1. Weight balance trees (Nievergelt & Reingold 73)

2. Definition A binary tree.
Define the balance at an internal node v to be
where s(v) = #(leaf descendants of v)
Tree T is of bounded balance α if for every v, α ≤ ρ(v) ≤ 1- α
¼ < α ≤ 1- (√2/2) =

3. Weight balance trees - example
5/14
2/5 ⅔

4. Obvious properties The depth is O(log n)
As we go from a node to a child the # of leaves under us decrease by a fraction smaller than (1-α)
≤ w
≤ (1-α) w

5. Updates
Nodes on the path to the point where we update may go out of balance ⅔
2/5
5/14

6. Updates (Cont.)
5/14
6/14 ⅓
2/5 ⅔ ⅓

7. Use rotations γ1 = ρ1 + ρ2 (1- ρ1) ρ1 y x γ2 = ρ1 + ρ2 (1- ρ1) ρ1
<===> A ρ2 x C y B C A B
a+b = ρ1 (a + b + c) + ρ2 (b + c) =
ρ1 (a + b + c) + ρ2 ((a + b + c) - a) =
ρ1 (a + b + c) + ρ2 ((a + b + c) - ρ1 (a + b + c) ) =
(ρ1 + ρ2 (1- ρ1) ) (a + b + c)

8. And double rotations ρ1 z x <===> ρ2 A y x y ρ3 z D A B C D B C
γ1 = ρ1 + ρ3 ρ2 (1- ρ1) ρ1 z x
<===>
γ3 =
1 - ρ3 ρ2
ρ2 (1- ρ3) ρ2 A y
γ2 =
ρ1 + ρ3 ρ2 (1- ρ1) ρ1 x y ρ3 z D A B C D B C

9. The rebalancing lemma (Blum & Mehlhorn)
Let v be a node such that
ρ(u)  (α, 1- α] for every descendant u of v
ρ(v) < α and
(insertion into right subtree of v)
(deletion from the left subtree of v) or
Let ρ2 be the balance of vr
There are constants d  (α, 1- α] and δ (> 0 if α < 1- (√2/2) ) such that
If ρ2 ≤ d a rotation rebalances the tree, i.e. γ1, γ2  ((1+ δ )α, 1- (1+ δ ) α]
If ρ2 > d a double rotation rebalances the tree, i.e γ1, γ2 , γ3  ((1+ δ )α, 1- (1+ δ ) α]

10. We will not prove the rebalancing lemma, its technical, you are encouraged to look at the proof…
Here is what distinguishes BB(α) trees

11. What is special about BB(α) trees ?
Thm:
Suppose we carry out a sequence of m updates on a tree of size at most n at any time (starting for the empty tree). Then even if a rotation at a node v takes O(s(v)) time, the total cost of all rotations over the sequence is O( m log n)).
Proof.

12. Let w = s(v) when a rotation occurs at v (v goes out of balance)
We show that the number of updates through v since the last rotation v was involved in, is at least δαw
Suppose ρ(v) ≤ α
Assume a insertion and b deletion happened through v since last rotation
Then
a insertions, b deletions v v 
ρ(v) w + b
w + b - a
ρ’(v) ≤ A’ B’ A B

13. ρ(v) w + b
w – a + b
α w + b
w - a + b ≤
(1+δ)α ≤
ρ’(v) ≤
(1+δ)α (w - a + b) ≤ α w + b
δα w ≤ (1+δ)α a + (1 - (1+δ)α ) b ≤ a + b

14. ( ) ) ( ) ( ( ) ( ) Divide the possible sizes of subtrees into groups1
1-α ( ) 2 1
w1 = --
1-α 1
1-α ( ) 2 1
1-α ( ) 3
w2 = -- …… 1
1-α ( ) i
i+1 ( 1 )
wi =
Node v is at level --
1-α
A node v is at level floor (log(1/1-α) s(v))
Maximum level is floor (log(1/1-α) m)

15. Let uℓ(v) be the # of updates that go through v when v is at level ℓ, ℓ-1, or ℓ+1
uℓ(v) = Total # of insertions and deletions in this subtree when s(v) is at level ℓ, ℓ-1, or ℓ+1 v
Let rℓ(v) be the # of rotations around v when v is at level ℓ (when v has about wℓ descendants)

16. We show that c wℓ rℓ(v) ≤ uℓ(v)
If each update through v when v is at level ℓ, ℓ-1, or ℓ+1 is charged (1/c) then we can pay wℓ for each rotation at v when v is at level ℓ 7 8
Total charges per update are O(log n)

17. So why c wℓ rℓ(v) ≤ uℓ(v) ?
If each update through v when v is at level ℓ, ℓ-1, or ℓ+1 is charged (1/c) then we can pay wℓ for each rotation at v when v is at level ℓ 7 8
Total charges per update are O(log n)