A) Briefly explain determining the rank of an element. <br> b) Describe in detail about interval tree.

a) Determining the Rank of an Element

The rank of an element in a set of elements is its position in the sorted list of those elements. In other words, it is the number of elements that are less than it in the set.

Here is a step-by-step approach to determine the rank of an element:

Step 1: Sort the elements in the set in ascending order.

Step 2: Find the position of the element in the sorted list. The position of the first element is 1, the position of the second element is 2, and so on.

Step 3: The rank of the element is its position in the sorted list.

For example, consider the set {7, 10, 3, 20, 15}. The sorted list is {3, 7, 10, 15, 20}. The rank of 7 is 2 because there is one element less than 7 in the set. The rank of 15 is 4 because there are three elements less than 15 in the set.

b) Interval Tree

An interval tree is a tree data structure to hold intervals. Specifically, it allows one to efficiently find all intervals that overlap with any given interval or point. It is often used for windowing queries, for instance, to find all roads on a computer graphics screen that intersect a given window.

Here is a detailed description of an interval tree:

1. Node Structure: Each node in an interval tree stores an interval [low, high]. The key of a node is the low value of the interval, and a max value is maintained in each node which is the maximum high value in the subtree rooted at the node.

2. Insertion: To insert an interval [low, high] we first make sure that the low value of the interval is used as key, then we proceed as in BST insertion and the max value of a node is updated during insertion.

3. Overlapping Intervals: To search for all intervals that overlap a given interval [low, high], we start from the root and do the following.

   • If the node's interval overlaps with [low, high], print it.
   • If the left child of the node is not empty and the max value in the left child is greater than low, then the left subtree may have an overlapping interval. So, move to the left child.
   • Else, move to the right child.

For example, consider the following intervals: [15, 20], [10, 30], [17, 19], [5, 20], [12, 15], [30, 40]. The interval tree would look like this:

                 [15, 20]
                /        \
             [10, 30]   [17, 19]
             /   \         \
          [5, 20] [12, 15] [30, 40]

In this tree, the max value of the root is 40, which is the maximum high value of all intervals. The left child of the root has max value 30, and the right child of the root has max value 40.