Maximum Score of Non-overlapping Intervals

Maximum Score of Non-overlapping Intervals

1. What is this problem about?

The Maximum Score of Non-overlapping Intervals interview question is a variation of the Weighted Interval Scheduling problem. You are given a set of intervals, each with a start time, end time, and a score. You need to select a subset of these intervals such that no two intervals overlap and the total score is maximized.

In some versions, you might be limited to selecting exactly k intervals.

2. Why is this asked in interviews?

Companies like Amazon and Sprinklr use the Maximum Score of Non-overlapping Intervals coding problem to test a candidate's ability to combine sorting, binary search, and dynamic programming. It's a fundamental problem that appears in various forms in resource allocation and scheduling. It evaluates if you can move from a greedy approach (which only works for non-weighted intervals) to a DP approach for the weighted version.

3. Algorithmic pattern used

The Array, Sorting, Binary Search, Dynamic Programming interview pattern is standard.

1. Sort all intervals by their end times.
2. Define dp[i] as the maximum score using a subset of the first i intervals.
3. For each interval i:
   • dp[i] = max(dp[i-1], score[i] + dp[latest_non_overlapping_interval]).
4. Use binary search to find the latest_non_overlapping_interval efficiently in $O(\log N)$.

4. Example explanation

Intervals:

• A: [1, 5, score 10]
• B: [2, 6, score 15]
• C: [6, 10, score 20]

Sorted by end time: A(5), B(6), C(10).

• dp[0] (A): 10.
• dp[1] (B): max(dp[0], 15 + dp[none]) = max(10, 15) = 15.
• dp[2] (C): max(dp[1], 20 + dp[A]) = max(15, 20 + 10) = 30. (Since A ends at 5 and C starts at 6, they don't overlap). Max score is 30.

5. Common mistakes candidates make

The most common mistake in the Maximum Score of Non-overlapping Intervals coding problem is not sorting by end time, which makes the subproblem definition much harder. Another error is using a linear search to find the previous interval, resulting in an $O(N^2)$ solution. Finally, failing to handle the "non-overlapping" boundary condition correctly (e.g., whether end == start is allowed) can lead to incorrect results.

6. Interview preparation tip

Master the "Sort + Binary Search + DP" template. This sequence of operations is extremely common for interval problems. Practice using your language's binary search library (like bisect in Python or lower_bound in C++) to find the right insertion point in a sorted list of end times.