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Components of an Algorithm

Objectives

- By the end of this lecture, students should:
- understand what an algorithms is
- appreciate the role of algorithm definitions in the problem solving process
- know the basic components of algorithms
- understand the difference between algorithms and programs
- be able to formulate algorithms for simple problems
- Reading: Walter Savitch. An Introduction to Problem Solving and Programming, Pearson, 2004. Sec. 1.1-1.3

How do we solve problems?

- We "just do"
- Guesswork-and-luck
- Trial-and-error
- Experience (possibly someone else's)
- Systematically !

The Problem-solving Process

"Doctor, my head hurts"

Patient has elevated pressure in anterior parietal lobe.

Analysis

Problem specification

1. Sterilize cranial saw

2. Anaesthetize patient

3. Remove top of skull

4. Get the big spoon...

5. etc., etc.

Design

Algorithm

Implementation

sterilize(saw,alcohol);

raise_hammer();

lower hammer(fast);

start(saw);

/* etc. etc. */

Program

Compilation

010011101011001010101010100101010101010011001010101010100101101001110101010101001001011101001111010101011111010101000110100001101...

Executable (solution)

Problem

Idea

Informal

Description

Refinement

Coding

Algorithm

Program

Testing & Debugging

Algorithm

- A sequence of instructions specifying the steps required to accomplish some task
- Named after:Muhammad ibn Musa al-Khwarizmi

of Khowarezm (now Khiva in Uzbekistan)

Circa 780-850 C.E. (Common Era)

Algorithm –History

Muhammad ibn Musa Al-Khwarizmi

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Al-Khwarizmi.html

- Book on arithmetic:
- Hindu numeration, decimal numbers, use of zero, method for finding square root
- Latin translation (c.1120 CE): “Algoritmi de numero Indorum”
- Book on algebra
- Hisab al-jabr w’al-muqabala

Algorithm – Working Definition

- A sequence of instructions describing how to do a task

[As opposed to actually executing

the instructions]

Algorithm -- Examples

- A cooking recipe
- Assembly instructionsfor a model
- The rules of how to play a game
- VCR instructions
- Descriptionof a martial arts technique
- Directions for driving from A to B
- A knitting pattern
- A car repair manual

Algorithm -- Examples

- A cooking recipe
- Assembly instructionsfor a model
- The rules of how to play a game
- VCR instructions
- Descriptionof a martial arts technique
- Directions for driving from A to B
- A knitting pattern
- A car repair manual

Algorithms are not Programs

- Algorithms are well-defined sequence of unambiguous instructions
- must terminate (to produce a result)
- Algorithm description relies on a well-defined“instruction language”
- Example: Manual AdditionDescribe the method! 123456

+ 789001912457

Algorithm – Examples (cont)

- Recipe for Almond and honey slice
- Recipe for Arroz con pollo

1/2 quantity Shortcrust Pastry

185 g unsalted butter

100 g castor sugar

5 tablespoons honey

50 ml cream

50 ml brandy or any other liqueur or spirit

300 g flaked almonds

Preheat oven for 200° C

Line a 30 cm 20 cm baking tray with baking paper, and then with pastry

Bake blind for 20 minutes, then remove weights and foil

Turn oven up to 220° C.

Bring remaining ingredients to a boil, stirring.

Spread evenly over pastry.

Bake until topping is bubbling and has caramelised evenly, about 15 minutes.

Cool before cutting into fingers or squares.

Almond and Honey SliceFrom: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

1/2 quantity Shortcrust Pastry

185 g unsalted butter

100 g castor sugar

5 tablespoons honey

50 ml cream

50 ml brandy or any other liqueur or spirit

300 g flaked almonds

Preheat oven for 200° C

Line a 30 cm 20 cm baking tray with baking paper, and then with pastry

Bake blind for 20 minutes, then remove weights and foil

Turn oven up to 220° C.

Bring remaining ingredients to a boil, stirring.

Spread evenly over pastry.

Bake until topping is bubbling and has caramelised evenly, about 15 minutes.

Cool before cutting into fingers or squares.

Almond and Honey SliceInstructions are given in the order in which they are performed (“executed”)

From: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.

Remove the chicken and to the juices in the casserole add garlic, onions and green peppers, and sauté until onion is golden.

Add bay leaf, whole tomatoes, and chicken broth.

When the broth boils add salt, saffron and rice.

Arrange chicken on rice, cover casserole and bake in a moderate oven (350°F) for 20 minutes or until the rice is tender.

Add beans and artichokes during last 10 minutes of cooking.

Correct Algorithm?From: “Arroz Con Pollo” in The Margaret Fulton Cookbook, Hamlyn, Sydney, 1968.

Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.

Remove the chicken and to the juices in the casserole add garlic, onions and green peppers, and sauté until onion is golden.

Add bay leaf, whole tomatoes, and chicken broth.

When the broth boils add salt, saffron and rice.

Arrange chicken on rice, cover casserole andbake in a moderate oven (350°F) for 20 minutes or until the rice is tender.

Add beans and artichokes during last 10 minutes of cooking.

Correct Algorithm?From: “Arroz Con Pollo” in The Margaret Fulton Cookbook, Hamlyn, Sydney, 1968.

Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.

Remove the chicken and to the juices in the casserole add garlic, onions and green peppers, and sauté until onion is golden.

Add bay leaf, whole tomatoes, and chicken broth.

When the broth boils add salt, saffron and rice.

Arrange chicken on rice, cover casserole andbake in a moderate oven (350°F) for 10 minutes.

Add beans and artichokes.

Cover, and bake for another 10 minutes or until rice is tender.

Correct Algorithm?Algorithm: A sequence of instructions describing how to do a task (or process)

Problem

Program

From Algorithms to ProgramsAlgorithms are not Programs

- always design the algorithm before you start to program,
- the idea comes before the coding.

Example

“run through the maze always sticking to the wall on your left”

repeat

if “no wall left”

then turn left, step ahead

else if “no wall ahead”

then step ahead

else turn right

until “at exit”

Algorithms and Languages

- the very same Algorithm may look very different when described (or implemented) in different (programming-) languages, e.g.

“Sum up the numbers between 1 and 10

by adding each of them to a total starting with 0”

s +/( 10) APL-styleint s:=0; for i = 0 to 10 do s:= s+i; end Pascal-style

Components of an Algorithm

- Input and Output Specification
- Variables and values
- Instructions
- Sequences
- Selections
- Repetitions
- Abstraction Mechanisms
- Objects
- Methods

Also required: Documentation

Values

- Represent quantities, amounts or measurements
- May be numerical or alphabetical(or of some other defined “type”)
- Example:
- Recipe ingredients

1/2 quantity Shortcrust Pastry

185 g unsalted butter

100 g castor sugar

5 tablespoons honey

50 ml cream

50 ml brandy or any other liqueur or spirit

300 g flaked almonds

Preheat oven for 200° C

Line a 30 cm 20 cm baking tray with baking paper, and then with pastry

Bake blind for 20 minutes, then remove weights and foil

Turn oven up to 220° C.

Bring remaining ingredients to a boil, stirring.

Spread evenly over pastry.

Bake until topping is bubbling and has caramelised evenly, about 15 minutes.

Cool before cutting into fingers or squares.

Almond and Honey SliceFrom: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

1/2 quantity Shotcrust Pastry

185 g unsalted butter

100 g castor sugar

5 tablespoons honey

50 ml cream

50 ml brandy or any other liqueur or spirit

300 g flaked almonds

Preheat oven for 200° C

Line a 30 cm 20 cm baking tray with baking paper, and then with pastry

Bake blind for 20 minutes, then remove weights and foil

Turn oven up to 220° C.

Bring remaining ingredients to a boil, stirring.

Spread evenly over pastry.

Bake until topping is bubbling and has caramelised evenly, about 15 minutes.

Cool before cutting into fingers or squares.

Almond and Honey SliceFrom: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

Restrictions on Variables

- Variables may be restricted to contain a specific type of value

Components of an Algorithm

- Values and Variables
- Instruction (a.k.a. primitive)
- Sequence (of instructions)
- Selection (between instructions)
- Repetition (of instructions)
- Abstractions (with Objects and Methods)
- Documentation (beside instructions)

Instructions(Primitives)

- Some action that is
- simple
- unambiguous
- that the system knows about...
- ...and should be able to actually do

Instructions – Examples

Directions to perform specific actions on values and variables.

- Take off your shoes
- Count to 10
- Cut alongdotted line
- Knit1
- Purl 2
- Pullrip-cord firmly
- Sift 10 grams of arsenic

Instructions -- Application

- Some instructions can only be applied to a specific type of values or variables
- Examples:

Chop

Instructions(Primitives) -- Recommendations

- When writing an algorithm, make each instruction simple and unambiguous
- Example:

Cut chicken into pieces.

Heat olive oil in a casserole dish.

Brown the chicken pieces in the casserole dish.

Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.

Components of an Algorithm

- Values and Variables
- Instruction (a.k.a. primitive)
- Sequence (of instructions)
- Selection (between instructions)
- Repetition (of instructions)
- Abstractions (with Objects and Methods)
- Documentation (beside instructions)

Instruction (Primitives)

A “sequence” of simple instructions

- When writing an algorithm, make the instructions simple and unambiguous.
- Example:

Cut chicken into pieces.

Heat olive oil in a casserole dish.

Brown the chicken pieces in the casserole dish.

Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.

Sequence

- A series of instructions
- ...to be carried out one after the other...
- ...without hesitation or question
- Example:
- How to cook a Gourmet MealTM

Sequence -- Example

1. Open freezer door

2. Take out Gourmet Meal™

3. Close freezer door

4. Open microwave door

5. Put Gourmet Meal™ on carousel

6. Shut microwave door

7. Set microwave on high for 5 minutes

8. Start microwave

9. Wait 5 minutes

10. Open microwave door

11. Remove Gourmet Meal™

12. Close microwave door

Components of an Algorithm

- Values and Variables
- Instruction (a.k.a. primitive)
- Sequence (of instructions)
- Selection (between instructions)
- Repetition (of instructions)
- Abstractions (with Objects and Methods)
- Documentation (beside instructions)

Selection

- An instruction that decides which of two possible sequences is executed
- The decision is based on a single true/false condition
- Examples:
- Car repair
- Reciprocals

Selection Example -- Car Repair

if (motor turns)

then

{

CheckFuel

CheckSparkPlugs

CheckCarburator

}

else

{

CheckStarterMotor

CheckEngineBlock

}

Selection Example –Car Repair (cont)

if (motor turns)

then

{

CheckFuel

CheckSparkPlugs

CheckCarburator

}

else

{

CheckStarterMotor

CheckEngineBlock

}

Should be a true or false condition.

Selection Example --Car Repair (cont)

if (motor turns)

then

{

CheckFuel

CheckSparkPlugs

CheckCarburator

}

else

{

CheckStarterMotor

CheckEngineBlock

}

Sequence if the condition is true.

Selection Example --Car Repair (cont)

if (motor turns)

then

{

CheckFuel

CheckSparkPlugs

CheckCarburator

}

else

{

CheckStarterMotor

CheckEngineBlock

}

Sequence if the condition is false.

Selection Example -- Reciprocals

Examples:

Reciprocal of 2: 1/2

Reciprocal of -3/4:1/(-3/4) = -4/3

Reciprocal of 0:“undefined”

Q. Give an algorithm for computing the reciprocal of a number.

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

Selection Example – Reciprocals (cont)Q. Give an algorithm for computing the reciprocal of a number.

Selection Example-- Reciprocals

Algorithm:

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

Num is a variablewhose value depends on the actual number the user provides.

Selection Example – Reciprocals (cont)

Algorithm:

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

Condition depends on the value of Num

Selection Example – Reciprocals (cont)

Algorithm:

input Num

if (Numis not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

For a given value of Num, only one of these two sequences can be executed

Selection Example – Reciprocals (cont)

Algorithm:

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

Executed if Num is not equal to 0

Selection Example – Reciprocals (cont)

Algorithm:

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

Executed if Numis equal to 0

Selection -- Exercise

Will the following algorithms produce the same output?

Algorithm 1:

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

else

{

output "infinity"

}

Algorithm 2:

input Num

if (Num is not equal 0)

then

{

output 1/Num

}

output "infinity"

Selection – Several Conditions

- What if several conditions need to be satisfied?

if ( today is Wednesday and the time is 10.00am )

then

{

Go to CSE1301 Lecture

}

else

{

Go to Library

}

Solution 1

Selection – Several Conditions (cont)

if ( today is Wednesday )

then

{

if ( the time is 10.00am )

then

{

Go to CSE1301 Lecture

}

}

else

...etc...etc...etc...

Often called a “nested selection”

Solution 2

Selection – At Least One of Several Conditions

- What if at least one of several conditions needs to be satisfied?

if ( I feel hungry or the time is 1.00pm or my mate has his eye on my lunch )

then

{

Eat my lunch now

}

- Values and Variables
- Instruction (a.k.a. primitive)
- Sequence (of instructions)
- Selection (between instructions)
- Repetition (of instructions)
- Abstractions (with Objects and Methods)
- Documentation (beside instructions)

Repetition

- Repeat an instruction...
- ...while (or maybe until) some condition occurs
- Test the condition each time before repeating the instruction
- Also known as iterationor loop
- Example:
- Algorithm for leaving the maze

Example

“run through the maze always sticking to the wall on your left”

repeat

if “no wall left”

then turn left, step ahead

else if “no wall ahead”

then step ahead

else turn right

until “at exit”

Danger: endless loop if termination test not correct

Components of an Algorithm

- Values and Variables
- Instruction (a.k.a. primitive)
- Sequence (of instructions)
- Selection (between instructions)
- Repetition (of instructions)
- Abstractions (with Objects and Methods) => Object-orientation (next lectures)
- Documentation (beside instructions)

Documentation

- Records what the algorithm does
- Describes how it does it
- Explains the purpose of each component of the algorithm
- Notes restrictions or expectations
- Example:
- Getting a date

Documentation -- Example

Think of something romantic to do

decide on time and location

Work through address book to look for a person

initialise booking to “unsuccessful”

until (successfully booked)

{

get next Name in little black book

AskOnDate(Name, Time, Location)

DetermineBookingSuccess

}

Assumes that I will find someone in the book before it runs out

SighWithRelief

Components of an Algorithm

- Values and Variables
- Instruction (a.k.a. primitive)
- Sequence (of instructions)
- Selection (between instructions)
- Repetition (of instructions)
- Abstractions (with Objects and Methods) => Object-orientation (next week)
- Documentation (beside instructions)

Analysis of Algorithms

we are interested in...

- Well-definedness (otherwise it’s not an algorithm)
- Correctness(does it fulfill the specification?)
- Termination(does it always end?)
- Resource consumption (time and memory)

Example

picking the lightest item...

- put first item on scale
- memorize weight
- take item from scale in hand
- repeat
- put next item on scale
- compare to memorized weight
- take lighter item in hand; put other item away
- until no more items left

...is this well-defined?

...can we be sure that this always terminates with the correct item in hand?

The Software Development Process

- Define the problem clearly
- Analyse the problem thoroughly
- Design an algorithm carefully
- Code the algorithm efficiently
- Test the code thoroughly
- Document the system lucidly

Summary

- Problem Solving Process
- Algorithms
- Components of Algorithms
- Values and Variables
- Instructions
- to be continued…(with Abstractions)
- Documentation
- Analysis of Algorithms

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