Balanced Trees: Understanding 2-3-4 Trees and Red-Black Trees, Lecture notes of Data Structures and Algorithms

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2-3-4 Trees and Red-

Black Trees

2-3-4 Trees Revealed

• Nodes store 1, 2, or 3 keys and have 2, 3, or 4 children, respectively
• All leaves have the same depth

b e h n r

a (^) c d f g (^) i l m p s x

--- (^) log ( N + 1 ) ≤ height ≤ log( N + 1 )

k

b e h n r

• That means if d = N 1/2, we get a height of 2
• However, searching out the correct child

on each level requires O(log N 1/2^ ) by binary search

• 2 log N1/2^ = O(log N) which is not as good

as we had hoped for!

• 2-3-4-trees will guarantee O(log N) height

using only 2, 3, or 4 children per node

Why 2-3-4?

• Why not minimize height by maximizing children in a “d-tree”?
• Let each node have d children so that we

get O(log (^) d N) search time! Right? log dN = log N/log d

Insertion into 2-3-4 Trees

• Insert the new key at the lowest internal node reached in the search
• What about a 4-node?
• We can’t insert another key!

g (^) d d g

• 3-node becomes 4-node
• 2-node becomes 3-node

f^ d g^ d f g

Splitting the Tree

As we travel down the tree, if we encounter any 4-node we will break it up into 2-nodes. This guarantees that we will never have the problem of inserting the middle element of a former 4- node into its parent 4-node.

a (^) f i l p r x

a (^) f i l p r

g

g

n

c (^) t

x

c n t

Whoa, cowboy

a (^) f i l p r x

a (^) f i l p r x

c (^) t

n

n

c t

g

g

Whoa, cowboy

Time Complexity of Insertion

in 2-3-4 Trees

Time complexity:

• A search visits O(log N) nodes
• An insertion requires O(log N) node splits
• Each node split takes constant time
• Hence, operations Search and Insert each take time O(log N)

Notes:

• Instead of doing splits top-down, we can perform them bottom-up starting at the in- sertion node, and only when needed. This is called bottom-up insertion.
• A deletion can be performed by fusing nodes (inverse of splitting), and takes O(log N) time

Beyond 2-3-4 Trees

What do we know about 2-3-4 Trees?

• Balanced
• O(log N) search time
• Different node structures

Can we get 2-3-4 tree advantages in a binary

tree format???

Welcome to the world of Red-Black Trees!!!

Siskel

Ebert

Roberto

2-3-4 Tree Evolution

Note how 2-3-4 trees relate to red-black trees

2-3-4 (^) Red-Black

Now we see red-black trees are just a way of representing 2-3-4 trees!

or

More Red-Black Tree

Properties

N # of internal nodes L # leaves (= N + 1) H height B black height

Property 1: 2 B^ ≤ N + 1 ≤ 4 B

Property 2:

This implies that searches take time O(logN)!

Property 3:

--- (^) log ( N + 1 ) ≤ B ≤ log( N + 1 )

log ( N + 1 ) ≤ H ≤ 2 log( N + 1 )

Let: n be the new node p be its parent g be its grandparent

Insertion - Plain and Simple

Case 1: Incoming edge of p is black

g

p

n

STOP!

Pretty easy, huh?

Well... it gets messier...

No violation

Restructuring

We call this a “ right rotation ”

• No further work on tree is necessary
• Inorder remains unchanged
• Tree becomes more balanced
• No two consecutive red edges!

Case 2: Incoming edge of p is red, and

its sibling is black

g

p

n

g

p

n

Uh oh!

Much Better!

Double Rotations

What if the new node is between its parent and grandparent in the inorder sequence?

We must perform a “double rotation”

(which is no more difficult than a “single” one)

g

p

n

p g

n

This would be called a “left-right double rotation”

Last of the Rotations

And this would be called a “right-left double rotation”

g

p

n

g p

n