leetcode-study

Below is a comprehensive study guide that explains how to recognize intervals problems and outlines the spectrum of techniques—from the most foundational to the more specialized methods—with concrete examples drawn from our collection.

1. Identifying Intervals Problems

Intervals problems are characterized by inputs that represent ranges or segments, and their objectives often involve merging, inserting, or covering these ranges. Here are key clues to help you quickly identify an intervals problem:

Input Data Structure:
- Pairs or Ranges:
  The input is typically a list (or array) of pairs, each pair representing a start and an end value. For example, the interval list in the merge-intervals problem looks like [[1,3],[2,6],[8,10],[15,18]].
- Sorted or Unsorted:
  Problems may specify that intervals are already sorted (as in insert-interval) or require you to sort them yourself.
- Single-Dimensional Ranges:
  Some problems, like summary-ranges, deal with a sorted array of numbers that must be grouped into continuous intervals.
Common Problem Objectives:
- Merging Overlaps:
  You may be asked to combine overlapping intervals into a single range—for example, merging [1,3] and [2,6] into [1,6] as seen in merge-intervals.
- Inserting and Merging:
  When a new interval is added to a set of non-overlapping intervals, you often need to merge any overlapping intervals. This is the goal of the insert-interval problem.
- Coverage and Optimization:
  Some problems model scenarios like “bursting” balloons with arrows, where each balloon is represented by an interval. The objective is to cover as many intervals as possible with each arrow, as seen in minimum-number-of-arrows-to-burst-balloons.
- Range Summarization:
  Other problems, such as summary-ranges, require you to group contiguous numbers into ranges and output them as a formatted string (either as a single number or a “start->end” format).
Verbal Clues and Keywords:
Look for words like “merge,” “insert,” “burst,” “cover,” “summarize,” or “range” in the problem statement. Real-world analogies such as scheduling meetings, covering intervals with resources, or compressing ranges are also strong signals that you’re dealing with an intervals problem.

In summary, if your problem’s input consists of pairs (or sequences) representing start and end values and the task involves combining, inserting, or summarizing these ranges, you’re most likely facing an intervals problem.

2. Most Common to Least Common Techniques and Approaches to Solving Intervals Problems

Once you’ve identified that you’re facing an intervals problem, deciding on the right strategy is the next step. Below is a list of techniques arranged from the most frequently used and foundational to those that are more specialized or multi-case in nature:

A. Sorting as a Preprocessing Step

Why It’s Essential:
Sorting brings structure to your intervals by grouping related ranges together. This makes it much easier to detect overlaps, merge intervals, or insert new ones while preserving order.
Examples and References:
- Merge Intervals:
  In the merge-intervals problem, the intervals are sorted by their start time so that as you iterate through them, overlapping intervals appear consecutively.
- Minimum Number of Arrows:
  In minimum-number-of-arrows-to-burst-balloons, the intervals (balloons) are first sorted by their end coordinate. This allows a greedy algorithm to shoot each arrow at the end coordinate, maximizing the number of balloons burst with a single arrow.
- Insert Interval:
  Because the insert-interval problem provides sorted intervals, you can efficiently partition them into segments that fall completely before, overlapping with, or after the new interval.

B. Greedy Algorithms

Why They Work:
Greedy strategies make the best local decision at each step, which is often sufficient for intervals problems because covering or merging overlapping intervals locally usually yields a global optimum.
Examples and References:
- Minimum Number of Arrows:
  After sorting by end coordinates, the greedy solution in minimum-number-of-arrows-to-burst-balloons fires an arrow at the end of the first balloon and skips over all subsequent balloons that the arrow covers.
- Merge Intervals:
  As you traverse the sorted intervals, you greedily merge overlapping intervals by extending the current interval’s end to the maximum value encountered (see merge-intervals).

C. Two-Pointer and Single-Pass Iteration

Why It’s Common:
Once intervals are sorted, often a single pass with one or two pointers is enough to process them efficiently. This technique helps you move through the list and make decisions about merging or grouping without extra passes.
Examples and References:
- Merge Intervals:
  The single-pass iteration in merge-intervals compares the current interval with the last interval in the result list to determine whether to merge.
- Summary Ranges:
  In summary-ranges, a pointer is used to traverse the sorted list of numbers to group contiguous sequences quickly.
- Insert Interval:
  A common approach in insert-interval is to traverse the list with a pointer and partition the intervals into those entirely before the new interval, those overlapping, and those after.

D. In-Place Merging and Updating

Why It’s Useful:
For problems that require minimal extra space, updating intervals in place to extend or merge them is a highly effective technique.
Examples and References:
- Merge Intervals:
  When two intervals overlap, the solution (see merge-intervals) updates the end of the current merged interval in place, avoiding the need for a temporary structure.
- Insert Interval:
  In insert-interval, overlapping intervals are merged by updating the new interval’s start and end bounds to encompass all overlapping intervals.

E. Conditional or Case-Based Multi-Phase Processing

Why It’s Sometimes Necessary:
When intervals naturally fall into distinct categories based on their relation to a target interval (or each other), dividing the problem into phases can simplify the logic and ensure that every case is handled correctly.
Examples and References:
- Insert Interval:
  The insert-interval problem is typically broken into three phases:
  1. Append intervals that end before the new interval starts.
  2. Merge intervals that overlap with the new interval.
  3. Append intervals that start after the new interval ends.
- Summary Ranges:
  In summary-ranges, the process involves one phase of grouping contiguous sequences and another phase of conditionally formatting these sequences (outputting a single number versus a range string).

Practical References

Merge Intervals (merge-intervals/solution.md)
Demonstrates the power of sorting by start time followed by a single pass to merge intervals.
Insert Interval (insert-interval/solution.md)
Exemplifies a multi-phase approach with clear segmentation: non-overlapping before, merging overlaps, and non-overlapping after.
Minimum Number of Arrows to Burst Balloons (minimum-number-of-arrows-to-burst-balloons/solution.md)
Uses sorting by end coordinate and a greedy algorithm to cover as many intervals (balloons) as possible with each arrow.
Summary Ranges (summary-ranges/solution.md)
Uses a pointer-based single pass to group contiguous numbers and then format them into the desired output format.

Summary

When you face an intervals problem, start by examining the input structure. If you see ranges or pairs that denote start and end values and operations such as merging, inserting, or covering are requested, you’re likely dealing with intervals. Most solutions begin with sorting the input as a foundational step. From there, applying greedy choices, two-pointer or single-pass iterations, and sometimes even multi-phase conditional processing will typically yield an optimal solution. Reviewing the examples and techniques from the problems—such as merge-intervals, insert-interval, minimum-number-of-arrows-to-burst-balloons, and summary-ranges—will give you a strong intuition for tackling similar challenges.

Happy coding and best of luck mastering intervals problems!