Maximum Score of Non-overlapping Intervals

This problem involves selecting up to four non-overlapping intervals from a list of weighted intervals. The goal is to maximize the sum of the selected weights while ensuring no overlap between the intervals. Using dynamic programming and sorting techniques will help efficiently solve this problem while keeping track of the best possible score for different interval choices.

Problem Statement

You are given a list of intervals, where each interval is represented by three integers: the left boundary, right boundary, and weight. You are tasked with choosing up to four non-overlapping intervals such that their total weight is maximized.

Return the lexicographically smallest array of indices corresponding to the chosen intervals. Two intervals are considered overlapping if they share any points, including boundaries. You must make use of dynamic programming and sorting to efficiently select the optimal set of intervals.

Examples

Example 1

Input: intervals = [[1,3,2],[4,5,2],[1,5,5],[6,9,3],[6,7,1],[8,9,1]]

Output: [2,3]

You can choose the intervals with indices 2, and 3 with respective weights of 5, and 3.

Example 2

Input: intervals = [[5,8,1],[6,7,7],[4,7,3],[9,10,6],[7,8,2],[11,14,3],[3,5,5]]

Output: [1,3,5,6]

You can choose the intervals with indices 1, 3, 5, and 6 with respective weights of 7, 6, 3, and 5.

Constraints

• 1 <= intevals.length <= 5 * 104
• intervals[i].length == 3
• intervals[i] = [li, ri, weighti]
• 1 <= li <= ri <= 109
• 1 <= weighti <= 109

Solution Approach

Dynamic Programming with State Transitions

Use dynamic programming to keep track of the maximum score achievable by selecting up to 4 intervals. State transitions will depend on whether the current interval overlaps with previously selected ones. For each interval, update the possible maximum score while respecting the non-overlap condition.

Sorting Intervals for Efficient Selection

Sort the intervals by their right boundary to ensure that when choosing an interval, no overlap occurs with previously selected ones. This allows the dynamic programming approach to consider the most optimal non-overlapping intervals in sequence.

Binary Search for Faster State Transitions

To optimize the search for non-overlapping intervals, binary search can be used to quickly find the right-most non-overlapping interval that can be selected. This improves the performance by reducing unnecessary comparisons.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the sorting step, which is O(n log n), and the dynamic programming updates, which involve binary search. Overall, the complexity is O(n log n), where n is the number of intervals. The space complexity is O(n), used for storing the dynamic programming states and interval indices.

What Interviewers Usually Probe

• Focus on dynamic programming state transitions and their efficiency.
• Ensure understanding of sorting and binary search integration for this problem.
• Evaluate the candidate's ability to handle large inputs within time limits.

Common Pitfalls or Variants

Common pitfalls

• Failing to handle the overlap condition correctly, especially when intervals share boundaries.
• Using an inefficient approach without leveraging binary search or sorting, leading to poor performance.
• Not accounting for lexicographical order when multiple sets of intervals yield the same score.

Follow-up variants

• Allowing more than four intervals to be selected while maximizing the score.
• Considering a variant where intervals have different constraints on the weight values or boundaries.
• Introducing a constraint where the intervals must follow a specific order in their left and right boundaries.

FAQ

How do I approach the Maximum Score of Non-overlapping Intervals problem?

Start by sorting the intervals, then use dynamic programming to track the maximum possible score while avoiding overlap between intervals. Binary search can help optimize the transition states.

What is the key pattern to solve this problem?

This problem relies on the state transition dynamic programming pattern, where the state depends on selecting non-overlapping intervals and maximizing their weights.

What are some common mistakes when solving this problem?

Common mistakes include failing to handle boundary overlaps correctly, not using binary search for efficient state transitions, and not returning the lexicographically smallest set of intervals.

How do I ensure the correct order of intervals when the score is the same?

Ensure that your final solution returns the lexicographically smallest array of indices by considering the sorted order of the intervals when ties occur.

Can this problem be optimized further beyond dynamic programming?

The main optimization comes from using sorting and binary search in combination with dynamic programming. Beyond this, further improvements are limited unless the problem constraints change.