To understand the concept of merging intervals, let's break it down from first principles, starting with what intervals are and then moving on to how and why we merge them.

What are Intervals?

An interval is a range between two endpoints, often representing a continuous set of values. In most contexts, especially in computer science and mathematics, an interval is defined by its starting point and its ending point. For example, the interval [1, 5] represents all numbers from 1 to 5, inclusive of 1 and 5.

Intervals are widely used in scheduling problems, data range queries, and time-based operations, among other applications, because they succinctly represent a continuous range of values or times.

The Problem Statement

The problem of merging intervals involves taking a list of intervals and combining those that overlap into single, wider intervals. This is often necessary to simplify the set of intervals into a minimal representation without losing any coverage of the range they collectively span.

Why Merge Intervals?

Merging intervals helps in:

• Reducing Complexity: It simplifies the set of intervals to a minimal form, making it easier to process or understand.
• Efficiency: It reduces the number of intervals to consider in computations, which can improve the performance of algorithms that operate on them.
• Clarity: It provides a clearer overview of the ranges covered by the intervals, which is particularly useful in scheduling and resource allocation problems.

How to Merge Intervals

Let's consider a step-by-step method to merge intervals:

1. Sort the Intervals: First, sort the intervals by their starting points. This step is crucial because it ensures that we consider intervals in a sequence where potential overlaps are easier to detect and handle.

2. Iterate and Merge: Next, iterate through the sorted intervals, and for each interval:

   • If the current interval does not overlap with the previously merged interval (or if there's no previously merged interval), add it to the result list as a new interval.
   • If the current interval overlaps with the previous one, merge them. Merging typically involves updating the end of the previous interval to the maximum end of both overlapping intervals.

Definition of Overlap

Two intervals overlap if one starts before the other one ends and ends after the other one starts. Mathematically, interval A = [A.start, A.end] overlaps with interval B = [B.start, B.end] if A.start <= B.end and A.end >= B.start.

Example

Given intervals: [[1,3], [2,6], [8,10], [15,18]]

1. Sort by Starting Point: The intervals are already sorted.
2. Merge:
   • [1,3] and [2,6] overlap, merge them into [1,6].
   • [8,10] does not overlap with [1,6], so it remains separate.
   • [15,18] also stands alone without overlap.

Final Merged Intervals: [[1,6], [8,10], [15,18]]

Implementation

In programming, this problem is typically solved by defining a data structure to represent an interval (if not already defined, such as in Python's range), sorting the list of such structures, and then iterating through them to find and merge overlaps as described.

This approach fundamentally relies on sorting to arrange intervals in a linear order by their start times, which makes it much easier to identify and merge overlapping intervals in a single pass through the list, resulting in a time complexity that is often dominated by the sorting step, usually O(n log n) for n intervals.