1. Tree Structures for Set of Intervals
Advanced Data Structures
Rajani Pingili

2. Tree Structures for Sets of Intervals:
Interval Trees
Segment Trees
Trees for the Union of Intervals
Orthogonal Range Trees
Higher-Dimensional Segment Trees
kd-Trees

3. Interval Trees:
Interval trees are invented by Edelsbrunner and McCreight.
The structure of interval tree stores a proper set of intervals. It returns the query key values for all of the intervals.
A set of intervals stored in interval trees.
Each interval contains a key node and it is associated with node.
Source: by

4. Segment Trees: Segment trees are invented by Bentley.
It is a static data structure.
The primary task performed by segment tree is the same as that done by an interval tree: keeping track of a set of n intervals, here assumed to be half-open, and listing for a given query key all the intervals
It is based on the canonical interval decomposition is a framework on which number of more general tasks can be performed.

5. Segment Trees:
Theorem: “The canonical interval decomposition is a representation of the interval as union of disjoint node intervals. Any search path for a value in the interval will go through exactly one node that belongs to the canonical interval decomposition. “
It is easy to construct the canonical interval decomposition.

6. Segment Trees: 1. Each time the node interval of the current node is entirely contained in [xi, xj [, we take that node into our representation and stop following that path down because all nodes below are redundant; 2. Each time the node interval of the current node partially overlaps [xi, xj [, we follow both paths down; 3. Each time the node interval of the current node is disjoint from [xi, xj [, we stop following that path down.

7. Trees for the Union of Intervals:

8. Orthogonal Range Trees:
Orthogonal range trees were independently discovered by Bentley, Lee and Wong, Lueker and Willard.
Orthogonal range searching is more useful for database index structures.
The main aim of the orthogonal range trees is that in order to solve the d- dimensional orthogonal range-searching problem, we build a balanced search tree for the key values that occur in the first coordinate of the data points.
Orthogonal range trees can be applied to solve computational Geometry Problems

9. Orthogonal Range Trees:

10. kd-Trees:
The kd-tree is another structure that supports orthogonal range searching.
It is quite popular in practical applications and conceptually easy to understand and implement; but it is unsatisfactory because its worst- case performance is much worse than orthogonal range trees.
It is widely used in database structures.
The application of kd-tree is Astronomy

11. kd-Trees:

12. References:
Advanced Data Structures, Peter Brass, 1st edition, Cambridge University Press
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