and space complexity quickly using problem constraints.