Introduction to Intervals

About the pattern

The Interval pattern is a powerful way to reason about problems involving ranges of values, whether time spans, numeric ranges, or geometric spans. Each interval is defined by a start and an end; for example, $[10, 20]$ represents everything from $10$ through $20$.

Note: Two intervals overlap if the start of one is less than or equal to the end of the other.

This pattern is widely used in scheduling, managing resources, and timelines. By analyzing how intervals interact, we can:

• Merge overlapping intervals into one.

• Insert a new interval into the correct place.

• Find intersections between intervals.

• Detect gaps to identify free time.

• Measure coverage or resource usage.

The Intervals pattern is useful because it helps simplify complex, real-world problems. Instead of analyzing dozens of small events separately, we compress them into ranges for easier planning and decision-making.

• Eliminate redundancy: Merge overlapping events into one.

• Find efficiencies: Identify free slots to schedule new tasks.

• Simplify planning: Represent complicated timelines with clean, nonoverlapping intervals.

When working with multiple intervals, they can relate to each other in a few key ways:

• Nonoverlapping: $[1, 3]$ and $[5, 8]$ ...