1. Chapter 14: Augmenting Data Structures

2. Dynamic order statistics
QUERY: OS(x,i) – the node containing the i-th smallest element in the subtree rooted at x
How to augment so that the following can be done efficiently:
The query itself
Maintaining augmented data (under Insert, Delete)

3. Added field to red-black trees: size(x): # nodes in the subtree rooted at x
Process the query
Determine the rank of an element x
Count size[left[x]]+1.
When climbing left (since right child of parent) increment count by size[left[parent[x]]]+1

4. Maintaining subtree sizes
Since the main tool for insert/delete
 Theorem: Query, Insert, Delete still take O(log n) time each

5. Interval trichotomy for 2 closed intervals i and i’
i and i’ can overlap in 4 ways or to not overlap in 2 ways:
Each way is represented by conditions on (some subset of) low(i),high(i),low(i’),high(i’)

6. Interval trees Red Black tree by left(i) of intervals
INTERVAL-SEARCH(T,i) – find a node in T whose interval overlaps interval i or report: none exists
Start with root. If i does not overlap current node x of T, then if max[left[x]]>=low(i) proceed to left[x]; else proceed to right[x]

7. Theorem: INTERVAL-SEARCH is correct
Proof:
Case (a). INTERVAL-SEARCH goes right. No interval on left subtree can overlap I.
Case (b). INTERVAL-SEARCH goes left. If no interval in left the subtree overlaps i, also no interval of of the right subtree could overlap i.