Post on 29-Mar-2015
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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
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How do we solve problems?
• We "just do"
• Guesswork-and-luck
• Trial-and-error
• Experience (possibly someone else's)
• Systematically !
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sterilize(saw,alcohol);raise_hammer();lower hammer(fast);start(saw);/* etc. etc. */
The Problem-solving Process
Problem specification
Algorithm
Program
Executable (solution)
Analysis
Design
Implementation
Compilation
"Doctor, my head hurts"
Patient has elevated pressure in anterior parietal lobe.
1. Sterilize cranial saw2. Anaesthetize patient3. Remove top of skull4. Get the big spoon...5. etc., etc.
010011101011001010101010100101010101010011001010101010100101101001110101010101001001011101001111010101011111010101000110100001101...
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Problem Idea
Algorithm Program
Informal Description Refinement
Coding
Testing & Debugging
...it’s not a linear process
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• 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
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Algorithm –History
Muhammad ibn Musa Al-Khwarizmihttp://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
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• A sequence of instructions describing how to do a task
Algorithm – Working Definition
[As opposed to actually executing
the instructions]
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Algorithm -- Examples• A cooking recipe• Assembly instructions for a model• The rules of how to play a game• VCR instructions• Description of a martial arts technique• Directions for driving from A to B• A knitting pattern• A car repair manual
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Algorithm -- Examples• A cooking recipe• Assembly instructions for a model• The rules of how to play a game• VCR instructions• Description of a martial arts technique• Directions for driving from A to B• A knitting pattern• A car repair manual
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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+ 789001
912457
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Algorithm – Examples (cont)
• Recipe for Almond and honey slice
• Recipe for Arroz con pollo
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Almond and Honey Slice
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.
From: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.
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Almond and Honey Slice
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.
From: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.
Instructions are given in the order in which they are performed (“executed”)
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Correct Algorithm?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.
From: “Arroz Con Pollo” in The Margaret Fulton Cookbook, Hamlyn, Sydney, 1968.
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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.
From: “Arroz Con Pollo” in The Margaret Fulton Cookbook, Hamlyn, Sydney, 1968.
Correct Algorithm?
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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 10 minutes.
Add beans and artichokes.
Cover, and bake for another 10 minutes or until rice is tender.
Correct Algorithm?
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From Algorithms to Programs
Problem
ProgramProgram
AlgorithmAlgorithm: A sequence of instructions describing how to do a task (or process)
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Algorithms are not Programs
• always design the algorithm before you start to program,
• the idea comes before the coding.
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Example
How to get out?
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Example
repeat
if “no wall left”
then turn left, step ahead
else if “no wall ahead”
then step ahead
else turn right
until “at exit”
“run through the maze always sticking to the wall on your left”
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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-style
int s:=0; for i = 0 to 10 do s:= s+i; end Pascal-style
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Components of an Algorithm
• Input and Output Specification• Variables and values• Instructions
– Sequences– Selections– Repetitions
• Abstraction Mechanisms – Objects– Methods
Also required: Documentation
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Values
• Represent quantities, amounts or measurements
• May be numerical or alphabetical (or of some other defined “type”)
• Example:– Recipe ingredients
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Almond and Honey Slice
1/2 quantity Shortcrust Pastry
185 g unsalted butter100 g castor sugar5 tablespoons honey50 ml cream50 ml brandy or any other
liqueur or spirit300 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.
From: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.
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Almond and Honey Slice
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.
From: Stephanie Alexander, The Cook’s Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.
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Variables
This jarcan contain
10 cookies
50 grams of sugar
3 slices of cake
etc.
ValuesVariable
• Are containers for values – places to store values
• Example:
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Restrictions on Variables
• Variables may be restricted to contain a specific type of value
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Components of an AlgorithmValues and Variables• Instruction (a.k.a. primitive)• Sequence (of instructions)• Selection (between instructions)• Repetition (of instructions)• Abstractions (with Objects and Methods)
• Documentation (beside instructions)
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Instructions (Primitives)
• Some action that is – simple– unambiguous– that the system knows about...– ...and should be able to actually do
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Instructions – Examples
• Take off your shoes
• Count to 10
• Cut along dotted line
• Knit 1
• Purl 2
• Pull rip-cord firmly
• Sift 10 grams of arsenic
Directions to perform specific actions on values and variables.
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Instructions -- Application
• Some instructions can only be applied to a specific type of values or variables
• Examples:
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Instructions (Primitives) -- Recommendations
• When writing an algorithm, make each instruction simple and unambiguous
• Example:
Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.
Cut chicken into pieces.
Heat olive oil in a casserole dish.
Brown the chicken pieces in the casserole dish.
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Components of an AlgorithmValues and VariablesInstruction (a.k.a. primitive)• Sequence (of instructions)• Selection (between instructions)• Repetition (of instructions)• Abstractions (with Objects and Methods)
• Documentation (beside instructions)
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Instruction (Primitives)• When writing an algorithm, make the
instructions simple and unambiguous.
• Example:
Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil.
Cut chicken into pieces.
Heat olive oil in a casserole dish.
Brown the chicken pieces in the casserole dish.
A “sequence” of simple instructions
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Sequence
• A series of instructions
• ...to be carried out one after the other...
• ...without hesitation or question
• Example:
– How to cook a Gourmet MealTM
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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
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Components of an AlgorithmValues and VariablesInstruction (a.k.a. primitive)Sequence (of instructions)• Selection (between instructions)• Repetition (of instructions)• Abstractions (with Objects and Methods)
• Documentation (beside instructions)
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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
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Selection Example -- Car Repairif (motor turns) then {
CheckFuelCheckSparkPlugsCheckCarburator
}else {
CheckStarterMotorCheckEngineBlock
}
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Selection Example –Car Repair (cont)
Should be a true or false condition.
if (motor turns) then {
CheckFuelCheckSparkPlugsCheckCarburator
}else {
CheckStarterMotorCheckEngineBlock
}
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Selection Example --Car Repair (cont)
Sequence if the condition is true.
if (motor turns) then {
CheckFuelCheckSparkPlugsCheckCarburator
}else {
CheckStarterMotorCheckEngineBlock
}
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Selection Example --Car Repair (cont)
Sequence if the condition is false.
if (motor turns) then {
CheckFuelCheckSparkPlugsCheckCarburator
}else {
CheckStarterMotorCheckEngineBlock
}
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Selection Example -- Reciprocals
Q. Give an algorithm for computing the reciprocal of a number.
Examples:
Reciprocal of 2: 1/2
Reciprocal of -3/4: 1/(-3/4) = -4/3
Reciprocal of 0: “undefined”
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Selection Example – Reciprocals (cont)
Q. Give an algorithm for computing the reciprocal of a number.
Algorithm: input Num if (Num is not equal 0) then { output 1/Num } else {
output "infinity" }
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Selection Example-- Reciprocals
input Num if (Num is not equal 0) then { output 1/Num } else {
output "infinity" }
Algorithm:
Num is a variable whose value depends on the actual number the user provides.
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Selection Example – Reciprocals (cont)
input Num if (Num is not equal 0) then { output 1/Num } else {
output "infinity" }
Algorithm:
Condition depends on the value of Num
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Selection Example – Reciprocals (cont)
input Num if (Num is not equal 0) then { output 1/Num } else {
output "infinity" }
Algorithm:
For a given value of Num, only one of these two sequences can be executed
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Selection Example – Reciprocals (cont)
input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity"
}
Algorithm:
Executed if Num is not equal to 0
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Selection Example – Reciprocals (cont)
input Num if (Num is not equal 0) then { output 1/Num } else {
output "infinity" }
Algorithm:
Executed if Num is equal to 0
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Selection -- Exercise
input Num if (Num is not equal 0) then { output 1/Num } else {
output "infinity" }
Will the following algorithms produce the same output?
Algorithm 1: input Num
if (Num is not equal 0)
then
{
output 1/Num
}
output "infinity"
Algorithm 2:
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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
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Selection – Several Conditions (cont)
Solution 2
Often called a “nested selection”
if ( today is Wednesday ) then {
if ( the time is 10.00am ) then { Go to CSE1301 Lecture } } else ...etc...etc...etc...
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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
}
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Components of an AlgorithmValues and VariablesInstruction (a.k.a. primitive)Sequence (of instructions)Selection (between instructions)• Repetition (of instructions)• Abstractions (with Objects and Methods)
• Documentation (beside instructions)
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Repetition
• Repeat an instruction...– ...while (or maybe until) some condition
occurs– Test the condition each time before
repeating the instruction• Also known as iteration or loop• Example:
– Algorithm for leaving the maze
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Example
repeatif “no wall left”
then turn left, step ahead
else if “no wall ahead”then step ahead
else turn rightuntil “at exit”
“run through the maze always sticking to the wall on your left”
Danger: endless loop if termination test not correct
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Components of an AlgorithmValues and VariablesInstruction (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)
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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
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Documentation -- ExampleThink of something romantic to dodecide on time and location
Work through address book to look for a personinitialise 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 outSighWithRelief
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Components of an AlgorithmValues and VariablesInstruction (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)
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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)
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Examplepicking the lightest item...
1. put first item on scale2. memorize weight3. take item from scale in hand4. repeat 5. put next item on scale6. compare to memorized weight7. take lighter item in hand; put other item away8. until no more items left
...is this well-defined?
...can we be sure that this always terminates with the correct item in hand?
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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
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Summary• Problem Solving Process• Algorithms• Components of Algorithms
– Values and Variables– Instructions– to be continued…(with Abstractions)
• Documentation• Analysis of Algorithms