Order statistics a dynamic version

Order statistics a dynamic version rank and select

The dictionary ADT • Insert(x,D) • Delete(x,D) • Find(x,D): Returns a pointer to x if x ∊ D, and a pointer to the successor or predecessor of x if x is not in D

Suppose we want to add to the dictionary ADT • Select(k,D): Returns the kthlargestelement in the dictionary: An element x such that k-1 elements are smaller than x

Select(5,D) 89 90 19 20 21 4 26 34 67 70 73 77

Can we still use a red-black tree ? 4 19 20 21 26 34 67 70 73 77 89 90

For each node v store # of leaves in the subtree of v 12 4 8 4 2 2 4 2 2 2 2 4 19 20 21 26 34 67 70 73 77 89 90

Select(7,T) 12 4 8 4 2 2 4 2 2 2 2 4 19 20 21 26 34 67 70 73 77 89 90

Select(7,T) 12 Select(3, ) 4 8 4 2 2 4 2 2 2 2 4 19 20 21 26 34 67 70 73 77 89 90

Select(7,T) 12 4 8 Select(3, ) 4 2 2 4 2 2 2 2 4 19 20 21 26 34 67 70 73 77 89 90

Select(7,T) 12 4 8 4 2 2 4 Select(1,) 2 2 2 2 4 19 20 21 26 34 67 70 73 77 89 90

Select(i,T) Select(i,T): Select(i,root(T)) Select(k,v): if v is a leaf thenifk = 1then return velse error if k ≤ (v.left).sizethen return Select(k,v.left)else return Select(k – (v.left).size),v.right) O(logn) worst case time

Rank(x,T) • Return the index of x in T

Rank(x,T) x Need to return 9

12 4 8 4 2 2 4 2 2 2 2 4 19 20 21 26 34 67 70 73 77 89 90 x Sum up the sizes of the subtrees to the left of the path

Insertion and deletions • Consider insertion, deletion is similar

Insert 12 8 4 2

Insert (cont) 13 9 5 3 2

Easy to maintain through rotations x y <===> y A C x A B C B size(x) ← size(B) + size(C) size(y) ← size(A) + size(x)

Summary • Insertion and deletion and other dictionary operations still take O(log n) time

Order statistics a dynamic version