SUPERIOR UNIVERSITY LAHORE
Faculty of Computer Science & IT
Design and Analysis of Algorithm
Report: AVL Tree
[Design and Analysis of Algorithm]
Group Members
Student Name
Student ID
Class
Section
Muhammad Nawaz
BCSM-F18-022
BSCS
5B
Abubakar Mehboob
BCSM-F18-041
BSCS
5B
Ahsan Naseer
BCSM-F18-031
BSCS
5B
H. Abdullah Ansari
BCSM-F18-452
BSCS
5B
Teacher: [Dr. Imran Khan (PhD. Scholar)]
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Contents
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History-………………………………………………………………………………………… 3
Abstract………………………………………………………………………………………… 3
Introduction…………………………………………………………………………………… 1-4
Variations of AVL Tree…………………………………………………………………… 4
o RAVL……………………………………………………………………………………… 4
o WAVL…………………………………………………………………………………… 4
How AVL tree Work………………………………………………………………………… 5
o Balancing the Tree………………………………………………………………… 5
o Rotations………………………………………………………………………………. 5
▪ Left Rotation………………………………………………………………. 6
▪ Right Rotation……………………………………………………………. 6
▪ Left-Right Rotation……………………………………………………. 7
▪ Right-Left Rotation……………………………………………………. 7
o Operations……………………………………………………………………………8
▪ Searching………………………………………………………………….… 8
▪ Insertion……………………………………………………………………. 8-9
▪ Deletion………………………………………………………………….…. 9-10
Why we use AVL Tre……………………………………………………………….………. 10
o Searching………………………………………………………………………….……. 10
o Storing……………………………………………………………………………………. 10
Advantages……………………………………………………………………………………… 11
Current Application………………………………………………………………….…… 11-12
Algorithm………………..……………………………………………………………………. ….. 12
Conclusion……………………………………………………………………………………. ….. 12
References……………………………………………………………………………….……… 13
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History of AVL Tree:
It was introduced in 1962. The AVL tree is named after its two Soviet inventors, Georgy
Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm
for the organization of information". that is, sibling nodes can have hugely differing
numbers of descendants.
Abstract:
AVL tree is the first dynamically balancing binary search tree that are to be invented. It is used
in many different areas in the real world that we are interacting today with. This paper discusses
the variations and methodologies of AVL tree, the basic operations of AVL tree such as
insertion, deletion and searching, the purpose of AVL tree including what can be used to
solve with AVL tree, the advantages and disadvantages of AVL tree and future, current
application of AVL tree
Introduction:
AVL tree is a “height balancing binary search tree and is also known as self-balancing tree”.
AVL tree is the first dynamically balanced trees to be proposed. In 1962, Adelson Velski &
Lendis, the two creators of AVL tree published the concept of AVL tree in the paper “An
algorithm for the organization of information”. Hence, it was given the name AVL. In this
paper, it only described the algorithm on rebalancing the tree after an insertion and updating
the height of the tree. An algorithm on rebalancing after deletion was lacking and it only
appeared later in 1965 when another author submitted a technical report on it. In AVL tree,
the height of the two child subtree nodes can only differ by 1 and this is known as the
property of AVL tree. Every node maintains an extra information known as “Balance
Factor”.
Binary Search Tree (BST) is sometime skewed because it cannot control the order in
which data comes for insertion. Similarly, AVL tree also cannot control the order in which
data arrives for insertion, but it can re-arrange data in tree. All major operations on a Binary
Search Tree including insertion, deletion and searching depend linearly on its height. For
both average and worst cases, the time complexity for performing insert, search and delete
operations in AVL tree is O (log n) where n is the number of nodes in the tree. The insert and
delete operations is slower and tedious as rotations are required to re-balance the tree.
Therefore, AVL tree are not preferred for applications that involves frequent insert and delete
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operations. In 1973, an experiment was conducted by Scoggs to investigate the performance
of AVL tree and to compare the actual costs and timings involved in performing the basic
operations. It was found out that for large trees, searching is most expensive follow by
deletion then by insertion (Scoggs, 1973).
AVL tree is often compared with red-black trees as the time complexity is O (log n) for
both. Red, Black (RB) trees are also balanced, but comparatively less balanced then AVL
tree. For this reason, AVL tree appeared to be faster than red-black tree in search operations.
Although, AVL tree appear to be more objective when compare with RB tree, but it causes
more rotations for insertion and deletion. Moreover, the RB tree does not need to be rebalanced as frequently as AVL tree.
Variations of AVL tree
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Relaxed AVL tree (RAVL)
Weak AVL tree (WAVL)
Relaxed AVL tree (RAVL):
The first variation of AVL tree is RAVL tree (relaxed AVL tree) and is only re-balanced after
insertion, but not after deletion. Thus, deletion can create trees that are not AVL tree. The
time complexity in RAVL tree for performing insert, search and deletion operations is O(h
+1), where h Is the height of the tree
Weak AVL tree (WAVL):
The second variation of AVL tree is WAVL tree (weak AVL tree). This new kind of balanced
binary tree was proposed by Haupler, San & Tarjan in their paper “Rank-Balanced Tree”
in 2015. WAVL is designed to combine the properties of AVL tree and Red Black (RB) tree.
If no deletion occurs, WAVL tree is the same as the RAVL tree. The time complexity in WAVL
tree for performing insert and deletion operations at most 2 operations take O (1). Most of the
re-balancing in WAVL take place at the bottom of the tree
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How AVL Tree Work?
Balance Factor
For every insert, delete and search operations, the depth of the balance tree is equivalent to
O(log N), take O(log N) time and every node must have left and right subtrees of the same
height. By default, nodes with no children have a height of 0.
AVL tree checks the height of the left and right subtree and ensure that the height or of the tree
is not greater than 1. “The difference between the height of left and right subtree of the node is
known as the balance factor”. The balance factor for every node is either -1,0 or 1. The balance
factor of the tree is denoted as follows.
Balance Factor = heightOfLeftSubtree - heightOfRightSubtree
These three diagrams above illustrates balance and unbalanced trees. In figure 1, the tree is
equally balanced as the balance factor is 0. In Figure 2 & 3, the tree is unbalanced because
the balance factor is 2 as illustrated above. In AVL trees, if the balance factor is greater than
1, the three need to be balance by using one of the four rotation techniques
Rotations
There are four rotations
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Left Rotation
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Right Rotation
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Left-Right Rotation
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Right Left Rotation
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Left Rotation
When a node is inserted into the right subtree of the right subtree and the tree become
unbalanced and single left rotation is performed. A simple diagram will be illustrated to explain
this process.
Figure 4 represent the right unbalanced tree. The balance factor of this tree is 2, thus it needs
to be rotated. In Figure 5, left Rotation is performed by making 2 the left subtree of 1. Figure
6 represents the balanced tree after left rotation.
Right Rotation
Single right rotation required when a node is inserted into the left subtree of the left subtree
and the tree become unbalanced. A simple diagram will be illustrated to explain this process.
Figure 7 represent the left unbalanced tree. The balance factor of this tree is 2, thus it needs to
be rotated. In Figure 8, left Rotation is performed by making 1 the left subtree of 2. Figure 9
represents the balanced tree after right rotation
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Left-Right Rotation
The first type of double rotation is a left-right rotation. “Left-Right rotation is a combination
of left rotation followed by right rotation”. A simple diagram will be illustrated to explain
this process.
In Figure-10, Z is the root node and node X is a left subtree of Z and node Y is a right subtree
of X. The tree is not balanced as the balance factor of the tree is 2. For this scenario, a double
rotation is required. Firstly, left rotation is performed by making Y the left subtree of Z and
making X the left subtree of Y as shown in Figure-11. Then, right rotation is performed. Figure12 show the balanced tree after performing left-right rotation.
Right Left Rotation
The second type of double rotation is a right-left rotation. “Right-Left rotation is a
combination of right rotation followed by left rotation”. A simple diagram will be illustrated
to explain this process.
In Figure-11, Z is the root node and node X is a right subtree of Z and node Y is a left subtree
of X. The tree is not balanced as the balance factor of the tree is 2. For this scenario, a double
rotation is required. Firstly, right rotation is performed by making Y the right subtree of Z and
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making X the right subtree of Y as shown in Figure-13. Then, right rotation is performed.
Figure-14 show the balanced tree after performing left-right rotation.
Operations
Searching
The time complexity for search operation in AVL tree is O(log n). The search operation
is AVL tree is similar as search in binary search tree. This means that all descendants to the
right of a node are greater than the node and all descendants to the left of a node are less than
the node. Steps to perform search operations in AVL tree will be described below.
Firstly, the element to search provided by the user is read. Then, this element is compare with
the root node value in the tree. If it matches, this value will be return to the user. If the values
do not match, compare search element with the root node value. If search element is greater
than the node value, continue the search process in right subtree and if not, continue from the
left subtree. Repeat the process until the search element is found or completed with a leaf node.
(Sathya, 2017).
Insertion
The time complexity for insert operation in AVL tree is O (log n). While inserting a
new number, insert operation may cause a balance factor of a node to become 2 or -2 and
required rotations to re-adjust the nodes and re-balance the tree. In AVL tree, new node
is inserted as a leaf and the insert will be done recursively. The procedure for insertion is
described below.
The element is inserted into the tree according to the concept of Binary Search Tree (BST).
The difference with BST is that the balance factor of every node is check after every insertion
to ensure that the tree is balanced. The next element will only be added if the balance factor is
either -1, 0 or 1
Generally, there are 4 cases for insertions (Poonia, 2014)). These are.
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Outside Cases (only single rotation is required)
o Insertion into left-subtree of left child
o Insertion into right-subtree of right child
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Inside Cases (double rotation is required
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o Insertion into right-subtree of left child
o Insertion into left-subtree of right child
Example
A simple example will be illustrated below with a construction of an AVL tree by inserting
number 1,2,3,4,5,6 in order.
In A, the tree become unbalanced after inserting 3. As 3 is inserted into right-subtree of the
right child, a single left rotation is performed to re-balance the tree. In C, thee tree become
unbalanced after inserting 5. As 5 is also inserted into right-subtree of the right child, a single
left rotation is performed to re-balance the tree. F represents the balanced tree after inserting
number 1 to 6 in order.
Deletion
The delete operation is more complex than the insert operation. The element is deleted into
the tree according to the concept of Binary Search Tree (BST). The difference with BST is
that the balance factor of every node is check after every delete operation to ensure that the
tree is balanced.
There are 3 cases to consider before performing the delete operation. Let ‘Y” be the
element to delete.
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Case 1 → if “Y” is a leaf, delete Y. (Figure-1)
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Case 2 → if Y has one child, use the child to replace the Y. (Figure-2)
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Case 3 → if Y have two children, replace Y with its in-order predecessor
(right most child of left-subtree) and delete recursively. (Figure-3)
Why we use AVL tree ?
Searching
AVL tree can be used for many purposes. The first and most common usage of AVL tree is in
search intensive applications as searching takes O (log n) in AVL tree as the tree is balanced.
AVL tree is used to create index of search keywords for a bunch of documents (see current
application of AVL tree). As an example, it is used in designing databases where insertions
and deletions are not frequent but require frequent search operations for items present in
there. AVL tree is also implemented in creating search engines to provide faster search
results. This is because search engine requires to response to query as fast as possible. As an
example, search engine in fingerprint databases used AVL tree as the fingerprints discovered
in crime scenes need to be compared and searched faster in fingerprint databases in a more
reliable nature. (Elmadani, 2016).
Storing
Another usage of AVL tree is for storing information in an efficient and sorted manner.
As an example, an AVL tree can be coded together with hash table for storing and retrieving
data. AVL tree can also be implemented to store data read from an input file. Another important
usage of AVL tree is in classification function of data mining.
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Advantages
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The first advantage of AVL tree is that it provides faster searching operations
compare to other counterparts such as red-black tree. This allow the user to complete
their task much faster than other search operations.
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This is vital in coding to ensure that projects are completed according to schedule. In
Addition, it is essential when user needs to complete faster searches and in a more
reliable nature.
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AVL tree has the capability of performing searching, insertion and deletion in a more
efficient manner when compare with binary search tree.
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The second advantage of AVL tree is due to its self-balancing capabilities. One of the
concerns for professional is to ensure that that the trees are balanced. This is because
if the tree is not balanced, the time to perform operations such as searching, inserting,
and deleting will be longer. (Nyln, 2016).
Current Application
The following are the current applications of AVL tree today.
AVL tree is mostly used in databases for indexing large records to improve searching.
Another application of AVL tree is in memory management subsystem of Linux kernel to
search memory. The next application of AVL tree is in dictionary search engines. Data
structure that may be built once without further reconstruction such as language dictionaries
or program dictionaries uses AVL tree. It can be seen that AVL tree is commonly used in
searching as searching in AVL tree takes O (log n) only.
Another important use of AVL tree is in data mining. Classification is one of the main
functions of data mining. It can be used to solve the issue arise with data mining. The concern
with data mining in large databases is scalability and efficiency. Decisions trees are widely
used with classification method. As classification model construction process is performed on
huge data, the algorithms may result in very bushy or meaningless results
Biometric authentication such as fingerprint can be used to verify a person and in crime
scene. Sets of fingerprints are searched in large databases and compared in real time. To
reduce consuming time, this require fast search engine in searching or retrieving from large
fingerprint databases. Experiments have been conducted to find out the best searching
methods among queue, SQL, graph search, hash, binary and AVL tree as methods for
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searching and retrieving from large databases According to results, it was found out the AVL
tree algorithm is the best method for searching (Elmadani, 2016).
Algorithm:
Conclusion
Although the features offered by AVL tree such as fast searching and O (log N) time
complexity for insert and delete operations may appeared appealing, it is not widely used today.
This is due to its costly insertion and deletion that required frequent rotations, and harder in
coding, debugging, and implementing, AVL tree is replaced with red-black tree.
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References
Algorithm & Data Structures Note 5. (N.D) Retrieved from,
https://www.inf.ed.ac.uk/teaching/courses/inf2b/algnotes/note05.pdf
Shafiq,H. (2015). C#Corner : How a search engine works. Retrieved from http://www.csharpcorner.com/UploadFile/e68d54/search-engine-using-2-3-trees-and-hash-table/
Elmadani,A. (2016). AVL TREE AN EFFICIENT RETRIEVAL ENGINE IN CLASSIFIED FINGERPRINT
DATABASE. International Educational Applied Scientific Research Journal, 1 (1), 30-35.
Retrieved from http://ieasrj.com/journal/index.php/ieasrj/article/view/11/9
Grossman,D. (2010). Lecture 8: AVL Delete; Memory Hierarchy[Powerpoint Slides] Retrieved
from https://courses.cs.washington.edu/courses/cse332/10sp/lectures/lecture8.pdf
Bhukya,D., & Ramachandram,S. (2010). Decision Tree Induction: An Approach for Data
Classification Using AVL-Tree. International Journal of Computer and Electrical Engineering,
2(4), 660-665. Retrieved from http://www.ijcee.org/papers/208-E269.pdf
Haeupler,B., Sen,S., & Tarjan,R . (2013). Rank Balanced Tree:Remarks. ACM Transactions on
Algorithms 1-21. Retrieved from
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.296.7295&rep=rep1&type=pdf
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