Write a brief note on interval trees.

Interval trees are a data structure used to efficiently store and query intervals. They are similar to binary search trees, but they are specifically designed for intervals. Interval trees allow for efficient searching, insertion, and deletion of intervals.

Structure of an Interval Tree

An interval tree is a binary search tree where each node contains an interval and a value. The interval is the range of values that the node represents, and the value is the data associated with the node. The left and right children of a node contain intervals that are completely contained within the interval of the parent node.

Searching in an Interval Tree

To search for an interval in an interval tree, we start at the root node and compare the search interval to the interval of the root node. If the search interval is completely contained within the interval of the root node, then we have found the node that contains the search interval. If the search interval intersects the interval of the root node, then we search the left and right children of the root node. We continue this process until we find a node that contains the search interval or until we reach a leaf node.

Insertion in an Interval Tree

To insert an interval into an interval tree, we first find the node that contains the interval. If the interval intersects the interval of the node, then we split the node into two nodes, with one node containing the left part of the interval and the other node containing the right part of the interval. We then insert the new interval into the appropriate node.

Deletion in an Interval Tree

To delete an interval from an interval tree, we first find the node that contains the interval. If the interval is completely contained within the interval of the node, then we simply delete the node. If the interval intersects the interval of the node, then we split the node into two nodes, with one node containing the left part of the interval and the other node containing the right part of the interval. We then delete the interval from the appropriate node.

Interval trees are a powerful data structure for storing and querying intervals. They are efficient for searching, insertion, and deletion, and they can be used to solve a variety of problems, such as finding the intersection of two intervals or finding the union of a set of intervals.