Problem Description

The goal of this problem is to find the "right interval" for a given set of intervals. Given an array intervals, where each element intervals[i] represents an interval with a start_i and an end_i, we need to identify for each interval i another interval j where the interval j starts at or after the end of interval i, and among all such possible j, it starts the earliest. This means that the start of interval j (start_j) must be greater than or equal to the end of interval i (end_i), and we want the smallest such start_j. If there is no such interval j that meets these criteria, the corresponding index should be -1.

The challenge is to write a function that returns an array of indices of the right intervals for each interval. If an interval has no right interval, as mentioned -1 will be the placeholder for that interval.

Intuition

To approach this problem, we utilize a binary search strategy to efficiently find the right interval for each entry. Here's how we arrive at the solution:

1. To make it possible to return indices, we augment each interval with its original index in the intervals array.
2. We then sort the augmented intervals according to the starting points. The sort operation allows us to apply binary search later since binary search requires a sorted sequence.
3. Prepare an answer array filled with -1 to assume initially that there is no right interval for each interval.
4. For each interval, use binary search (bisect_left) to efficiently find the least starting interval j that is greater than or equal to the end of the current interval i.
5. If the binary search finds such an interval, update the answer array at index i with the index of the found interval j.
6. After completing the search for all intervals, the answer array is returned.

Implementing binary search reduces the complexity of finding the right interval from a brute-force search which would be O(n^2) to O(n log n) since each binary search operation takes O(log n) and we perform it for each of the n intervals.

Learn more about Binary Search and Sorting patterns.

Solution Approach

The solution to this problem involves the following steps, which implement a binary search algorithm:

1. Augment Intervals with Indices: First, we update each interval to include its original index. This is achieved by iterating through the intervals array and appending the index to each interval.

   for i, v in enumerate(intervals):
       v.append(i)

2. Sort Intervals by Start Times: We then sort the augmented intervals based on their start times. This allows us to leverage binary search later on since it requires the list to be sorted.

   intervals.sort()

3. Initialize Answer Array: We prepare an answer array, ans, with the same length as the intervals array and initialize all its values to -1, which indicates that initially, we assume there is no right interval for any interval.

   ans = [-1] * n

4. Binary Search for Right Intervals: For each interval in intervals, we perform a binary search to find the minimum start_j value that is greater than or equal to end_i. To do this, we use the bisect_left method from the bisect module. It returns the index at which the end_i value could be inserted to maintain the sorted order.

   for _, e, i in intervals:
       j = bisect_left(intervals, [e])

5. Updating the Answer: If the binary search returns an index less than the number of intervals (n), it means we have found a right interval. We then update the ans[i] with the original index of the identified right interval, which is stored at intervals[j][2].

   if j < n:
       ans[i] = intervals[j][2]

6. Return the Final Array: After the loop, the ans array is populated with the indices of right intervals for each interval in the given array. This array is then returned as the final answer.

   return ans

This approach efficiently uses binary search to minimize the time complexity. The key to binary search is the sorted nature of the intervals after they are augmented with their original indices. By maintaining a sorted list of start times and employing binary search, we're able to significantly reduce the number of comparisons needed to find the right interval from linear (checking each possibility one by one) to logarithmic, thus enhancing the performance of the solution.

Example Walkthrough

Let's walk through this approach with a small example. Consider the list of intervals intervals = [[1,2], [3,4], [2,3], [4,5]].

1. First, we augment each interval with its index.

   augmented_intervals = [[1, 2, 0], [3, 4, 1], [2, 3, 2], [4, 5, 3]]

2. Next, we sort the augmented intervals by their start times.

   sorted_intervals = [[1, 2, 0], [2, 3, 2], [3, 4, 1], [4, 5, 3]]

3. We initialize the answer array with all elements set to -1.

   ans = [-1, -1, -1, -1]

4. Now, using a binary search, we look for the right interval for each interval in sorted_intervals.

   For interval [1, 2, 0]:

   • The end time is 2.
   • The binary search tries to find the minimum index where 2 could be inserted to maintain the sorted order, which is index 1 (interval [2, 3, 2]).
   • Thus, the right interval index is 2.

   For interval [2, 3, 2]:

   • The end time is 3.
   • The binary search finds index 2 (interval [3, 4, 1]).
   • The right interval index is 1.

   For interval [3, 4, 1]:

   • The end time is 4.
   • The binary search finds index 3 (interval [4, 5, 3]).
   • The right interval index is 3.

   For interval [4, 5, 3]:

   • The end time is 5, and there's no interval starting after 5.
   • The binary search returns an index of 4, which is outside the array bounds.
   • There's no right interval, so the value remains -1.

5. After performing the binary search for all intervals, we update the answer array with the right intervals' indices.

   ans = [2, 1, 3, -1]

6. The ans array, which keeps track of the right interval for each interval, is returned as the final answer. In our example, this is [2, 1, 3, -1], indicating that the interval [1,2] is followed by [2,3], [3,4] follows [2,3], and [4,5] follows [3,4]. The last interval, [4,5], has no following interval that satisfies the conditions.

This approach efficiently finds the right intervals for each given interval with reduced time complexity.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code consists of two major operations: sorting the intervals list and performing binary search using bisect_left for each interval.

1. Sorting the intervals list using the sort method has a time complexity of O(n log n), where n is the number of intervals.
2. Iterating over each interval and performing a binary search using bisect_left has a time complexity of O(n log n) since the binary search operation is O(log n) and it is executed n times, once for each interval.

Thus, the combined time complexity of these operations would be O(n log n + n log n). However, since both terms have the same order of growth, the time complexity simplifies to O(n log n).

The space complexity of the code is O(n) for the following reasons:

1. The intervals list is expanded to include the original index position of each interval, but the overall space required is still linearly proportional to the number of intervals n.
2. The ans list is created to store the result for each interval, which requires O(n) space.
3. Apart from the two lists mentioned above, no additional space that scales with the size of the input is used.

Therefore, the total space complexity is O(n).

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