Introduction to Algorithms

Jiafen Liu

Sept. 2013

Today’s Tasks

Balanced Search Trees

• Red-black trees – Height of a red-black tree

– Rotations

– Insertion

Balanced Search Trees

• Balanced search tree: A search-tree data structure for which a height of O(lgn) is guaranteed when implementing a dynamic set of n items.

• Examples: – AVL Tree– 2-3-4 Tree– B Tree– Red-black Tree

Red-black trees

• This data structure requires an extra 1-bit color field in each node.

• Red-black properties: – Every node is either red or black. – The root and leaves (NIL’s) are black. – If a node is red, then its parent is black.– All simple paths from any node x to a

descendant leaf have the same number of black nodes = black-height(x).

Example of a red-black tree

• Convention: black-height of x does not include x itself.

Example of a red-black tree

• It could have a bunch of blacks, but it will never repeat two reds in a row.

Goals of red-black trees

• there are a couple of goals that we are trying to achieve. – These properties should force the tree to have

logarithmic height, O(lgn) height.– The other desire we have from these properties

is that they are easy to maintain.• We can create a tree in the beginning that has this

property.• A perfectly balanced binary tree with all nodes black

will satisfy those properties. • The tricky part is to maintain them when we make

changes to the tree.

Height of a red black tree

• Let's look at the height of a red black tree. And we will start to see where these properties come from.

• Theorem. A red-black tree with n keys has height h ≤ 2 lg(n + 1) .– Can be proved by induction, as in the book

P164.– Another informal way: intuition.

Intuition

• Merge red nodes into their black parents.

• This process produces another type of balanced search tree: 2-3-4 tree.– Any guesses why it's called a 2-3-4 tree?– Another nice property:

• All of the leaves have the same depth. Why?

• By Property 4

height of merged tree h'

• Now we will prove the height of merged tree h' .

• The first question is how many leaves are there in a red-black tree?

– #(internal nodes) +1 – It can also be proved by induction. Try it.

Height of Red-Black Tree

• The number of leaves in each tree is n + 1

• 2h' ≤ n + 1 ≤ 4h'

• h' ≤ lg(n + 1)

• How to connect h' and h?

• We also have h' ≥ 1/2 h

• h ≤ 2 lg(n + 1).

Query operations

• Corollary. The queries SEARCH, MIN, MAX, SUCCESSOR, and PREDECESSOR all run in O(lgn) time on a red-black tree with n nodes.

Modifying operations

• The operations INSERT and DELETE cause modifications to the red-black tree.

• How to INSERT and DELETE a node in Red Black Tree?– The first thing we do is just use the BST

operation. – They will preserve the binary search tree

property, but don't necessarily preserve balance.

Modifying operations

• The operations INSERT and DELETE cause modifications to the red-black tree.

• How to INSERT and DELETE a node in Red Black Tree?– The second thing is to set color of new

internal node, to preserve property 1.– We can color it red, and property 3 does not

hold.– The good news is that property 4 is still true.

How to fix property 3?

• We are going to move the violation of property 3 up the tree.

• IDEA: Only red-black property 3 might be violated. Move the violation up the tree by recoloring until it can be fixed with rotations and recoloring.

• we have to restructure the links of the tree via “rotation”.

Rotations

• Rotations maintain the inorder ordering of keys.

a α, b β, c γ ∈ ∈ ∈ ⇒ a ≤ A ≤ b ≤ B ≤ c.• A rotation can be performed in O(1) time.

– Because we only change a constant number of pointers.

Implement of Rotation

• P166

Example

• Insert 15

Example

• Insert 15

• Color it red

Example

• Insert 15

• Color it red

• Handle Property 3 by recoloring

Example

• Insert 15

• Color it red

• Handle Property 3 by recoloring

Example

• Insert 15

• Color it red

• Handle Property 3 by recoloring

• Failed！

RIGHT-ROTATE

• RIGHT-ROTATE(18)

• It turns out to be?

RIGHT-ROTATE

• We are still in trouble between 10 and 18. But made this straighter.

• It doesn't look more balanced than before.• We can not resolve by recoloring. Then?

LEFT-ROTATE

• LEFT-ROTATE(7)

• It turns out to be?

LEFT-ROTATE

• LEFT-ROTATE(7) and recoloring at the same time

• It satisfy all the 4 properties

Pseudocode of insertionRB-INSERT(T, x)

TREE-INSERT(T, x)

color[x] ← RED // only RB property 3 can be violated

while x ≠ root[T] and color[x] = RED

do if p[x] = left[ p[ p[x] ]

then y ← right[ p[ p[x] ] // y = aunt or uncle of x

if color[y] = RED

then Case 1 ⟨ ⟩

Case 1

• Let denote a subtree with a black root. • All have the same black-height.

• Push C’s black onto A and D, and recurse. • Caution: We don't know whether B was the right

child or the left child. In fact, it doesn't matter.

How?

Pseudocode of insertionRB-INSERT(T, x)

TREE-INSERT(T, x)

color[x] ← RED // only RB property 3 can be violated

while x ≠ root[T] and color[x] = RED

do if p[x] = left[ p[ p[x] ]

then y ← right[ p[ p[x] ] // y = aunt/uncle of x

if color[y] = RED

then Case 1 ⟨ ⟩

else if x = right[p[x]] //y is red, x,y have a zigzag

then Case 2 ⟨ ⟩ // This falls into case 3

Case 2

• Now we have a zigzig.

• we may want a straight path between x.

How?

Pseudocode of insertionRB-INSERT(T, x)

TREE-INSERT(T, x)

color[x] ← RED // only RB property 3 can be violated

while x ≠ root[T] and color[x] = RED

do if p[x] = left[ p[ p[x] ]

then y ← right[ p[ p[x] ] // y = aunt/uncle of x

if color[y] = RED

then Case 1 ⟨ ⟩

else if x = right[p[x]]

then Case 2 ⟨ ⟩ // This falls into case 3

⟨Case 3 ⟩ //straight path

Case 3

• Done! No more violations of RB property 3 are possible.

• Property 4 is also preserved.

How?

Pseudocode of insertionRB-INSERT(T, x)

TREE-INSERT(T, x)

color[x] ← RED // only RB property 3 can be violated

while x ≠ root[T] and color[x] = RED

do if p[x] = left[ p[ p[x] ]

then y ← right[ p[ p[x] ] // y = aunt/uncle of x

if color[y] = RED

then Case 1 ⟨ ⟩

else if x = right[p[x]]

then Case 2 ⟨ ⟩ // This falls into case 3

⟨Case 3 ⟩

else “then” clause with “left” and “right” swapped ⟨ ⟩

color[root[T]] ← BLACK

Analysis

• Go up the tree performing Case 1, which only recolors nodes.

• If Case 2 or Case 3 occurs, perform 1 or 2 rotations, and terminate.

• Running time of insertion?– O(lgn) with O(1) rotations.

• RB-DELETE — same asymptotic running time and number of rotations as RB-INSERT (see textbook).