The document describes a course on advanced data structures. It provides information on the instructor, teaching assistant, topics to be covered including AVL and Red Black Trees, objectives of learning deletions, insertions and searches in logarithmic time. It also lists credits to other professors and researchers. The document then goes into details about balanced binary search trees, describing properties of AVL Trees and Red Black Trees to ensure the tree remains balanced during operations.

1. CS 6213 – Advanced Data Structures

2.  Instructor
Prof. Amrinder Arora
amrinder@gwu.edu
Please copy TA on emails
Please feel free to call as well
 TA
Iswarya Parupudi
iswarya2291@gwmail.gwu.edu
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3. Record / Struct
Basics
Arrays / Linked
Lists / Stacks /
Queues
Graphs / Trees
/ BSTs
Trie, B-Tree
CS 6213
Advanced
Splay Trees
Union Find
Generics
Applications
Multi Threading
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4.  Objectives
 Deletions, Insertions, Searches, all in log n time
 Credits:
 Prof. Roger Crawfis (Ohio State)
 Sarah Buchanan (UCF)
 Rafee Memon (CMU)
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5.  Requiring balancing at the root is
not enough.
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6.  Impose some properties (different properties based on different
implementations)
 Perform rotations (localized operations) that maintain BST
property and adjust the height
 Left Rotation, Right Rotation
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7. Left-Rotate(T, x)
x

Right-Rotate(T, y)
y

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y


x


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8.  Rotations are the basic tree-restructuring operation for almost all
balanced search trees.
 Rotation takes a BST and a node,
 Changes pointers to change the local structure, and
 Won’t violate the binary-search-tree property.
 Left rotation and right rotation are inverses.
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9. 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
Left-Rotate(T, x)
y
x
8.
else if x = left[p[x]]


Right-Rotate(T, y) x
y
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
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10.  The pseudo-code for Left-Rotate assumes that
 right[x]  nil[T ], and
 root’s parent is nil[T ].
 Left Rotation on x, makes x the left child of y, and
the left subtree of y into the right subtree of x.
 Pseudocode for Right-Rotate is symmetric:
exchange left and right everywhere.
 Time: O(1) for both Left-Rotate and Right-Rotate,
since a constant number of pointers are modified.
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11.  Invented by Adelson-Velskii and Landis.
 A binary search tree with a balance condition.
 Easy to maintain.
 Ensures the depth is O(log n)
 Balance condition – for every node in the tree, the height of the
left and right subtrees can differ by at most 1.
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12.  Definition: A tree rooted at node v with L and R as left and right
subtrees is an AVL tree if:
 | height(L) - height(R)| ≤ 1
 L is an AVL tree; and
 R is an AVL tree.
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13.  Are these AVL trees?
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14. 5
7
8
2
1
4
7
8
2
1
3
4
3
5
 We note that both of them are still binary search
trees, although not both are AVL trees
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15.  Mathematical definition: A tree rooted at node n is an AVL tree if:
 |height(L) - height(R)| ≤ 1; and
 L is an AVL tree; and
 R is an AVL tree.
 Using this definition, how can we
check if a tree is a valid AVL tree?
 Definition is recursive,
so how about recursively?
 How about bottom up traversal?
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16.  public boolean isAVL:
For a tree rooted at node n:
 Base case?
 Could check for leaf node, but for elegance…
 n is null → for simplicity, return valid
 Constraint?
 Check that heights of sub-trees are within 1 of each other (assume we
have a getHeight function)
 Recursive case?
 Check that left and right sub-trees are AVL trees
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17. public boolean isAVL(Node n)
{
if (n == null) return true; // base case
int heightL = getHeight(n.leftChild);
int heightR = getHeight(n.rightChild);
int heightDiff = Math.abs(heightL – heightR);
return (isAVL(n.leftChild)
&& isAVL(n.rightChild)
&& heightDiff <= 1);
}
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18.  When we do an Insertion
 Update all the balancing information for the nodes on the path back
to the root.
 As we are updating the balance information on the path up to the root, we
may find a node whose new balance violates the AVL condition.
 What if the insertion violated the AVL tree property?
 We do a simple modification to the tree known as a rotation.
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19.  4 cases, but two are symmetrical with the other two
 1) Left outer subtree ↔ 4) Right outer subtree
 2) Left inner subtree ↔ 3) Right inner subtree
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20.  Let us call the node that must be rebalanced α
 Since any node has at most 2 children, and a
height imbalance requires that α’s 2 subtrees’
height differ by 2, there are 4 violation cases:
1)
An insertion into the left subtree of the left
child of α.
2)
An insertion into the left subtree of the right
child of α.
4)
An insertion into the right subtree of the right
child of α.
α
An insertion into the right subtree of the left
child of α.
3)
α
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α
α
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21. 
Outside cases (left-left or right-right), fixed by a single rotation:
1) An insertion into the left subtree of the left child of α.
4) An insertion into the right subtree of the right child of α.
6
6
α
8
2
2
7
α
1
5
7
8
9
4
3
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22. 
Inside cases (right-left or left-right), fixed by a double rotation:
2) An insertion into the right subtree of the left child of α.
3) An insertion into the left subtree of the right child of α.
7
6
α
9
2
2
7
α
1
6
8
9
8
4
5
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23.  We want X up a level and Z down a level.
 We want k2 as the new root.
 Pretend the tree is flexible, grab hold of k2, letting gravity take hold.
 k3 > k2 (by BST property), so k3 becomes k2’s right subtree.
 X and Z can remain attached to the same point,
 and Y becomes k3’s subtree to maintain BST property.
k
k
3
2
k
k
Z
2
k
1
3
X
1
X
k
Y
Y
Z
X
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24.  Symmetric case
k
k
2
1
k
X
k
1
k
3
2
k
Y
3
Z
Y
Z
Z
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25.  Place k2 as the new root.
 This forces k1 to be k2’s left child and k3 to be its right child.
 Then the 4 subtrees A,B,C,D fall according to BST rules.
k
k
3
k
3
2
k
k
Z
1
1
k
X
2
Y
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k
k
1
D
3
k
A
A
2
B
B
C
D
C
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26.  A Double Rotation is like 2 single rotations.
 Example: Left-right double rotation.
k
Single Left
Rotation at K1
3
k
D
2
k
k
2
A
3
k
D
1
k
1
B
C
A
B
C
k
2
k
1
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3
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B
C
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D

27.  In an insert there is at most one node that needs to be
rebalanced.
 But in a delete there may be multiple nodes to be rebalanced.
 Technically only one rebalance that happens at a node, but once that
happens it may affect the ancestral nodes.
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28.  If the node is a leaf, remove it.
 Retrace back up the tree starting with the parent of deleted node to
the root, adjusting the balance factor as needed.
 If it’s not a leaf, replace with the largest in its left subtree
(inorder predecessor) and remove that node.
 You could also replace with the smallest in its right subtree (inorder
successor)
 If the predecessor was not a leaf, you may need to adjust the
pointers.
 After deletion, retrace the path back up the tree, starting with
the parent of the replacement, to the root, adjusting the balance
factor as needed.
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29. 32
16
8
48
24
40
28
36
56
44
52
60
58
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30. 32
16
48
24
40
28
36
56
44
52
60
58
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31. 32
24
16
48
28
40
36
56
44
52
60
58
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32. 48
32
56
24
16
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52
40
28
36
44
60
58
62
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33.  Actual Implementation for Binary Search Trees
 O(log n) average height – Why is that? (We didn’t prove it for
sure.)
 If we initially have an AVL tree:
 Insertion requires at most 2 rotations
 Deletion requires at most O(log n) rotations
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34.  A variation of binary search trees to ensure that the tree is
somewhat balanced.
 Height is O(lg n), where n is the number of nodes.
 Operations take O(lg n) time in the worst case.
 Red-Black Tree Properties:
 Every node is either red or black.
 The root is black.
 Every leaf (nil) is black.
 If a node is red, then both its children are black.
 For each node, all paths from the node to descendant leaves contain
the same number of black nodes.
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35.  Binary search tree + 1 bit per node: the attribute color, which is
either red or black.
 All other attributes of BSTs are inherited:
 key, left, right, and p.
 All empty trees (leaves) are colored black.
 We use a single sentinel, nil, for all the leaves of red-black tree T, with
color[nil] = black.
 The root’s parent is also nil[T]
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36. 26
Remember: every internal
node has two children, even
though nil leaves are not
usually shown.
41
17
30
47
38
50
nil[T]
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37.  Height of a node:
 h(x) = number of edges in a longest path to a leaf.
 Black-height of a node x, bh(x):
 bh(x) = number of black nodes (including nil[T ])
on the path from x to leaf, not counting x.
 Black-height of a red-black tree is the black-height of its root.
 By Property 5, black height is well defined.
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38. h=4
26 bh=2
 Height of a node:
h(x) = # of edges in a
longest path to a
leaf.
 Black-height of a node bh(x) = #
17
h=1
bh=1
h=3
41 bh=2
of black nodes on path from x to
leaf, not counting x.
 How are they related?
 bh(x)
h=2 30
bh=1
≤ h(x) ≤ 2 bh(x)
h=1
bh=1
38
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nil[T]
h=2
47 bh=1
h=1 50
bh=1
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39. Consider a node x in an RB tree: The longest descending path from
x to a leaf has length h(x), which is at most twice the length of the
shortest descending path from x to a leaf.
Proof:
# black nodes on any path from x = bh(x) (prop
5)
 # nodes on shortest path from x, s(x). (prop 1)
But, there are no consecutive red (prop 4),
and we end with black (prop 3), so h(x) ≤ 2
bh(x).
Thus, h(x) ≤ 2 s(x). QED
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40.  Lemma 13.1: A red-black tree with n internal nodes has height at
most
2 lg(n+1).
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41. 1.
2.
3.
4.
5.
Every node is either red or black.
The root is black.
Every leaf (nil) is black.
If a node is red, then both its children are black.
For each node, all paths from the node to
descendant leaves contain the same number of
black nodes.
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42.  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.
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43.  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…
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44. p[p[z]]
new z
C
C
p[z]
y
A
D
z


A

D


B


z is a right child here.
Similar steps if z is a left child.

B


 p[p[z]] (z’s grandparent) must be black, since z and p[z] are both red.
 Make p[z] and y black
 Make p[p[z]] red
 now z and p[z] are not both red.
 restores property 5.
 The next iteration has p[p[z]] as the new z (i.e., z moves up 2 levels).
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45. B
C
p[z]
z

A

A
 y
B

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
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 no more iterations.
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46. C
C
p[z]
p[z]
 y
A
 y
B
z

z
B



 Left rotate around p[z], p[z] and z switch roles

A

 now z is a left child, and both
z and p[z] are red.
 This now becomes case 2.
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47.  Symmetric to cases 2 and 3 in that parent of z is right child.
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48.  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.
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49.  Deletion, like insertion, should preserve all the RB properties.
 First do regular BST deletion
 Find the node
 If not a leaf, find the predecessor
 If predecessor is a node with a single child, then adjust the pointers
 Now, we need to fix any violations of RB properties that may
result.
 The properties that may be violated depends on the color of the
deleted node.
 Red – We are lucky.
 Black? We need to consider 3 cases
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50.  The parent is red, and the children of sibling node are black
Recolor parent to black and sibling to red
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51.  The parent is black, and the children of sibling node are black
 Unlike Case 1, we cannot finish in one move, but the problem can
be shifted “upwards”. B is now “double black”.
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52.  The parent is red, and the children of sibling node are not both black
 We can do a rotation
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53.  Deletion of elements in array A
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54.  Review ArrayList.java source code of Java.
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