The document discusses algorithms and data structures, focusing on binary search trees (BSTs). It provides the following key points:
- BSTs are an important data structure for dynamic sets that can perform operations like search, insert, and delete in O(h) time where h is the height of the tree.
- Each node in a BST contains a key, and pointers to its left/right children and parent. The keys must satisfy the BST property - all keys in the left subtree are less than the node's key, and all keys in the right subtree are greater.
- Rotations are a basic operation used to restructure trees during insertions/deletions. They involve reassigning child
3. Binary search trees
● Binary search trees are an important data
structure for dynamic sets.
● Accomplish many dynamic-set operations in
O(h) time, where h=height of tree.
● we represent a binary tree by a linked data
structure in which each node is an object.
● T:root points to the root of tree T .
4. Binary search trees
● Each node contains the attributes
■ key (and possibly other satellite data).
■ left: points to left child.
■ right: points to right child.
■ p: points to parent. T.root.p = NIL.
5. Binary search trees
● Stored keys must satisfy the binary-search-
tree property.
■ If y is in left subtree of x, then y.key <= x.key.
■ If y is in right subtree of x, then y.key >= x.key.
6. Binary search trees
(a) A binary search tree on 6 nodes with height 2.
(b) A less efficient binary search tree with height 4 that
contains the same keys.
8. Binary search trees
● The binary-search-tree property allows us to
print out all the keys in a binary search tree in
sorted order by a simple recursive algorithm,
called an inorder tree walk.
It takes ‚
time to
walk an n-node
binary search tree
9. Binary search trees
● A: prints elements in sorted (increasing) order
● This is called an inorder tree walk
■ Preorder tree walk: print root, then left, then right
■ Postorder tree walk: print left, then right, then root
10. Tree traversal
● Used to print out the data in a tree in a certain
order
● Pre-order traversal
■ Print the data at the root
■ Recursively print out all data in the left subtree
■ Recursively print out all data in the right subtree
11. Preorder, Postorder and Inorder
● Preorder traversal
■ node, left, right
■ prefix expression
○ ++a*bc*+*defg
16. insert
● Proceed down the tree as you would with a find
● If X is found, do nothing (or update something)
● Otherwise, insert X at the last spot on the path traversed
● Time complexity = O(height of the tree)
17. delete
● When we delete a node, we need to consider
how we take care of the children of the deleted
node.
■ This has to be done such that the property of the
search tree is maintained.
18. delete
Three cases:
(1) the node is a leaf
■ Delete it immediately
(2) the node has one child
■ Adjust a pointer from the parent to bypass that node
19. delete
(3) the node has 2 children
■ replace the key of that node with the minimum element at the right
subtree
■ delete the minimum element
○ Has either no child or only right child because if it has a left child, that
left child would be smaller and would have been chosen. So invoke
case 1 or 2.
●
Time complexity = O(height of the tree)
20. Balanced Binary Search Trees
A binary search tree can implement any of the basic dynamic-set
operations in O(h) time. These operations are O(lgn) if tree is
“balanced”.
There has been lots of interest in developing algorithms to keep binary
search trees balanced, including
1st type: insert nodes as is done in the BST insert, then rebalance tree
Red-Black trees
AVL trees
Splay trees
2nd type: allow more than one key per node of the search tree:
2-3 trees
2-3-4 trees
B-trees
21. Red-Black Trees (RBT)
Red-Black tree: BST in which each node is colored red or black.
Constraints on the coloring and connection of nodes ensure that
no root to leaf path is more than twice as long as any other, so
tree is approximately balanced.
Each RBT node contains fields left, right, parent, color, and key.
L
E
F
T
PARENT
KEY
COLOR
R
I
G
H
T
22. Red-Black Trees
● Red-black trees:
■ Binary search trees augmented with node color
■ Operations designed to guarantee that the height
h = O(lg n)
● First: describe the properties of red-black trees
● Then: prove that these guarantee h = O(lg n)
● Finally: describe operations on red-black trees
23. Red-Black Properties
● The red-black properties:
1. Every node is either red or black
2. Every leaf (NULL pointer) is black
○ Note: this means every “real” node has 2 children
3. If a node is red, both children are black
○ Note: can’t have 2 consecutive reds on a path
4. Every path from node to descendent leaf contains
the same number of black nodes
5. The root is always black
24. Red-Black Trees
● Put example on board and verify properties:
1.
2.
3.
4.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf contains
the same number of black nodes
5. The root is always black
● black-height: # black nodes on path to leaf
■ Label example with h and bh values
25. Black Height bh(x)
Black-height of a node x: bh(x) is the number of black nodes (including
the NIL leaf) on the path from x to a leaf, not counting x itself.
2 20
2 18
1 17
0
0
1 22
1 19
0
Every node has a
black-height, bh(x).
For all NIL leaves,
bh(x) = 0.
1 21 1 25
0
0
0 0
0
For root x,
bh(x) = bh(T).
26. Height of a Red-black Tree
h=4
26 bh=2
● Example:
● Height of a node:
■ Number of edges in a
longest path to a leaf.
● Black-height of a node
17
h=1
bh=1
h=2 30
bh=1
bh(x) is the number of
black nodes on path
from x to leaf, not
counting x.
nil[T]
h=3
41 bh=2
h=1
bh=1
38
h=2
47 bh=1
h=1 50
bh=1
27. Height of Red-Black Trees
● What is the minimum black-height of a node
●
●
●
●
with height h?
A height-h node has black-height h/2
Proof : By property 4,
h/2 nodes on the path from the node to a leaf
are red.
Hence
are black.
28. Height of Red-Black Trees
● The subtree rooted at any node x contains >= 2bh(x)_1
internal nodes.
● Proof :By induction height of x =0x is a leafbh(x)=0.
The subtree rooted at x has 0 internal nodes. 20 -1 = 0.
● Let the height of x be h and bh(x) = b.
● Any child of x has height h -1 and
● black-height either b (if the child is red) or
● b -1 (if the child is black)
● By induction each child has >= 2bh(x)-1-1 internal nodes
● Thus, the subtree rooted at x contains >= 2(2bh(x)-1-1)+1
●
= 2bh(x)-1(internal Nodes)
29. Height of Red-Black Trees
● Theorem: A red-black tree with n internal
nodes has height h 2 lg(n + 1)
● How do you suppose we’ll prove this?
● Proof: The subtree rooted at any node x contains
● >= 2bh(x)_1 internal nodes.
● A height-h node has black-height h/2
n 2h/2 -1
n + 1 2h/2
lg(n+1) h/2 h 2lg(n+1)
● Thus
●
●
30. RB Trees: Worst-Case Time
● So we’ve proved that a red-black tree has
O(lg n) height
● Corollary: These operations take O(lg n) time:
■ Minimum(), Maximum()
■ Successor(), Predecessor()
■ Search()
● Insert() and Delete():
■ Will also take O(lg n) time
■ But will need special care since they modify tree
31. Red-Black Trees: An Example
● Color this tree:
7
5
9
12
Red-black properties:
1.
Every node is either red or black
2.
Every leaf (NULL pointer) is black
3.
If a node is red, both children are black
4.
Every path from node to descendent leaf
contains the same number of black nodes
5.
The root is always black
32. Red-Black Trees:
The Problem With Insertion
● Insert 8
■ Where does it go?
7
5
9
12
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
33. Red-Black Trees:
The Problem With Insertion
● Insert 8
■ Where does it go?
■ What color
7
5
should it be?
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
9
8
12
34. Red-Black Trees:
The Problem With Insertion
● Insert 8
■ Where does it go?
■ What color
7
5
should it be?
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
9
8
12
35. Red-Black Trees:
The Problem With Insertion
● Insert 11
■ Where does it go?
7
5
9
8
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
12
36. Red-Black Trees:
The Problem With Insertion
● Insert 11
■ Where does it go?
■ What color?
7
5
9
8
12
11
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
37. Red-Black Trees:
The Problem With Insertion
● Insert 11
■ Where does it go?
■ What color?
○ Can’t be red! (#3)
7
5
9
8
12
11
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
38. Red-Black Trees:
The Problem With Insertion
● Insert 11
■ Where does it go?
■ What color?
○ Can’t be red! (#3)
○ Can’t be black! (#4)
1.
2.
3.
4.
5.
7
5
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
9
8
12
11
39. Red-Black Trees:
The Problem With Insertion
● Insert 11
■ Where does it go?
■ What color?
○ Solution:
recolor the tree
1.
2.
3.
4.
5.
7
5
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
9
8
12
11
40. Red-Black Trees:
The Problem With Insertion
● Insert 10
■ Where does it go?
7
5
9
8
12
11
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
41. Red-Black Trees:
The Problem With Insertion
● Insert 10
■ Where does it go?
■ What color?
7
5
9
8
12
11
1.
2.
3.
4.
5.
Every node is either red or black
Every leaf (NULL pointer) is black
If a node is red, both children are black
Every path from node to descendent leaf
contains the same number of black nodes
The root is always black
10
42. Red-Black Trees:
The Problem With Insertion
● Insert 10
■ Where does it go?
7
5
■ What color?
○ A: no color! Tree
is too imbalanced
○ Must change tree structure
to allow recoloring
■ Goal: restructure tree in
O(lg n) time
9
8
12
11
10
43. RB Trees: Rotation
● Our basic operation for changing tree structure
is called rotation:
y
x
x
rightRotate(y)
C
A
y
leftRotate(x)
A
B
B
C
● Does rotation preserve inorder key ordering?
● What would the code for rightRotate()
actually do?
46. RB Trees: Rotation
Left-Rotate (T, x)
1. y right[x] // Set y.
2. right[x] left[y] //Turn y’s left subtree into x’s right subtree.
3. if left[y] nil[T ]
4.
then p[left[y]] x
5. p[y] p[x]
// Link x’s parent to y.
6. if p[x] == nil[T ]
7.
then root[T ] y
8.
else if x == left[p[x]]
9.
then left[p[x]] y
10.
else right[p[x]] y
11. left[y] x
// Put x on y’s left.
12. p[x] y
47. RB Trees: Rotation
right-Rotate (T, y)
1. x right[y]
2. left[y] right[x]
3. If right[x] nil[T ]
4.
then p[right[x]] y
5. p[x] p[y]
6. if p[y] == nil[T ]
7.
then root[T ] x
8.
else if y == left[p[y]]
9.
then left[p[y]] x
10.
else right[p[y]] x
11. right[x] y
12. p[y] x
50. Red-Black Trees: Insertion
● Remember:
1. Insert nodes one at a time, and after every
Insertion balance the tree.
2. Every node inserted starts as a Red node.
3. Consult the cases, Every time two Red nodes
touch must rebalance at that point.
4. The root will always be Black.
51. Red-Black Trees: Insertion
● If we insert, what color to make the new node?
■ Red? Might violate property 3(If a node is red, both
children are black).
■ Black? Might violate property 4(Every path from node
to descendent leaf contains the same number of black nodes).
Insertion: the basic idea
■ Insert x into tree, color x red
■ Only r-b property 3 might be violated (if p[x] red)
○ If so, move violation up tree until a place is found where it
can be fixed
■ Total time will be O(lg n)
52. Reminder: Red-black Properties
1. Every node is either red or black
2. Every leaf (NULL pointer) is black
3. If a node is red, both children are black
4. Every path from node to descendent leaf
contains the same number of black nodes
5. The root is always black
53. Insertion in RB Trees
● Insertion must preserve all red-black properties.
● Should an inserted node be colored Red? Black?
● Basic steps:
■ Use Tree-Insert from BST (slightly modified) to
insert a node x into T.
○ Procedure RB-Insert(x).
■ Color the node x red.
■ Fix the modified tree by re-coloring nodes and
performing rotation to preserve RB tree property.
○ Procedure RB-Insert-Fixup.
54. Algorithm: Insertion
We have detected a need for balance when z is red and his parent too.
• If z has a red uncle: colour the parent and uncle black, and
grandparent red.
z
Balanced trees: Red-black trees
55. Algorithm: Insertion
We have detected a need for balance when z is red and his parent too.
• If z has a red uncle: colour the parent and uncle black, and
grandparent red.
• If z is a left child and has a black uncle: colour the parent black and
the grandparent red, then rotateRight(z.parent.parent)
Balanced trees: Red-black trees
56. Algorithm: Insertion
We have detected a need for balance when z is red and his parent too.
• If z has a red uncle: colour the parent and uncle black, and
grandparent red.
right child and has black uncle: colour the parent black and
• If z is a left child and has a a black uncle, then rotateLeft(z.parent)
andgrandparent red, then rotateRight(z.parent.parent)
the
Balanced trees: Red-black trees
57. Algorithm: Insertion
Let’s insert 4, 7, 12, 15, 3 and 5.
4
7
1
2
Double red violation!
It also shows it’s
unbalanced…
Balanced trees: Red-black trees
58. Algorithm: Insertion
Let’s insert 4, 7, 12, 15, 3 and 5.
7
4
What should we do?
Nothing, no
double red.
3
12
5
15
Double red violation.
We can’t have a better
balance, and there is a
red uncle…
Balanced trees: Red-black trees
59. Insertion
RB-Insert(T, z)
1.
y nil[T]
2.
x root[T]
3.
while x nil[T]
4.
do y x
5.
if key[z] < key[x]
6.
then x left[x]
7.
else x right[x]
8.
p[z] y
9.
if y = nil[T]
10.
then root[T] z
11.
else if key[z] < key[y]
12.
then left[y] z
13.
else right[y] z
RB-Insert(T, z) Contd.
14. left[z] nil[T]
15. right[z] nil[T]
16. color[z] RED
17. RB-Insert-Fixup (T, z)
How does it differ from the TreeInsert procedure of BSTs?
Which of the RB properties might
be violated?
Fix the violations by calling RBInsert-Fixup.
60. Insertion – Fixup
● Which property might be violated?
1. OK(Every node is either red or black)
2. If z is the root, then there’s a violation. Otherwise,
OK(The root is black)
3. OK(Every leaf (NIL) is black)
4. If z.p is red, there’s a violation: both z and z.p are
red(If a node is red, then both its children are black)
● OK(For each node, all simple paths from the node to descendant
leaves contain the same number of black nodes)
● Remove the violation by calling RB-INSERT-FIXUP:
61. Insertion – Fixup
● Problem: we may have one pair of consecutive
reds where we did the insertion.
● Solution: rotate it up the tree and away…
Three cases have to be handled…
62. Insertion – Fixup
RB-Insert-Fixup (T, z)
1.
while color[p[z]] == RED
2.
do if p[z] == left[p[p[z]]]
3.
then y right[p[p[z]]]
4.
if color[y] == RED
5.
then color[p[z]] BLACK // Case 1
6.
color[y] BLACK
// Case 1
7.
color[p[p[z]]] RED // Case 1
8.
z p[p[z]]
// Case 1
63. Insertion – Fixup
RB-Insert-Fixup(T, z) (Contd.)
9.
else if z == right[p[z]] // color[y] RED
10.
then z p[z]
// Case 2
11.
LEFT-ROTATE(T, z)
// Case 2
12.
color[p[z]] BLACK
// Case 3
13.
color[p[p[z]]] RED
// Case 3
14.
RIGHT-ROTATE(T, p[p[z]]) // Case 3
15.
else (if p[z] = right[p[p[z]]])(same as 10-14
16.
with “right” and “left” exchanged)
17. color[root[T ]] BLACK
64. Insertion – Fixup
A node z ’after insertion. Because both z and its parent z.p are red, a
violation of property 4 occurs. Since z’s uncle y is red, case 1 in the
code applies. We recolor nodes and move the pointer z up the tree
65. Insertion – Fixup
Once again, z and its parent are both red, but z’s uncle y
is black. Since z is the right child of z.p, case 2 applies.
We perform a left rotation,
66. Insertion – Fixup
Now, z is the left child of its parent, and case 3 applies.
Recoloring and right rotation yield the tree
68. Correctness
Loop invariant:
● At the start of each iteration of the while loop,
■ z is red.
■ If p[z] is the root, then p[z] is black.
■ There is at most one red-black violation:
○ Property 2: z is a red root, or
○ Property 4: z and p[z] are both red.
69. Correctness – Contd.
● Initialization:
● Termination: The loop terminates only if p[z] is
black. Hence, property 4 is OK.
The last line ensures property 2 always holds.
● Maintenance: We drop out when z is the root (since
then p[z] is sentinel nil[T ], which is black). When we
start the loop body, the only violation is of property 4.
■ There are 6 cases, 3 of which are symmetric to the other 3.
We consider cases in which p[z] is a left child.
■ Let y be z’s uncle (p[z]’s sibling).
70. Case 1 – uncle y is red
p[p[z]]
new z
C
C
p[z]
y
A
D
z
A
B
D
B
z is a right child here.
Similar steps if z is a left child.
● 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).
71. Case 1 – uncle y is red
We take the same action whether z is a right child or z is a left child.
Each subtree has a black root, and each has the same black-height.
The code for case 1 changes the colors of some nodes, preserving property
5: all downward simple paths from a node to a leaf have the same number
of blacks. The while loop continues with node z’s grandparent z.p.p as
the new z. Any violation of property 4 can now occur only between the new
z, which is red, and its parent, if it is red as well.
72. Case 2 – y is black, z is a right child
C
C
p[z]
p[z]
y
A
y
B
z
z
B
A
● 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.
73. Case 3 – y is black, z is a left child
B
C
p[z]
y
B
z
A
A
C
● 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.
74. Cases 2 and 3 of the procedure RB-INSERT-FIXUP
As in case 1, property 4 is violated in either case 2 or case 3 because z
and its parent z.p are both red. Each subtree has a black root , and each
has the same black-height. We transform case 2 into case 3 by a left
rotation, which preserves property 5: all downward simple paths from a
node to a leaf have the same number of blacks. Case 3 causes some
color changes and a right rotation, which also preserve property 5. The
while loop then terminates, because property 4 is satisfied: there are no
longer two red nodes in a row.
75. Algorithm Analysis
● O(lg n) time to get through RB-Insert up to the
call of RB-Insert-Fixup.
● Within RB-Insert-Fixup:
■ 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.
■ Note: there are at most 2 rotations overall.
76. Deletion
● Deletion, like insertion, should preserve all the
RB properties.
● The properties that may be violated depends on
the color of the deleted node.
■ Red – OK. Why?
■ Black?
● Steps:
■ Do regular BST deletion.
■ Fix any violations of RB properties that may result.
77. Deletion
RB-Delete(T, z)
1.
if left[z] == nil[T] or right[z] == nil[T]
2.
then y z
3.
else y TREE-SUCCESSOR(z)
4.
if left[y] == nil[T ]
5.
then x left[y]
6.
else x right[y]
7.
p[x] p[y] // Do this, even if x is nil[T]
78. Deletion
RB-Delete (T, z) (Contd.)
8. if p[y] == nil[T ]
9.
then root[T ] x
10. else if y == left[p[y]]
11.
then left[p[y]] x
12.
else right[p[y]] x
13. if y == z
14. then key[z] key[y]
15. copy y’s satellite data into z
16. if color[y] == BLACK
17. then RB-Delete-Fixup(T, x)
18. return y
The node passed to
the fixup routine is the
lone child of the
spliced up node, or
the sentinel.
79. RB Properties Violation
● If the delete node is red?
Not a problem – no RB properties violated
● If y is black, we could have violations of redblack properties:
■ Prop. 1. OK.
■ Prop. 2. If y is the root and x is red, then the root has
become red.
■ Prop. 3. OK.
■ Prop. 4. Violation if p[y] and x are both red.
■ Prop. 5. Any path containing y now has 1 fewer black
80. RB Properties Violation
● Prop. 5. Any path containing y now has 1 fewer black
node.
■ Correct by giving x an “extra black.”
■ Add 1 to count of black nodes on paths containing x.
■ Now property 5 is OK, but property 1 is not.
■ x is either doubly black (if color[x] == BLACK) or red &
black (if color[x] == RED).
■ The attribute color[x] is still either RED or BLACK. No
new values for color attribute.
■ In other words, the extra blackness on a node is by virtue of
x pointing to the node.
● Remove the violations by calling RB-Delete-Fixup.
81. Deletion – Fixup
RB-Delete-Fixup(T, x)
1.
while x root[T ] and color[x] == BLACK
2.
do if x == left[p[x]]
3.
then w right[p[x]]
4.
if color[w] == RED
5.
then color[w] BLACK
// Case 1
6.
color[p[x]] RED
// Case 1
7.
LEFT-ROTATE(T, p[x])
// Case 1
8.
w right[p[x]]
// Case 1
82. RB-Delete-Fixup(T, x) (Contd.)
/* x is still left[p[x]] */
9.
if color[left[w]] == BLACK and color[right[w]] == BLACK
10.
then color[w] RED
// Case 2
11.
x p[x]
// Case 2
12.
else if color[right[w]] == BLACK
13.
then color[left[w]] BLACK
// Case 3
14.
color[w] RED
// Case 3
15.
RIGHT-ROTATE(T,w)
// Case 3
16.
w right[p[x]]
// Case 3
17.
color[w] color[p[x]]
// Case 4
18.
color[p[x]] BLACK
// Case 4
19.
color[right[w]] BLACK
// Case 4
20.
LEFT-ROTATE(T, p[x])
// Case 4
21.
x root[T ]
// Case 4
22.
else (same as then clause with “right” and “left” exchanged)
23. color[x] BLACK
83. Deletion – Fixup
● Idea: Move the extra black up the tree until x points
to a red & black node turn it into a black node,
● x points to the root just remove the extra black, or
● We can do certain rotations and recolorings and
finish.
● Within the while loop:
■ x always points to a nonroot doubly black node.
■ w is x’s sibling.
■ w cannot be nil[T ], since that would violate property 5 at
p[x].
● 8 cases in all, 4 of which are symmetric to the other.
84. Case 1 – w is red
p[x]
B
w
x
A
D
B
D
x
C
E
A
E
new
w
C
● w must have black children.
● Make w black and p[x] red (because w is red p[x] couldn’t have
been red).
● Then left rotate on p[x].
● New sibling of x was a child of w before rotation must be
black.
● Go immediately to case 2, 3, or 4.
85. Case 2 – w is black, both w’s children
p[x]
are black new x
c
B
w
x
A
B
D
A
C
E
c
D
C
E
● Take 1 black off x ( singly black) and off w ( red).
● Move that black to p[x].
● Do the next iteration with p[x] as the new x.
● If entered this case from case 1, then p[x] was red new x is
red & black color attribute of new x is RED loop
terminates. Then new x is made black in the last line.
86. Case 3 – w is black, w’s left child is
c
red, w’s right child is black
c
B
w
x
A
B
x
D
A
C
E
new w
C
D
E
● Make w red and w’s left child black.
● Then right rotate on w.
● New sibling w of x is black with a red right child case 4.
87. Case 4 – w is black, w’s right child is
red
c
B
w
x
A
D
B
D
C
c’
x
E
E
A
C
● Make w be p[x]’s color (c).
● Make p[x] black and w’s right child black.
● Then left rotate on p[x].
● Remove the extra black on x ( x is now singly black) without
violating any red-black properties.
● All done. Setting x to root causes the loop to terminate.