Itc lec5-24+sep+2012

Introduction To Computer Science (ITC)

Instructor: Adeela [email protected], [email protected]

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National University of Computer and Emerging Sciences, Islamabad

Lecture 05Introduction to Algorithms

24 September 2012

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Algorithm

Algorithm is a step by step procedure for solving a problem

It represents a process that a computer carries out, in order to complete a well defined task

The objective of computer science is to solve problems by developing, analyzing, and

implementing algorithmic solutions

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Al-Khwarizmi Principle

• All complex problems can be broken into simpler sub-problems.

• Solve a complex problem by breaking it down into smaller sub-problems and then solve them (in a specified order), one at a time.

• When all the steps are solved, the original problem itself has also been solved.

• This process is called Algorithm. (Originating from al-Khwārizmī)

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Computer Programming

• Computer is a powerful tool• It is not intelligent!• In order to use computer to solve our problems,

we must tell it what we want done and the order in which we want it done.

• These instructions are called computer program. • Writing these instructions is called computer

programming.• The person giving these instructions is called a

computer programmer.

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Computer Programming

• Analyze the problem

• Develop a sequence of instructions for solving the problem.

• Communicate it to the computer.

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Phases of the software life cycle

• Requirement definition

• Analysis and design

• Coding

• Testing

• Implementation

• Maintenance

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Problem Solving Techniques

• Ask questions

• Look for things that are similar

• Means-ends analysis

• Divide and Conquer

• Merging solutions

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Ask Questions• Ask questions until you have developed a clear

understanding of the problem.• Who, What, Why, Where, When.

– What do I have to work with? (my data or input)

– How much data is there?

– What should my output look like?

– How many times is the process going to be repeated?

– What are the exceptions to the main course?

– What special error condition might come up?

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Look for things that are Similar

• Do not reinvent the wheel!

• Draw Analogies

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Means-Ends analysis

• Starting point and ending state are known.• You need to devise the transformation

function.• Ends are defined – you need to analyze

your means of getting between them.• Lahore to Islamabad

– What are the options?– Narrow down the options?– Figure out the details?

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Divide and Conquer

• Same as the Al-khwarizmi Principle.

• Breakup the large problem into smaller units and then solve them one at a time.

• Building block approach.

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Divide and Conquer

Hard Problem

Hard Sub-problemEasy Sub-problem Easy Sub-problem

Easy Sub-problemEasy Sub-problem

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Merging Solution

• Sometimes merging two independent solutions solves the problem more efficiently?

• Calculate Average– Count values– Sum Values– Divide sum by count

• Alternative approach – calculate partial sum as you count

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Problem Solving Techniques

• What is the unknown?– What is required?

• What are the data?– What is given?

• What is the condition?– By what condition the unknown is linked to the

data?

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Conversion from Fahrenheit to Celsius

• Output– Temperature in Celsius (C)

• Inputs– Temperature in Fahrenheit (F)

• Process32)(F

9

5C

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Calculate and print the average grade of 3 tests for the entire class

• Input– 3 test scores for each student

• output– Average of 3 tests for each student

• Process1. Get three scores2. Add them together3. Divide by three to get the average4. Print the average5. Repeat step 1 to 4 for next student6. Stop if there are no more students

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A flowchart is a visual or graphical representation of an algorithm.

The flowchart employs a series of blocks and arrows, each of which represents a particular operation or step in the algorithm.

The arrows represent the sequence in which the operations are implemented.

Flow Charts

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Flowcharts – Most Common Symbols

SymbolSymbol NameName FunctionFunction

TerminalTerminal Represents the beginning or Represents the beginning or end of a end of a program.program.Flow-lineFlow-line Represents the flow of logic.Represents the flow of logic.

ProcessProcess Represents calculations Represents calculations or dataor data

manipulation.manipulation.

Input/OutputInput/Output Represents inputs or Represents inputs or outputs of data outputs of data and information.and information.

DecisionDecision Represents a comparison, Represents a comparison, question, question, or decision that or decision that determines determines alternative paths alternative paths to be followed.to be followed.

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Flowcharts – An Example Find the solution of a quadratic equation Ax2+Bx+C=0, given A, B and C.

START

INPUT A, B, C

CalculateR = SQRT(B2-4AC)

AA

AA

X1 = (-B+R)/(2A)X2 = (-B-R)/(2A)

PRINTA, B, C, X1, X2

END

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Flow Charting

Expresses the flow of processing in a structured pictorial format.

Processing Steps

Input and Output Steps

Decision

Flow of

data

ConnectorsTerminator

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Flow chart for Converting Fahrenheit into Celsius

Get temp. in ‘F’

Print ‘C’

Calculate )32(F

95

C

Stop

Begin

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Add them together

Divide the result by three

More students?Yes

StopNo

Flow chart for calculating average of three scores

Get three scores

Print the average

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STARTSTART

INPUT INPUT A, BA, B

Add A to B Add A to B and store in Cand store in C

OUTPUTOUTPUT CC

ENDEND

Comparison of Algorithm representations in Natural language, flowchart and Pseudo-code

Step 1: Begin the calculations

Step 2: Input two values A and B

Step 3: Add the values

Step 4: Display the result

Step 5: End the calculation

BEGIN Adder Input A and B C = A + B PRINT CEND Adder

Natural languageNatural language FlowchartFlowchart Pseudo-codePseudo-code

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Algorithm Representation(Natural Languages)

• English or some other natural language.• Are not particularly good:

–too verbose–unstructured–too rich in interpretation (ambiguous)–imprecise

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Algorithm Representation(Using Programming Language)

{

int I, m, Carry;

int a[100], b[100], c[100];

cin >> m;

for ( int j = 0 ; k <= m-1 ; j++ ) {

cin >> a[j];

cin >> b[j];

}

Carry = 0;

i = 0;

while ( i < m ) { …

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Programming Languages

• Are not particularly good either

– Too many implementation details to worry

about

– Too rigid syntax

• Easy to lose sight of the real task

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Pseudo-code

• We need a compromise between the two:

Pseudo-code• Computer scientists use pseudo-code to

express algorithms:– English like constructs (or other natural

language), but– modeled to look like statements in typical

programming languages.

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Pseudo-code Primitives

Three basic kind of operations:• Sequential

– Computation ( Set … )– Input/Output ( Get ... / Print ... )

• Conditional– If … Else– If …

• Iterative / looping– Repeat ...– While ...

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Performs a computation and stores the result.

Computation

Example:Example:

Set the value of Set the value of C to (A + B)C to (A + B)

Set the value of Set the value of locationlocation to 0 to 0

Set the value of Set the value of GPA to (sum / count)GPA to (sum / count)

General format:General format: Set the value of <variable> to <expression>

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VariablesA variable is a named storage. - A value can be stored into it, overwriting

the previous value

- Its value can be copied ExamplesExamples::

Set the value of A to 3

The variable A holds the value 3 after its execution

Set the value of A to (A+1)

Same as: add 1 to the value of A ( A is now 4)

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Not too Strict on Syntax

• Pseudo-code is kind of a programming language without a rigid syntax, for example we can write:– Set the value of A to (B+C)

• as – Set A to (B+C)

• or even:• Set the value of sum to 0• Set the value of GPA to 0

• as • Set sum and GPA to 0

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Sequential Operations - Input/OutputSequential Operations - Input/Output

Outside worldOutside world

InputInput

OutputOutput

•The Computer needs to communicate with the outside world:

INPUT operations allow the computing agent to receive from the outside world data values to use in subsequent computations.OUTPUT operations allow the computing agent to communicate results of computations to the outside world.

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InputGeneral format:

The computing agent (computer) suspends executions and waits for an input value.

Get a value for <variable>

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Input - Examples

• Examples:– Get value for grade– Get values for N, M

• Can write:– Get value for N1

– ...– Get value for N100

• as– Get value for N1,..., N100

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OutputGeneral format:

The computing agent (computer) displays the value of the variable(s).

Print the value of <variable>

Print the message, "<text>"

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Output - Examples

• Examples:– Print the value of grade– Print the message, "Hello"

• Can write:– Print the value of N1

– ...– Print the value of N100

• as– Print the values of N1,..., N100

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Example

• Write an algorithm to calculate the average of three numbers.

Steps Operations

1 Get values for N1, N2, and N3

2 Set the value of Average to (N1+N2+N3)/3

3 Print the value of Average

4 Stop

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Conditional Operations

If <condition> then

operations for the then-part

Else

operations for the else-part

1. Evaluate <condition> expression to see whether it is true or false.

2. If true, then execute operations in then-part

3. Otherwise, execute operations in else-part.

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Conditions, or Boolean Expressions

• A condition is one whose value is true or false, for example:

3 > 2 is greater than (true)

3 = 2 is equal to (false) A > 2 is true if A’s value is

greater than 2 (at the time this is executed), false otherwise.

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E1 or E2E1 or E2

true if at least one of them is true; false true if at least one of them is true; false otherwise.otherwise.

E.g. 3 > 2 or 2 > 3 is true E.g. 3 > 2 or 2 > 3 is true

E1 and E2E1 and E2

true if both are true; false otherwise true if both are true; false otherwise

E.g. 3 > 2 and 2 > 3 is falseE.g. 3 > 2 and 2 > 3 is false

not Enot E

true if E is false, false if E is true true if E is false, false if E is true

Conditions may be compoundedConditions may be compounded

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1. Get a value for A

2. If A = 0 then

3. Print the message, “The input is zero”

Else

4. Print the message, “The input is not zero”

ExampleExample

1. Get a value for grade

2. If grade < 1 or grade > 9 then

3. Print the message, “Invalid grade”

Else

4. Set the value of total to (grade + total)

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Iterative Operation - While

While <condition> remains true do steps i to j

step i: operation

step i+1: operation

…

step j: operation1.1. Evaluate <condition>Evaluate <condition>2.2. If condition is true, execute steps i to j, If condition is true, execute steps i to j,

then go back to i.then go back to i.3. Otherwise, if condition is false,continue 3. Otherwise, if condition is false,continue

execution from step j+1.execution from step j+1.

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1 Get a value for count

2 While count < 10 do

3 Set square to (count * count)

4 Print the values of count and square

5 Add 1 to count

6 Stop

Example

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What happens when it gets executed?

If count starts with 7, we get printout

7 49

8 64

9 81

What if count starts with 11?

Nothing is printed, loop is executed 0 times.

While Loops

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Set value of A equal to 1

While A > 0

Print message, “Enter an integer”

Get a value for A

End of the loop

Stop

What does the following algorithm do?What does the following algorithm do?

ExerciseExercise

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Tracing an algorithmTracing an algorithm

The current values of algorithm variables at

various points during execution can be

known by tracing the algorithm with a table

called Trace Table

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Trace TableTrace TableProblem: Determine the value of the variable x and y after the following algorithm is executedPseudocode:x = 5Y = 7If x = 5 then

y= 8else

y= 0if y = 7 then

x = 6else

x = 3if x = y then

y = 0

Nesting???

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… … ContinuedContinued

834.3

834.2

854.1

853

752

-51

yXStep

No.

Trace table for algorithm

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Find the phone number of a given Name in an (unsorted) list of names and their phone numbers

Names Phone numbers

N1 T1

N2 T2

…

N1000 T1000

Sequential Search: an Example

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Sequential Search: an Example

123456789

1011

553614442563521463541236452361442563551123441155521364528975541258

Name to fin

d:

Smith, J

ohn

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1. Get value for Name

2. Get values for N1,…,N1000

3. Get values for T1,…,T1000

4. If Name = N1 then print the value of T1

5. If Name = N2 then print the value of T2

…

1002. If Name = N999 then print the value of T999

1003. If Name = N1000 then print the value of 1000

1005. Stop

Sequential Search: 1st Attempt

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Get values for Name, N1,…, N1000, T1,…, T1000

Set the value i to 1 and the value of Found to 0

While Found = 0 AND i <= 1000 do

If Name = Ni then

Print the value of Ti

Set the value of Found to 1

Else

Add 1 to the value of I

EndIF

End of While loop

Stop

Sequential Search: Using a Loop

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Location A1 A2 A3 A4 A5 A6 A7

Value 5 2 8 4 8 6 4

The largest is 8 at location 3The largest is 8 at location 3

Idea (sketch): Go through the entire list, at each Idea (sketch): Go through the entire list, at each iteration find the largest-so-far and record its iteration find the largest-so-far and record its locationlocation

Given a list of variables A1, A2, …, An, find the Given a list of variables A1, A2, …, An, find the largest value and its (first) location largest value and its (first) location

Selection: Find the Largest Number

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Location A1 A2 A3 A4 A5 A6 A7

Value 5 2 8 4 8 6 4

To begin with, To begin with, set set largest-so-farlargest-so-far to (the value of) to (the value of) A1A1set set locationlocation to 1to 1set set ii to 2 to 2

i

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Location A1 A2 A3 A4 A5 A6 A7

Value 5 2 8 4 8 6 4

CompareCompare A1A1 and and A2A2 largest-so-farlargest-so-far still holds the value of still holds the value of A1A1set set ii to to ii+1+1

i

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Location A1 A2 A3 A4 A5 A6 A7

Value 5 2 8 4 8 6 4

Compare Compare A1A1 and and A3A3largest-so-farlargest-so-far now holds the value of now holds the value of A3A3locationlocation is 3 is 3set set ii to to ii+1+1

i

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Location A1 A2 A3 A4 A5 A6 A7

Value 5 2 8 4 8 6 4

Continue the similar process until Continue the similar process until ii = 8 = 8

i

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Get a value for n, the size of the listGet values for A1, A2, …, An, the list to be searchedSet largest_so_far to A1 and set location to 1Set the value of i to 2While i is less or equal to n do If Ai > largest_so_far then Set the value of largest_so_far to Ai

Set the value of location to I EndIF

Add 1 to the value of iEnd of While loopPrint the values of largest_so_far and location

Selection: Find The Largest Number