DAA Introduction - in Chapter - I and lecture

DAA Introduction

1. Unit - 1Unit - 1
Design and Analysis of AlgorithmDesign and Analysis of Algorithm
INTRODUCTIONINTRODUCTION
Dr. K. RANJITH SINGHDr. K. RANJITH SINGH
Assistant ProfessorAssistant Professor
Department of Computer ScienceDepartment of Computer Science
Karpagam Academy of Higher Education Karpagam Academy of Higher Education
(Deemed to be University)(Deemed to be University)
CoimbatoreCoimbatore

2. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-2
What is an algorithm?What is an algorithm?
An An algorithmalgorithm is a sequence of unambiguous instructions is a sequence of unambiguous instructions
for solving a problem, i.e., for obtaining a required for solving a problem, i.e., for obtaining a required
output for any output for any legitimatelegitimate input in a finite amount of input in a finite amount of
time.time.
“computer”
problem
algorithm
input output

3. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-3
AlgorithmAlgorithm

An An algorithmalgorithm is a sequence of unambiguous is a sequence of unambiguous
instructions for solving a problem, i.e., for instructions for solving a problem, i.e., for
obtaining a required output for any legitimate obtaining a required output for any legitimate
input in a finite amount of time.input in a finite amount of time.
• Can be represented various forms
• Unambiguity/clearness
• Effectiveness
• Finiteness/termination
• Correctness

4. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-4
Historical PerspectiveHistorical Perspective

Euclid’s algorithm for finding the greatest common divisorEuclid’s algorithm for finding the greatest common divisor

Muhammad ibn Musa al-Khwarizmi – 9Muhammad ibn Musa al-Khwarizmi – 9
thth
century century
mathematician mathematician
www.lib.virginia.edu/science/parshall/khwariz.htmlwww.lib.virginia.edu/science/parshall/khwariz.html

5. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-5
Notion of algorithm and problemNotion of algorithm and problem
“computer”
algorithmic solution
(different from a conventional solution)
problem
algorithm
input
(or instance)
output

6. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-6
Example of computational problem: sortingExample of computational problem: sorting

Statement of problem:Statement of problem:
•Input:Input: A sequence of A sequence of nn numbers <a numbers <a
11, ,
aa
22, …, a, …, a
nn>>
•Output:Output: A reordering of the input sequence <a A reordering of the input sequence <a
´´
11, ,
aa
´´
22, …, a, …, a
´´
nn> so that > so that
aa
´´
ii ≤≤ aa
´´
jj whenever whenever ii < < jj

Instance: The sequence <5, 3, 2, 8, 3>Instance: The sequence <5, 3, 2, 8, 3>
Algorithms:Algorithms:
•Selection sortSelection sort
•Insertion sortInsertion sort
•Merge sortMerge sort
•(many others)(many others)

7. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-7
Selection SortSelection Sort

Input: array Input: array a[1],…,a[n]a[1],…,a[n]

Output: array Output: array aa sorted in non-decreasing order sorted in non-decreasing order

Algorithm:Algorithm:
for i=1 to n
swap a[i] with smallest of a[i],…,a[n]
• Is this unambiguous? Effective?
• See also pseudocode, section 3.1

8. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-8
Some Well-known Computational ProblemsSome Well-known Computational Problems

SortingSorting

SearchingSearching

Shortest paths in a graphShortest paths in a graph

Minimum spanning treeMinimum spanning tree

Primality testingPrimality testing

Traveling salesman problemTraveling salesman problem

Knapsack problemKnapsack problem

ChessChess

Towers of HanoiTowers of Hanoi

Program terminationProgram termination
Some of these problems don’t have efficient algorithms,
or algorithms at all!

9. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-9
Basic Issues Related to AlgorithmsBasic Issues Related to Algorithms
How to design algorithmsHow to design algorithms

How to express algorithmsHow to express algorithms
Proving correctnessProving correctness
Efficiency (or complexity) analysisEfficiency (or complexity) analysis
•Theoretical analysisTheoretical analysis
•Empirical analysisEmpirical analysis
Optimality Optimality

10. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-10
Algorithm design strategiesAlgorithm design strategies

Brute forceBrute force

Divide and conquerDivide and conquer

Decrease and conquerDecrease and conquer

Transform and conquerTransform and conquer

Greedy approachGreedy approach

Dynamic programmingDynamic programming

Backtracking and branch-and-boundBacktracking and branch-and-bound

Space and time tradeoffsSpace and time tradeoffs

11. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-11
Analysis of AlgorithmsAnalysis of Algorithms

How good is the algorithm?How good is the algorithm?
•CorrectnessCorrectness
•Time efficiencyTime efficiency
•Space efficiencySpace efficiency

Does there exist a better algorithm?Does there exist a better algorithm?
•Lower boundsLower bounds
•OptimalityOptimality

12. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-12
What is an algorithm?What is an algorithm?

Recipe, process, method, technique, procedure, routine,… Recipe, process, method, technique, procedure, routine,…
with the following requirements:with the following requirements:
1.1.FinitenessFiniteness
terminates after a finite number of stepsterminates after a finite number of steps
2.2.DefinitenessDefiniteness
rigorously and unambiguously specifiedrigorously and unambiguously specified
3.3.Clearly specified inputClearly specified input
valid inputs are clearly specifiedvalid inputs are clearly specified
4.4.Clearly specified/expected outputClearly specified/expected output
can be proved to produce the correct output given a valid inputcan be proved to produce the correct output given a valid input
5.5.EffectivenessEffectiveness
steps are sufficiently simple and basicsteps are sufficiently simple and basic

13. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-13
Why study algorithms?Why study algorithms?

Theoretical importanceTheoretical importance
•the core of computer sciencethe core of computer science

Practical importancePractical importance
•A practitioner’s toolkit of known algorithmsA practitioner’s toolkit of known algorithms
•Framework for designing and analyzing algorithms for new Framework for designing and analyzing algorithms for new
problemsproblems
Example: Google’s PageRank Technology

14. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-14
Euclid’s AlgorithmEuclid’s Algorithm
Problem: Find gcd(Problem: Find gcd(m,nm,n), the greatest common divisor of two ), the greatest common divisor of two
nonnegative, not both zero integers nonnegative, not both zero integers m m and and nn
Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ?
Euclid’s algorithm is based on repeated application of equalityEuclid’s algorithm is based on repeated application of equality
gcd(gcd(m,nm,n) = gcd() = gcd(n, m n, m mod mod nn))
until the second number becomes 0, which makes the problemuntil the second number becomes 0, which makes the problem
trivial.trivial.
Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12

15. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-15
Two descriptions of Euclid’s algorithmTwo descriptions of Euclid’s algorithm
Step 1 If Step 1 If nn = 0, return = 0, return mm and stop; otherwise go to Step 2 and stop; otherwise go to Step 2
Step 2 Step 2 Divide Divide mm by by n n and assign the value of the remainder toand assign the value of the remainder to r r
Step 3 Assign the value of Step 3 Assign the value of n n to to mm and the value of and the value of rr to to n. n. Go toGo to
Step 1. Step 1.

whilewhile nn ≠ 0 ≠ 0 do do
r ← m r ← m mod mod nn
m← n m← n
n ← rn ← r
returnreturn mm

16. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-16
Other methods for computing gcd(Other methods for computing gcd(m,nm,n))
Consecutive integer checking algorithmConsecutive integer checking algorithm
Step 1 Assign the value of min{Step 1 Assign the value of min{m,nm,n} to } to tt
Step 2 Step 2 Divide Divide mm by by t. t. If the remainder is 0, go to Step 3;If the remainder is 0, go to Step 3;
otherwise, go to Step 4 otherwise, go to Step 4
Step 3 Step 3 Divide Divide nn by by t. t. If the remainder is 0, return If the remainder is 0, return tt and stop; and stop;
otherwise, go to Step 4 otherwise, go to Step 4
Step 4 Decrease Step 4 Decrease t t by 1 and go to Step 2by 1 and go to Step 2
Is this slower than Euclid’s algorithm?
How much slower?
O(n), if n <= m , vs O(log n)

17. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-17
Other methods for gcd(Other methods for gcd(m,nm,n) [cont.]) [cont.]
Middle-school procedureMiddle-school procedure
Step 1 Find the prime factorization of Step 1 Find the prime factorization of mm
Step 2 Find the prime factorization of Step 2 Find the prime factorization of nn
Step 3 Find all the common prime factorsStep 3 Find all the common prime factors
Step 4 Compute the product of all the common prime factorsStep 4 Compute the product of all the common prime factors
and return it as gcd and return it as gcd(m,n(m,n))
Is this an algorithm?Is this an algorithm?
How efficient is it?How efficient is it?
Time complexity: O(sqrt(n))

18. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-18
Sieve of EratosthenesSieve of Eratosthenes
Input: Input: Integer Integer n n ≥≥ 2 2
Output: List of primes less than or equal to Output: List of primes less than or equal to nn
for for p p ← 2← 2 to to nn do do AA[[pp] ← ] ← pp
for for p p ← 2← 2 to to nn do do
if if AA[[pp] ]  0 // 0 //p p hasn’t been previously eliminated from the listhasn’t been previously eliminated from the list
j j ← ← pp** pp
while while j j ≤≤ n n dodo
AA[[jj] ] ← 0← 0 //mark element as eliminated//mark element as eliminated
j j ← ← jj + p+ p
Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time complexity: O(n)

19. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-19
Two main issues related to algorithmsTwo main issues related to algorithms

How to design algorithmsHow to design algorithms

How to analyze algorithm efficiencyHow to analyze algorithm efficiency

20. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-20
Algorithm design techniques/strategiesAlgorithm design techniques/strategies

Brute forceBrute force

Divide and conquerDivide and conquer

Decrease and conquerDecrease and conquer

Transform and conquerTransform and conquer

Space and time tradeoffsSpace and time tradeoffs

Greedy approachGreedy approach

Dynamic programmingDynamic programming

Iterative improvementIterative improvement

Backtracking Backtracking

Branch and boundBranch and bound

21. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-21
Analysis of algorithmsAnalysis of algorithms

How good is the algorithm?How good is the algorithm?
•time efficiencytime efficiency
•space efficiencyspace efficiency
•correctness ignored in this coursecorrectness ignored in this course

Does there exist a better algorithm?Does there exist a better algorithm?
•lower boundslower bounds
•optimalityoptimality

22. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-22
Important problem typesImportant problem types

sortingsorting

searchingsearching
string processingstring processing
graph problemsgraph problems

combinatorial problemscombinatorial problems

geometric problemsgeometric problems

numerical problemsnumerical problems

23. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-23
Sorting (I)Sorting (I)
Rearrange the items of a given list in ascending order.Rearrange the items of a given list in ascending order.
•Input: A sequence of n numbers <aInput: A sequence of n numbers <a
11, ,
aa
22, …, a, …, a
nn>>
•Output: A reordering <aOutput: A reordering <a
´´
11, ,
aa
´´
22, …, a, …, a
´´
nn> of the input sequence such that a> of the input sequence such that a
´´
11≤ ≤ aa
´´
2 2 ≤≤
… … ≤≤
aa
´´
n.n.
Why sorting?Why sorting?
•Help searchingHelp searching
•Algorithms often use sorting as a key subroutine.Algorithms often use sorting as a key subroutine.
Sorting keySorting key
•A specially chosen piece of information used to guide sorting. E.g., sort A specially chosen piece of information used to guide sorting. E.g., sort
student records by names.student records by names.

24. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-24
Sorting (II)Sorting (II)

Examples of sorting algorithmsExamples of sorting algorithms
•Selection sortSelection sort
•Bubble sortBubble sort
•Insertion sortInsertion sort
•Merge sortMerge sort
•Heap sort …Heap sort …

Evaluate sorting algorithm complexity: the number of key comparisons. Evaluate sorting algorithm complexity: the number of key comparisons.

Two propertiesTwo properties
•StabilityStability: A sorting algorithm is called stable if it preserves the relative order of : A sorting algorithm is called stable if it preserves the relative order of
any two equal elements in its input.any two equal elements in its input.
•In placeIn place : A sorting algorithm is in place if it does not require extra memory, : A sorting algorithm is in place if it does not require extra memory,
except, possibly for a few memory units.except, possibly for a few memory units.

25. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-25
Selection SortSelection Sort
AlgorithmAlgorithm SelectionSort(A[0..n-1])SelectionSort(A[0..n-1])
//The algorithm sorts a given array by selection sort//The algorithm sorts a given array by selection sort
//Input: An array A[0..n-1] of orderable elements//Input: An array A[0..n-1] of orderable elements
//Output: Array A[0..n-1] sorted in ascending order//Output: Array A[0..n-1] sorted in ascending order
for i for i  0 to n – 2 do 0 to n – 2 do
min min  i i
for j for j  i + 1 to n – 1 do i + 1 to n – 1 do
if A[j] < A[min] if A[j] < A[min]
min min  j j
swap A[i] and A[min]swap A[i] and A[min]

26. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-26
SearchingSearching

Find a given value, called a Find a given value, called a search keysearch key, in a given set., in a given set.

Examples of searching algorithmsExamples of searching algorithms
•Sequential searchSequential search
•Binary search …Binary search …
Input: sorted array a_i < … < a_j and key x;
m (i+j)/2;
while i < j and x != a_m do
if x < a_m then j  m-1
else i  m+1;
if x = a_m then output a_m;
Time: O(log n)

27. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-27
String ProcessingString Processing

A string is a sequence of characters from an alphabet. A string is a sequence of characters from an alphabet.

Text strings: letters, numbers, and special characters.Text strings: letters, numbers, and special characters.

String matching: searching for a given word/pattern in a String matching: searching for a given word/pattern in a
text.text.
Examples:
(i)searching for a word or phrase on WWW or in a
Word document
(ii)searching for a short read in the reference genomic
sequence

28. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-28
Graph ProblemsGraph Problems

Informal definitionInformal definition
•A graph is a collection of points called A graph is a collection of points called verticesvertices, some of , some of
which are connected by line segments called which are connected by line segments called edgesedges..
Modeling real-life problemsModeling real-life problems
•Modeling WWWModeling WWW
•Communication networksCommunication networks
•Project scheduling …Project scheduling …
Examples of graph algorithmsExamples of graph algorithms
•Graph traversal algorithmsGraph traversal algorithms
•Shortest-path algorithmsShortest-path algorithms
•Topological sortingTopological sorting

29. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-29
Fundamental data structuresFundamental data structures

listlist
•arrayarray
•linked listlinked list
•string string

stackstack

queuequeue

priority queue/heappriority queue/heap

graphgraph

tree and binary treetree and binary tree

set and dictionaryset and dictionary

30. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-30
Linear Data StructuresLinear Data Structures

ArraysArrays
•A sequence of n items of the same A sequence of n items of the same
data type that are stored data type that are stored
contiguously in computer memory contiguously in computer memory
and made accessible by specifying a and made accessible by specifying a
value of the array’s index.value of the array’s index.

Linked ListLinked List
•A sequence of zero or more nodes A sequence of zero or more nodes
each containing two kinds of each containing two kinds of
information: some data and one or information: some data and one or
more links called pointers to other more links called pointers to other
nodes of the linked list.nodes of the linked list.
•Singly linked list (next pointer)Singly linked list (next pointer)
•Doubly linked list (next + previous Doubly linked list (next + previous
pointers)pointers)

Arrays

fixed length (need preliminary
reservation of memory)

contiguous memory locations

direct access

Insert/delete

Linked Lists

dynamic length

arbitrary memory locations

access by following links

Insert/delete
…a1 ana2 .

31. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-31
Stacks and QueuesStacks and Queues

StacksStacks
•A stack of plates A stack of plates
–insertion/deletion can be done only at the top.insertion/deletion can be done only at the top.
–LIFOLIFO
•Two operations (push and pop)Two operations (push and pop)

QueuesQueues
•A queue of customers waiting for services A queue of customers waiting for services
–Insertion/enqueue from the rear and deletion/dequeue from Insertion/enqueue from the rear and deletion/dequeue from
the front.the front.
–FIFOFIFO
•Two operations (enqueue and dequeue)Two operations (enqueue and dequeue)

32. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-32
Priority Queue and HeapPriority Queue and Heap

Priority queues (implemented using heaps)

A data structure for maintaining a set of
elements, each associated with a key/priority,
with the following operations

Finding the element with the highest priority

Deleting the element with the highest priority

Inserting a new element

Scheduling jobs on a shared computer
9
6 8
523
965823

33. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-33
GraphsGraphs

Formal definitionFormal definition
•A graph A graph G = <V, E>G = <V, E> is defined by a pair of two sets: a is defined by a pair of two sets: a
finite set V of items called finite set V of items called verticesvertices and a set E of vertex and a set E of vertex
pairs called pairs called edgesedges..

Undirected Undirected and and directeddirected graphs ( graphs (digraphsdigraphs).).
What’s the maximum number of edges in an undirected What’s the maximum number of edges in an undirected
graph with |V| vertices?graph with |V| vertices?

Complete, dense,Complete, dense, andand sparsesparse graphs graphs
•A graph with every pair of its vertices connected by an A graph with every pair of its vertices connected by an
edge is called complete, Kedge is called complete, K
|V||V|
12
34

34. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-34
Graph RepresentationGraph Representation

Adjacency matrixAdjacency matrix
•n x n boolean matrix if |V| is n.n x n boolean matrix if |V| is n.
•The element on the ith row and jth column is 1 if there’s an The element on the ith row and jth column is 1 if there’s an
edge from ith vertex to the jth vertex; otherwise 0.edge from ith vertex to the jth vertex; otherwise 0.
•The adjacency matrix of an undirected graph is symmetric.The adjacency matrix of an undirected graph is symmetric.

Adjacency linked listsAdjacency linked lists
•A collection of linked lists, one for each vertex, that contain all A collection of linked lists, one for each vertex, that contain all
the vertices adjacent to the list’s vertex.the vertices adjacent to the list’s vertex.

Which data structure would you use if the graph is a 100-node Which data structure would you use if the graph is a 100-node
star shape?star shape?

0 1 1 1
0 0 0 1
0 0 0 1
0 0 0 0
234
4
4

35. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-35
Weighted GraphsWeighted Graphs

Weighted graphsWeighted graphs
•Graphs or digraphs with numbers assigned to the edges.Graphs or digraphs with numbers assigned to the edges.
12
34
6
8
5
7
9

36. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-36
Graph Properties -- Paths and ConnectivityGraph Properties -- Paths and Connectivity
PathsPaths
•A path from vertex u to v of a graph G is defined as a sequence of A path from vertex u to v of a graph G is defined as a sequence of
adjacent (connected by an edge) vertices that starts with u and ends adjacent (connected by an edge) vertices that starts with u and ends
with v.with v.
•Simple pathsSimple paths: All edges of a path are distinct.: All edges of a path are distinct.
•Path lengths: the number of edges, or the number of vertices – 1.Path lengths: the number of edges, or the number of vertices – 1.
Connected graphsConnected graphs
•A graph is said to be connected if for every pair of its vertices u and A graph is said to be connected if for every pair of its vertices u and
v there is a path from u to v.v there is a path from u to v.
Connected componentConnected component
•The maximum connected subgraph of a given graph.The maximum connected subgraph of a given graph.

37. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-37
Graph Properties -- AcyclicityGraph Properties -- Acyclicity

CycleCycle
•A simple path of a positive length that starts and A simple path of a positive length that starts and
ends a the same vertex.ends a the same vertex.

Acyclic graphAcyclic graph
•A graph without cyclesA graph without cycles
•DAG DAG (Directed Acyclic Graph)(Directed Acyclic Graph)
12
34

38. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-38
TreesTrees
TreesTrees
•A tree (or A tree (or free treefree tree) is a connected acyclic graph.) is a connected acyclic graph.
•Forest: a graph that has no cycles but is not necessarily connected.Forest: a graph that has no cycles but is not necessarily connected.
Properties of treesProperties of trees
•For every two vertices in a tree there always exists exactly one For every two vertices in a tree there always exists exactly one
simple path from one of these vertices to the other. simple path from one of these vertices to the other. Why?Why?
–Rooted treesRooted trees:: The above property makes it possible to select an The above property makes it possible to select an
arbitrary vertex in a free tree and consider it as the root of the arbitrary vertex in a free tree and consider it as the root of the
so called rooted tree.so called rooted tree.
–Levels in a rooted tree.Levels in a rooted tree.

|E| = |V| - 113
24
5
1
3
2
4 5
rooted

39. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-39
Rooted Trees (I)Rooted Trees (I)

AncestorsAncestors
•For any vertex For any vertex vv in a tree in a tree TT, all the vertices on the simple path , all the vertices on the simple path
from the root to that vertex are called ancestors.from the root to that vertex are called ancestors.

DescendantsDescendants
•All the vertices for which a vertex All the vertices for which a vertex vv is an ancestor are said to is an ancestor are said to
be descendants of be descendants of vv..

Parent, childParent, child and and siblingssiblings
•If If (u, v)(u, v) is the last edge of the simple path from the root to is the last edge of the simple path from the root to
vertex vertex vv, , uu is said to be the parent of is said to be the parent of vv and and vv is called a child is called a child
of of uu..
•Vertices that have the same parent are called siblings.Vertices that have the same parent are called siblings.

LeavesLeaves
•A vertex without children is called a leaf.A vertex without children is called a leaf.

SubtreeSubtree
•A vertex A vertex vv with all its descendants is called the subtree of with all its descendants is called the subtree of TT
rooted at rooted at vv..

40. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-40
Rooted Trees (II)Rooted Trees (II)

DepthDepth of a vertex of a vertex
•The length of the simple path from the root to the vertex.The length of the simple path from the root to the vertex.

HeightHeight of a tree of a tree
•The length of the longest simple path from the root to a leaf.The length of the longest simple path from the root to a leaf.
1
3
2
4 5
h = 2

41. Karpagam Academy of Higher Education Design & Analysis of Algorithms, 1-41
Ordered TreesOrdered Trees

Ordered treesOrdered trees
•An ordered tree is a rooted tree in which all the children of each An ordered tree is a rooted tree in which all the children of each
vertex are ordered.vertex are ordered.

Binary treesBinary trees
•A binary tree is an ordered tree in which every vertex has no more A binary tree is an ordered tree in which every vertex has no more
than two children and each children is designated s either a left child than two children and each children is designated s either a left child
or a right child of its parent.or a right child of its parent.

Binary search treesBinary search trees
•Each vertex is assigned a number.Each vertex is assigned a number.
•A number assigned to each parental vertex is larger than all the A number assigned to each parental vertex is larger than all the
numbers in its left subtree and smaller than all the numbers in its numbers in its left subtree and smaller than all the numbers in its
right subtree.right subtree.

loglog
22nn  h h  n – 1 n – 1, where h is the height of a binary tree and n the size., where h is the height of a binary tree and n the size.
9
6 8
523
6
3 9
258