CS2420: Lecture 31

1
CS2420 Lecture 31

• Vladimir Kulyukin
• Computer Science Department
• Utah State University

2
Outline

• Red-Black Tree
• Bottom-up Insertion
• Top-Down Insertion
• Section 12.2

3
Red-Black Tree Properties

1. Every node is colored black or red.
2. The root is black.
3. If a node is red, its children must be black.
4. Every path from a node to a NULL pointer must
   have the same number of black nodes.
5. The NULL pointers are black.

4
Red-Black Tree Example
30
15
70
10
20
85
60
5
50
65
80
90
40
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Red-Black Tree Properties

• No path is more than twice as long as any other
  path.
• If every path from the root to a NULL node
  contains B black nodes, the tree contains at
  least 2B 1 black nodes (Proof is by induction
  on B).
• The height of a red-black tree is at most
  2log(N1).

6
Red-Black Tree Bottom-Up Insertion

• A new node is always inserted as a leaf, because
  this is a BST.
• There are two choices for coloring the new node
  black or red.
• If we color the node black, we violate property
  4.
• If we color the node red, we may violate property
  3.
• Which should we choose black or red?

7
Red-Black Tree Bottom-Up Insertion

• A new node is colored red.
• Why? Because it is easier.
• We may have to do no work.

8
Red-Black Tree Three Basic Insertion Operations

• Color Flip (Color Change)
• Two Rotations
• Single rotation
• Double rotation

9
Red-Black Tree Rotations

• In terms of pointer manipulation, Red-Black tree
  rotations are the same as the AVL search tree
  rotations.
• Thus, there are two single rotations (right and
  left) and two double rotations (left right and
  right left).
• For implementation purposes, we need only the two
  single rotations.
• The rotations are symmetric.
• Unlike the AVL search tree rotations, the
  Red-Black tree rotations change node colors.

10
Single Right at G S is Black
G
S
P
X
C
D
E
X is inserted and colored red Xs parents
sibling, S, is black
A
B
11
Single Right at G S is Black
G
P
S
P
X
G
X
S
C
D
E
C
A
B
D
E
A
B
Before Rotation
After Rotation
12
Single Left at G S is Black
G
P
S
X
A
B
C
E
D
X is inserted and colored red Xs parents
sibling, S, is black
13
Single Left at G S is Black
G
P
P
S
X
G
X
C
S
C
A
B
D
E
D
E
A
B
Before Rotation
After Rotation
14
Double Rotation (LR) at G S is Black
X is inserted and colored red Xs parents
sibling, S, is black
G
S
P
X
A
D
E
B
C
15
Double Rotation (LR) at G S is Black
X
G
P
S
P
G
S
D
E
A
B
C
X
A
D
E
B
C
Before Rotation
After Rotation
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Double Rotation (RL) at G S is Black

• RL at G is symmetric to LR at G.
• We first rotate P right and then rotate G left.
• The color flips are the same as in LR.

17
Question

• When S is black, will the single and
• double rotations re-balance the tree?

18
Answer

• Yes, when S is black, both the single and double
  rotations re-balance the tree. Why? Because the
  number of black nodes on all paths remains the
  same.

19
Single Right at G S is Red
G
S
P
X
C
D
E
A
B
X is inserted and colored red Xs parents
sibling, S, is red
20
Single Right at G S is Red
P
G
P
S
X
G
S
A
B
C
X
C
D
E
D
E
A
B
Before Rotation
After Rotation
21
Single Left at G S is Red

• Single left at G when S is Red is symmetric to
  single right at G when S is Red.
• G goes down, P goes up.
• The color flips are the same.

22
Double Rotation (LR) at G S is Red
G
S
P
X
A
D
E
B
C
23
Double Rotation (LR) at G S is Red
X
G
S
P
G
P
A
B
X
A
D
E
C
S
B
C
D
E
Before Rotation
After Rotation
24
Question

• When S is red, will the single or
• double rotations re-balance the
• tree?

25
Answer

• When S is red, neither the single nor the double
  rotations re-balance the tree, because the root
  of the sub-tree is now red.
• Property 3 may be violated higher up.

26
Bottom-Up Insertion

• We insert a leaf, color it red, and then climb up
  the insertion path until we reach the root or
  perform a rotation that rebalances the tree.
• We may have to make two passes one down and one
  up the tree.

27
Red-Black Tree Top-Down Insertion

• Can we insert, change colors, and rotate in one
  pass down the tree?
• Yes, we can.
• How?
• We need to guarantee that when we insert a node
  (leaf), S is never red.
• Basically, we have to flip colors as we drill
  down the tree.

28
Top-Down Color Flip

• When a node X has two red children, X is colored
  red and its two children are colored black.

29
Top-Down Color Flip
X
X
XL
XR
XL
XR
Before Color Flip
After Color Flip
30
Question

• Is there anything that can go wrong
• during the top-down color flip?

31
Answer

• Yes, the color flip may introduce two consecutive
  red nodes.
• Can it introduce more than two consecutive red
  nodes?
• No! Because that would mean that property 3 was
  already violated before the insertion.

32
Question

• Can we fix property 3 violations?

33
Answer

• Yes!
• If a color flip results in two consecutive red
  nodes, we can apply either single or double
  rotation, whichever is appropriate.
• But, since we have ensured that S is black, the
  rotation will rebalance the tree.

34
Top-Down Insertion Example
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45
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85
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80
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40
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Top-Down Insertion Example
Flip?
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Top-Down Insertion Example
Flip? No!
30
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Top-Down Insertion Example
Flip? No!
30
Flip?
15
70
45
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85
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Top-Down Insertion Example
Flip? No!
30
Flip? No!
15
70
45
10
20
85
60
5
50
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80
90
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Top-Down Insertion Example
Flip? No!
30
Flip? No!
15
70
Flip?
10
20
85
60
45
5
50
65
80
90
40
55
40
Top-Down Insertion Example
Flip? No!
30
Flip? No!
15
70
Flip? No!
10
20
85
60
45
5
50
65
80
90
40
55
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Top-Down Insertion Example
Flip? No!
30
Flip? No!
15
70
Flip? No!
10
20
85
60
Flip?
5
50
65
80
90
45
40
55
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Top-Down Insertion Example
Flip? No!
30
Flip? No!
15
70
Flip? No!
10
20
85
60
Flip? Yes!
5
50
65
80
90
45
40
55
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Top-Down Insertion Example
30
15
70
2 Reds!!!
10
20
85
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80
90
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Question

• Which Rotation and Where?

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Single Right at 70 85 is Black
70
60
85
60
50
70
50
85
C
D
E
C
A
B
D
E
A
B
Before Rotation
After Rotation
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Top-Down Insertion After the Rotation
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10
70
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85
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80
90
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Top-Down Insertion Example
30
15
60
10
70
20
50
Flip?
5
85
65
40
55
45
80
90
48
Top-Down Insertion Example
30
15
60
10
70
20
50
Flip? No!
5
85
65
40
55
45
80
90
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Top-Down Insertion Example
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85
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80
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Red-Black Tree Implementation

• nullNode is a node used in place of a NULL
  pointer.
• nullNode is always colored black .
• header is a node that is used as a dummy root.
• The value of the header is -9999.
• The real root is the right child of the header.