Fundamentals of algorithms

Fundamentals of Algorithms
Dr. AMIT KUMAR @ JUET

Prerequisite
 Introduction to programming in ‘C’ and data
structure (with sufficient familiarity in the
usage of array, pointers, recursion, searching,
sorting, tree etc…. )

Purpose of this subject
A rigorous introduction to the design and analysis
of algorithms
 Not a lab or programming course
 Not a math course, either

Evaluation scheme: (Fundamental of Algorithms Theory)
 Test – I 15
 Test – II 25
 Test- III 35
 Assignments 5
 Quiz 5
 Tutorial/Problem Solving 10
 Attendance 05
Total 100

Course Content
 Course Content:
 Analysis of algorithm: Asymptotic Notation, Sorting and merging
Algorithm
 Tree and related data Structure: Heap, Priority Queues, B-Tree, AVL,
Splay Tree, Red-Black Tree, Threaded Tree
 Files: Classification, Record Organization, Retrieval System, External
Sorting
 Set, Dictionary: Design, Analysis, integration and applications
 Fundamental techniques: Divide and Conquer method, Dynamic
Programming, Introduction to Greedy Method
 Hashing: technique, collision resolution and analysis
 Text Processing: String operation, pattern matching algorithm, tries,
text compression, text similarity testing.

BOOKS and References:
 Textbook: Introduction to Algorithms, Cormen,
Leiserson, Rivest, Stein
 Aho, Hopcraft, Ullman: Data Structure and
Algorithms
 Kruse, Tonso, Leung: Data Structure and
program Design in C
 Sahni: Data structure and algorithm and
application in C++
 Weiss: Data Structure and Algorithm analysis in
C/C++

Design and Analysis of Algorithms
• Analysis: predict the cost of an algorithm in terms of resources
and performance
• Design: design algorithms which minimize the cost

Algorithms
An algorithm is a finite sequence of step by step,
discrete, unambiguous instructions for solving a
particular problem
 Has input data, and is expected to produce output data
 Each instruction can be carried out in a finite amount of
time in a deterministic way

Algorithms
 In simple terms, an algorithm is a series of instructions
to solve a problem (complete a task).
 Problems can be in any form
 Business
 Get a part from Vancouver to Ottawa by morning
 Allocate manpower to maximize profit
 Life
 I am hungry. How do I order pizza?
 Explain how to tie shoelaces to a five year old child

Algorithms (Criteria)
1. Input: Zero or more quantities are externally
supplied.
2. Output: At least one quantity is produced.
3. Definiteness: Each instruction is clear and
unambiguous.
4. Finiteness: If we trace out the instructions of an
algorithm, then for all cases, the algorithm
terminates after a finite number of steps.
5. Effectiveness: Every instruction must be very basic
so that it can be carried out, in principle, by a
person using only pencil and paper. It is not enough
that each operation be definite as in criteria 3, it
also must be feasible.

Algorithms
 We deal with data processing problems
 Translated to programs that can be run on a computer
 Since we can only input, store, process & output data
on a computer, the instructions in our algorithms will
be limited to these functions.

Algorithm Description
 Understand the problem before solving it
 Identify & name each Input/Givens
 Identify & name each Output/Results
 Assign a name to our algorithm (Name)
 Combine the previous 3 pieces of information into a
formal statement (Definition)
 Results := Name (Givens)

Method
 Once we have prepared the Algorithm Description, we
need to solve the problem
 We develop a series of instructions (limited to those
described previously) that, when executed, will
compute the desired Results from the Givens
(Method)

Writing of Algorithm
Algorithm is basically a sequence of instruction written
in simple English type (pseudo) language. The
algorithms is basically divided into two parts.
 Algorithm heading: It consist of name of algorithm,
problem description, input and output.
 Algorithm body: It consists of logical body of the
algorithm.

Assignment command
Syntax
X = 5Y + 16
On the left side of =, we put the name of a variable and on the right side
we put a value or an expression.
Each variable refers to a unique location in computer memory that contains
a value.
Interpretation
An assignement is executed in two steps :
1-evaluation of the expression found on the right side.
2-setting the returned value in the memory cell corresponding to variable.
Example
Let SideSize=15
Let Area=SideSizeSideSize

Assignment in computer science and
equality in mathematics
a) The following instructions are the same in mathematics.
A=B B=A
not in computer science.
Let A=B is different from Let B=A
b) In mathematics we work with relations.
A relation B=A+1 means that it is true all the time
In computer science, we work with assignments. We can have:
A=5
B=A+1
A=2
The relation B=A+1 is true only after the second instruction
and before the third one.

c) The instruction A=A+3 is false in mathematics.
In computer science Let A=A+3 means: the new value of A is
equal to the old one plus three.
d) The instruction A+5=3 is allowed in mathematics (it is an
equation).
Let A+5=3 has no meaning in computer science (the left side
must be a variable).
Assignment in computer science and
equality in mathematics

Input command
Syntax
Get variable
The variable must be from Givens
Interpretation
Here the user must give a value. The given value is
assigned to the variable.
Example
Get Size_Side

Output command
Syntax
Give variable
The variable must be from Results
Interpretation
The value of the variable is displayed.
Example
Give Area

Algorithm 1.1
 Write an algorithm to find the sum of three given
numbers
 NAME: SUM3
 GIVENS: N1, N2, N3
 RESULTS: Total
 DEFINITION: Total := SUM3(N1, N2, N3)
 -------------------------
 METHOD:
Get N1
Get N2
Get N3
Let Total = N1 + N2 + N3
Give Total

Algorithm 1.2
 Write an algorithm to find the result of a division
operation for the given two numbers X and Y
 NAME: Division
 GIVENS: X, Y
 RESULTS: Quotient
 DEFINITION: Quotient := Division(X, Y)
 -------------------------
 METHOD:
Get X
Get Y
Let Quotient = X/Y
Give Quotient

Algorithm 1.3
 Write an algorithm to find the sum and product of the
two given numbers
 NAME: SumTimes
 GIVENS:Num1, Num2
 RESULTS: Total, Product
 DEFINITION: Total & Product := SumTimes(Num1, Num2)
 -------------------------
 METHOD:
Get Num1
Get Num2
Let Total = Num1 + Num2
Let Product = Num1 * Num2
Give Total
Give Product

Algorithm 1.4
 Find the sum and average of three given numbers
 NAME:AVG3
 GIVENS:Num1, Num2, Num3
 RESULTS:Sum , Average
 DEFINITION:Sum & Average := AVG3(Num1, Num2, Num3)
 -------------------------
 METHOD:
Get Num1
Get Num2
Get Num3
Let Sum = Num1 + Num2 + Num3
Let Average = Sum /3
Give Sum
Give Average

Variables
 Observe that we have used names for the data
items in our Givens and Results
 Num1, Num2, Num3, Sum, Average in Algorithm 1.4
 Each name refers to a unique location in computer
memory (one or more adjacent bytes) that contains
a value
 Since that value can change as the instructions in
our algorithm are executed, we call each data item a
variable

Variables
 In our algorithm, when we use a variable name, we are
referring to the value stored in memory for that data
item
 Later in this lecture we will learn more about how to
define variables

Intermediates
 Occasionally, in an algorithm, we need to have a
variable (in addition to those representing Givens or
Results) to store a value temporarily
 These are intermediate variables and we identify them
in the Algorithm Description as Intermediates

Algorithm 1.5
 Given 3 assignment marks (out of 50, 20, 70), find the
average (calculated as a mark out of 100)
 General Concept
 How does one figure out the percentage of several
marks?
 Add them all up
 Divide by the maximum possible mark (50+20+70)
 Multiply by 100

Algorithm 1.5
 Given 3 assignment marks (out of 50, 20, 70), find the average,
calculated as a mark out of 100
 NAME: CalcMark
 GIVENS: A1, A2, A3
 RESULTS: Mark
 INTERMEDIATES: Total, MaxMark (Constant)
 DEFINITION: Mark := CalcMark(A1, A2, A3)
 -------------------------
 METHOD:
Set MaxMark = 140 (Constant)
Get A1
Get A2
Get A3
Let Total = A1 + A2 + A3
Let Mark = Total/MaxMark * 100
Give Mark

Algorithm 1.6
 Given a two digit number, find the
sum of its digits
 General Concept
 How can we break apart a number?
 41 = 4 Tens and 1 Ones
 so for the number 41, we want 4 + 1 = 5
 Use integer division
 DIV returns the integer part of a
division
 MOD returns the remainder of a
division
41 10 = 4
41 MOD 10 = 1

Algorithm 1.6
 Given a two digit number, find the sum of its digits
 NAME: SumDig
 GIVENS: N
 RESULTS: Sum
 INTERMEDIATES: Tens, Ones
 DEFINITION: Sum := SumDig(N)
 -------------------------
 METHOD:
Get N
Let Tens = N div10
Let Ones = N mod 10
Let Sum = Tens + Ones
Give Sum

Algorithm 1.7
 Write an algorithm which swaps the values of two
numbers
 Example 1
 Two car family. The wrong car is at the end of the driveway
 Move first car out on to the street
 Move second car out on to the street
 Move first car back in
 Move second car back in
 Example 2
 You are looking after a 3 year old. He wants milk and juice. You put the
milk in the blue glass and the juice in the red glass. The child is screaming
that you did it wrong.
 Get a third glass (white)
 Pour the milk from the blue glass to the white glass
 Pour the juice from the red glass to the blue glass
 Pour the milk from the white glass to the red glass

Algorithm 1.7
 Write an algorithm which swaps the values of two
numbers
 NAME: Swap
 GIVENS: X, Y
 RESULTS: X, Y
 INTERMEDIATES: Temp
 DEFINITION: Swap (X, Y)
 -------------------------
 METHOD:
Get X
Get Y
Let Temp = X
Let X = Y
Let Y = Temp
Give X
Give Y

Algorithm 1.8
 Write an algorithm which adds the given two numbers
(X and Y) and returns the sum in the given variable X
 NAME: AddXY
 GIVENS:X, Y
 RESULTS: X
 INTERMEDIATES: None
 DEFINITION: AddXY (X, Y)
 -------------------------
 METHOD:
Get X
Get Y
Let X = X+ Y
Give X

Tracing an Algorithm
 The purpose of tracing an algorithm is to ensure
that it works
 This is a paper test. As we will see later, it should
be completed before writing the computer code
 Tracing involves
 Executing the sequence of instructions with a sample
set of values
 Computing the value of each variable after each
instruction is executed
 Checking for the correct result

 Step 1 - Number every instruction in the algorithm
 Step 2 – Make a table
 The first column of the table indicates which instruction
has been executed
 Subsequent columns list all the variables of the
algorithm (Givens, Results, Intermediates)

 Step 3 – Complete the table
 Each column of the table represents a variable
 Start at the first line of your algorithm. Identify what
will happen to each variable as that instruction is
executed
 Change any values which the instruction changes and leave all
other columns blank

Trace 1.1
 Trace Algorithm 1.4 using the numbers 24, 31, and
35
METHOD:
(1) Get Num1
(2) Get Num2
(3) Get Num3
(4) Let Sum = Num1 + Num2 + Num3
(5) Let Average = Sum /3
(6) Give Sum
(7) Give Average
Line Num1 Num2 Num3 Sum Avg
1 24
2 31
3 35
4 90
5 30
6 output 90
7 output 30

Trace 1.2
 Trace Algorithm 1.5 with the numbers 40, 18, 26
METHOD:
(1) Set MaxMark =140
(2) Get A1
(3) Get A2
(4) Get A3
(5) Let Total = A1 + A2 + A3
(6) Let Mark = Total/MaxMark * 100
(7) Give Mark
Ln A1 A2 A3 MM Ttl Mark
1 140
2 40
3 18
4 26
5 84
6 60
7 output 60
NB THE ANSWER IS NOT 69 (40/50 + 18/20 + 26/70)/3

Trace 1.3
 Trace Algorithm 1.7 when X and Y have the values 25
and 88, respectively
METHOD:
(1) Get X
(2) Get Y
(3) Let Temp = X
(4) Let X = Y
(5) Let Y = Temp
(6) Give X
(7) Give Y
LN X Y Temp
1 25
2 88
3 25
4 88
5 25
6 output 88
7 output 25

Flow Charts
 Is a pictorial representation of Algorithms
 Can be created in MS Visio, MS word, Paint etc.
 Begin and End with an Oval
 Get/Give use a parallelogram
 Lets use a rectangle
 Flow is shown with arrows

Algorithm 1.1
NAME: SUM3
GIVENS: N1, N2, N3
RESULTS: Total
DEFINITION:
Total := SUM3(N1, N2, N3)
Start
SUM3
Get N1
Get N2
Get N3
Let Total = N1 + N2 + N3
Give Total
Finish
SUM3

Algorithm 1.2
NAME: Division
GIVENS: X, Y
RESULTS: Quotient
DEFINITION:
Quotient := Division(X, Y)
Start
DIVISION
Get X
Get Y
Let Quotient = X/Y
Give
Quotient
Finish
DIVISIONDr. AMIT KUMAR @ JUET

Algorithm 1.3
NAME: SumTimes
GIVENS:Num1, Num2
RESULTS: Total, Product
DEFINITION:
Total & Product :=
SumTimes(Num1, Num2)
Start
SUMTIMES
Get Num1
Get Num2
Let Total = Num1 + Num2
Let Product = Num1 * Num2
Give Total
Give Product
Finish
SUMTIMES

Algorithm 1.4
NAME:AVG3
GIVENS:Num1, Num2, Num3
RESULTS:Sum , Average
DEFINITION:
Sum & Average :=
AVG3(Num1, Num2, Num3)
Start
AVG3
Get Num1
Get Num2
Get Num3
Let Sum = Num1 + Num2 + Num3
Let Average = Sum/3
Give Suml
Give Average
Finish
AVG3

Algorithm 1.5
NAME: CalcMark
GIVENS: A1, A2, A3
RESULTS: Mark
INTERMEDIATES:
Total,
MaxMark (Constant)
DEFINITION:
Mark := CalcMark(A1, A2, A3)
Start
CALCMARK
Get A1
Get A2
Get A3
Let Total = A1 + A2 + A3
Let Mark = Total/MaxMark
Give Mark
Finish
CALCMARK
Set MaxMark =140

Algorithm 1.6
NAME: SumDig
GIVENS: N
RESULTS: Sum
INTERMEDIATES: Tens, Ones
DEFINITION:
Sum := SumDig(N)
Start
SUMDIG
Get N
Let Tens = N div 10
Let Ones = N mod 10
Let Sum = Tens + Ones
Give Sum
Finish
SUMDIG

Algorithm 1.7
NAME: Swap
GIVENS: X, Y
Change: X, Y
RESULTS: None
INTERMEDIATES: Temp
DEFINITION: Swap (X, Y)
Start
SWAP
Get X
Get Y
Let Temp = X
Let X = Y
Let Y = Temp
Give X
Give Y
Finish
SWAP

Algorithm 1.8
NAME: AddXY
GIVENS:X, Y
Change: X
RESULTS:None
INTERMEDIATES: None
DEFINITION: AddXY (X, Y)
Start
ADDXY
Get X
Get Y
Let X = X + Y
Give X
Finish
ADDXY

Fundamentals of algorithms

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