Discuss in detail about Interval Trees.

Interval Trees

An interval tree is a data structure used to store and efficiently search for intervals. It is a balanced binary search tree that is augmented with additional information to allow for efficient range queries. Interval trees can be used to solve a variety of problems, including:

• Finding all intervals that overlap a given interval
• Finding the minimum or maximum interval that contains a given point
• Counting the number of intervals that overlap a given interval

Structure

An interval tree is a binary search tree where each node stores an interval and some additional information. The additional information typically includes the minimum and maximum values of the intervals in the subtree rooted at that node.

Operations

The following operations can be performed on an interval tree:

• Insert: Insert a new interval into the tree.
• Delete: Delete an interval from the tree.
• Search: Find all intervals that overlap a given interval.
• Range query: Find the minimum or maximum interval that contains a given point.
• Count: Count the number of intervals that overlap a given interval.

Example

Here is an example of an interval tree:

        [1, 10]
       /        \
     [2, 5]      [7, 12]
    /      \     /      \
  [2, 3]   [4, 5]  [8, 10]  [11, 12]

The interval tree can be used to answer the following queries:

• Find all intervals that overlap the interval [3, 8].
• Find the minimum interval that contains the point 5.
• Count the number of intervals that overlap the interval [6, 9].

Applications

Interval trees have a variety of applications, including:

• Scheduling: Interval trees can be used to schedule events or tasks.
• Resource allocation: Interval trees can be used to allocate resources to tasks.
• Spatial indexing: Interval trees can be used to index spatial data, such as points, lines, and polygons.
• Text indexing: Interval trees can be used to index text documents.

Complexity

The following table shows the complexity of the various operations on an interval tree:

Operation	Time Complexity
Insert	O(log n)
Delete	O(log n)
Search	O(log n + k)
Range query	O(log n + k)
Count	O(log n + k)

where n is the number of intervals in the tree and k is the number of intervals that overlap the query interval.

Conclusion

Interval trees are a powerful data structure that can be used to efficiently store and search for intervals. They have a variety of applications, including scheduling, resource allocation, spatial indexing, and text indexing.