UOS lahore

1. Red Black TreeRed Black Tree
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2. Group Member
Balawal Hussain 269
Nasir Shahzad 287
Ali Raza 277
Umair Khan 293
Usman Aslam 299
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3. Red Black VS BNS Tree
Red black tree are new form of data
structure but inherit similarities from
binary search tree
 for example: red black tree have up to 2 for example: red black tree have up to 2
children per node. They must follow the
binary search tree.
 On left side is smaller than parent.
 On right side is greater than parent.
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4. 4

5. Binary search tree
• In BST, all left sub-tree element should be
less than root node and all right sub-tree
should be greater than root node, called
BST.BST.
• Parent node should not have more than
two children.
Parent<Right
Parent>left
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6. Binary search tree
30
25 50
Example: 30,25,50,27,40,66,5,3,10
25 50
5 27 40 66
3 10
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7. About Red black tree
Invented in Invented byInvented in
1972.
Invented by
Rudolf Bayer
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8. Red black tree
Red black tree properties:
1-Every node is either red or black.
2-The root is black.
3-Every leaf (NIL) is black.3-Every leaf (NIL) is black.
4-If a node is red, then both its children are
black.
5-For each node, all simple paths from the node
to descendant leaves contain the same
number of black nodes.
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9. Red black tree
19
12 35
2116 5621
30
3
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10. Black-Height of a Node
26
17 41
30 47
38 50
NIL NIL
h = 4
bh = 2
h = 3
bh = 2
h = 2
bh = 1
h = 1
h = 1
bh = 1
h = 2
bh = 1 h = 1
bh = 1
38 50NIL
NIL NIL NIL NIL
NIL
h = 1
bh = 1
bh = 1
Height of a node:
the number of edges in the longest path to a
leaf.
Black-height of a node x:
bh(x) is the number of black nodes
(including NIL) on the path from x to a leaf, not
counting x
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11. ROTATIONSROTATIONS
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12. RB tree Rotations
There are two type of rotations:
1-left rotation1-left rotation
2-right rotation
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13. Left Rotations
assumption for a left rotation on a node x:
The right child of x (y) is not NIL
Idea:
rotate around the link from x to y
Makes y the new root of the sub-tree
x becomes y’s left child
y’s left child becomes x’s right child 13

14. Example: LEFT-ROTATE
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15. Right Rotations
Assumptions for a right rotation on a node x:
The left child of y (x) is not NIL
Idea:
 Pivots around the link from y to x
 Makes x the new root of the subtree
 y becomes x’s right child
 x’s right child becomes y’s left child
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16. Insertion
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17. RB tree Insertion
As case in BST, first we find place where
we are to insert.
No red red child parent relation.
Red node can have only black childrenRed node can have only black children
Since inserted color is red, the black height
of the tree is remain unchanged.
 A new item is always inserted as a leaf in
the tree.
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18. Case 1 – uncle y is red
C
A D
B
  
z
y
C
A D
B
  
new z
p[z]
p[p[z]]
• p[p[z]] (z’s grandparent) must be black, since z and p[z] are both red
and there are no other violations of property 4.
• Make p[z] and y black  now z and p[z] are not both red. But
property 5 might now be violated.
• Make p[p[z]] red  restores property 5.
• The next iteration has p[p[z]] as the new z (i.e., z moves up 2 levels).
B
 
B
 
z is a right child here.
Similar steps if z is a left child.
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19. Case 2 – y is black, z is a right child
C
A
B


z
y
C
B
A 

z
y
p[z]
p[z]
• Left rotate around p[z], p[z] and z switch roles  now z is
a left child, and both z and p[z] are red.
• Takes us immediately to case 3.
B

 
A
 
z
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20. Case 3 – y is black, z is a left child
B
A
   
C
B
A 
 y
p[z]
C
z
• Make p[z] black and p[p[z]] red.
• Then right rotate on p[p[z]]. Ensures property 4 is
maintained.
• No longer have 2 reds in a row.
• p[z] is now black  no more iterations.
A
 
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21. RB tree Insertion
• Example:
Insert 1 1
• The root can’t be red.
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22. RB tree Insertion
• Example:
Change the color, according to 2 property.
1
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23. RB tree Insertion
• Example:
Insert 2
1
2
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24. RB tree Insertion
• Example:
Insert 3
1
2
3
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25. RB tree Insertion
• Example:
Case 3
1
2
31 3
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26. RB tree Insertion
• Example:
• Follow the property 2
1
2
31 3
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27. RB tree Insertion
• Example:
• Insert 4
1
2
31 3
4
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28. RB tree Insertion
• Example:
• Case 1
1
2
31 3
4
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29. RB tree Insertion
• Example:
• Color
1
2
31 3
4
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30. RB tree Insertion
• Example:
• Insert 5
1
2
31 3
4
5
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31. RB tree Insertion
• Example:
• Case 3
1
2
41 4
5
3
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32. RB tree Insertion
• Example:
• Color
1
2
41 4
53
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33. RB tree Insertion
• Example:
• Insert 6
1
2
41 4
53
6
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34. RB tree Insertion
• Example:
• Case 1/color
1
2
41 4
53
6
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35. Algorithm Analysis
O(lg n) time to get through RB-Insert up to
the call of RB-Insert.
Within RB-Insert:
– Each iteration takes O(1) time.
– Each iteration but the last moves z up 2 levels.
– O(lg n) levels  O(lg n) time.
– Thus, insertion in a red-black tree takes O(lg n)
time.
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36. Problems?????
• Let a, b, c be arbitrary nodes in sub-trees , ,  in the
tree below. How do the depths of a, b, c change when a
left rotation is performed on node x?
– a: increases by 1
– b: stays the same– b: stays the same
– c: decreases by 1
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37. Problems?????
• What is the ratio between the longest path
and the shortest path in a red-black tree?
- The shortest path is at least bh(root)- The shortest path is at least bh(root)
- The longest path is equal to h(root)
- We know that h(root)≤2bh(root)
- Therefore, the ratio is ≤2
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