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The Design and Analysis of Algorithms
Chapter 6:
Transform and Conquer
2-3-4 Trees, Red-Black Trees
Chapter 6.
2-3-4 Trees, Red-Black Trees


Basic Idea
2-3-4 Trees




Definition
Operations
Complexity
Red-Black Trees



Definition
Properties
Insertion
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Basic Idea
Disadvantages of Binary Search trees –
worst case complexity is O(N)
Solution to the problem:

AVL trees – keep the tree balanced

Multi-way search tree – decrease tree levels


2-3 and 2-3-4 trees

Disadvantages: nodes with different structure
Red-Black trees – use advantages of 2-3-4
trees with binary nodes
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Multi-Way Trees
k1 < k2 < … < kn-1
< k1
[k1, k2 )
 kn-1
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2-3-4 Trees - Definition

Three types of nodes:




2-node: contains one key, has two links
3-node: contains 2 ordered keys, has 3 links
4-node: contains 3 ordered keys, has 4 links
All leaves must be on the same level,
i.e. the tree is perfectly height-balanced.
This is achieved by allowing more than
one key in a node
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2-3-4 Trees - Example
J
D
B
P
L N
FH
K
M
R TW
O
Q
S
U V
X
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2-3-4 Trees - Operations
Search – straightforward: start
comparing with the root and
branch accordingly


Insert: The new key is inserted
at the lowest internal level
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Insert in a 2-node
 The
2-node becomes a 3-node .
M 
P

MP
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Insert in a 3-node
 The
3-node becomes a 4-node .
R 
MP

MPR
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Insert in a 4-node
Bottom-up Insertion: Promotion

The 4-node is split, and the middle element is
moved up – inserted in the parent node. The
process is called promotion and may
continue up the top of the tree.

If the 4-node is a root (no parent), then a new
root is created.

After the split the insertion proceeds as in
the previous cases.
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Insert in a 4-node - Example
N
GN
C

FG L
CF
L
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Top-down Insertion
In our way down the tree, whenever we
reach a 4-node, we break it up into two
2-nodes, and move the middle element up
into the parent node.
In this way we make sure there will be place
for the new key
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Complexity of Search and
Insert
Height of the tree:
A 2–3–4 tree with minimum number of keys will correspond to a
perfect binary tree
N ≥ 1 + 2 + … + 2h = 2 h+1 – 1
h ≤ log(N+1) – 1
A 2–3–4 tree with maximum number of keys will correspond to a
perfect 4-tree tree
N ≤ 3(1 + 4 + 42 + … + 4h) = 3. (4 h+1 -1)/3
4 (h+1) ≥ N + 1
h ≥ log4(N + 1) -1 = 1/2 log(N + 1) -1
Therefore h = Θ(log(N))
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Complexity of Search and
Insert
• 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)
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Red-Black Trees - Definition
 edges
are colored red or black

no two consecutive red edges on any
root-leaf path

same number of black edges on any
root-leaf path (=black height of the tree)

edges connecting leaves are black
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2-3-4 and Red-Black Trees
2-3-4 tree
red-black tree
2-node
3-node
2-node
two nodes connected with a red link (left or right)
G
GN
N
F
CF
L
C
L
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2-3-4 and Red-Black Trees
4-node
three nodes connected with red links
N
G NP
P
G
C
L
C
L
O
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2-3-4 and Red-Black Trees
2-3-4 tree
Red-black tree
or
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Red-Black Trees
1/2 log(N+1)  B  log(N + 1)
log(N+1)  H  2 log(N + 1)
where :
N is the number of internal nodes
L is the number of leaves (L = N + 1)
H - height
B - black height (count the black edges only)
This implies that searches take time O(logN)
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Red-Black Trees: Insertion

Perform a standard search to find the leaf where the
key should be added
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Replace the leaf with an internal node with the new key

Color the incoming edge of the new node red

Add two new leaves, and color their incoming edges
black
If the parent had an incoming red edge, we now have
two consecutive red edges. We must reorganize tree
to remove that violation. What must be done depends on
the sibling of the parent.

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Restructuring
Incoming edge of p is red and its sibling is black
single rotation
g - grandparent, p – parent, n – new node)
g
p

p
g
n
n
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Restructuring
Double Rotations: the new node is between its parent and
grandparent in the inorder sequence
g
p
n

p
g
n
Left-right double rotation
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Restructuring
Right-left double rotation
g
n

p
g
p
n
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Promotion: bottom up
rebalancing
Incoming edge of p is red and its sibling is also red
The black depth remains
unchanged for all of the
descendants of g
g
g

This process will continue
upward beyond g if
p
n
p
necessary: rename g as n
and repeat.
n
Promotions may continue
up the tree and are
executed O(log N) times.
The time complexity of an
insertion is O(logN).
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