algorithmic foundations comp108 comp108 algorithmic foundations basics prudence wong...

Algorithmic FoundationsCOMP108

COMP108Algorithmic FoundationsBasics

Prudence Wonghttp://www.csc.liv.ac.uk/~pwong/teaching/comp108/201112


Crossing Bridge @ Night

1 min

2 min

5 min

10 min

each time, 2 persons share a torchthey walk @ speed of slower person

Target: all cross the bridge

Can you do it in 17 mins?


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(Basics)

Module InformationDr Prudence Wong

Rm 3.18 Ashton Building, [email protected]

office hours: Wed 10-11am, Thu 3-4pm

DemonstratorsMr Mihai Burcea, Mr Jude-Thaddeus Ojiaku

ReferencesMain: Introduction to the Design and Analysis of

Algorithms. A. V. Levitin. Addison Wesley.

Reference: Introduction to Algorithms. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein. The MIT Press


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Module Information (2)Teaching, Assessments and Help

33 lectures, 11 tutorials

2 assessments (20%), 1 written exam (80%)

Office hours, email

Tutorials/LabsLocation :

Seminar Rooms G12 or 2.23 (theoretical) or

Labs 1, 2, 3 (practical)

Week 2: Seminar Rooms, G12 Ashton Bldg or 2.23 Holt Bldg

Class Tests: Weeks 7 & 11 (15 Mar & 3 May)


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Why COMP108?Pre-requisite for: COMP202, COMP218

(COMP202 is pre-requisite for COMP309)

Algorithm design is a foundation for efficient and effective programs

"Year 1 modules do not count towards honour classification…"

(Career Services: Employers DO consider year 1 module results)


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(Basics)

Aims To give an overview of the study of algorithms in

terms of their efficiency.

To introduce the standard algorithmic design paradigms employed in the development of efficient algorithmic solutions.

To describe the analysis of algorithms in terms of the use of formal models of Time and Space.

What do we mean by good?

How to achieve?

Can we prove?


Ready to start …

Learning outcomes

Able to tell what an algorithm is & have some understanding why we study algorithms

Able to use pseudo code to describe algorithm


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What is an algorithm?A sequence of precise and concise

instructions that guide you (or a computer) to solve a specific problem

Daily life examples: cooking recipe, furniture assembly manual(What are input / output in each case?)

Input Algorithm Output


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Why do we study algorithms?

Given a map of n cities & traveling cost between them.

What is the cheapest way to go from city A to city B?

The obvious solution to a problem may not be efficient

Simple solution Compute the cost of each

path from A to B Choose the cheapest one

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7 9

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A

B


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Shortest path to go from A to B

How many paths between A & B? involving 1 intermediate city?



A

B2



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How many paths between A & B? involving 3 intermediate cities?

A

B2

Number of paths= 2





Number of paths= 2 Number of paths= 2 + 3

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A

B

3





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A

B

Number of paths= 2 + 3





Number of paths= 2 + 3

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A

B

6

Number of paths= 2 + 3 + 6= 11





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A

B

For large n, it’s impossible to check all paths!We need more sophisticated solutions

TOO MANY!!





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A

B

There is an algorithm, called Dijkstra's algorithm,

that can compute this shortest path efficiently.


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Idea of Dijkstra's algorithm

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A

B10

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Go one step from A, label the neighbouring cities with the cost.


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Choose city with smallest cost and go one step from there.

A

B

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10


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If an alternative cheaper path is found, record this path and erase the more expensive one.

A

B

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This path must NOT be used because we find a cheaper alternative path

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Then repeat & repeat …


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A

B

Lesson to learn:Brute force algorithm may run slowly. We need more sophisticated algorithms.


How to represent algorithms … Able to tell what an algorithm is and have

some understanding why we study algorithms

Able to use pseudo code to describe algorithm


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Algorithm vs Program

Algorithms are free from grammatical rules Content is more important than form Acceptable as long as it tells people how to perform a

task

Programs must follow some syntax rules Form is important Even if the idea is correct, it is still not acceptable if

there is syntax error

An algorithm is a sequence of precise and concise instructions that guide a person/computer to solve a specific problem


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Compute the n-th powerInput: a number x & a non-negative integer n

Output: the n-th power of x

Algorithm: 1. Set a temporary variable p to 1.2. Repeat the multiplication p = p * x for n times.3. Output the result p.


Pseudo Code pseudo code:p = 1for i = 1 to n do p = p * xoutput p

C++:p = 1;for (i=1; i<=n; i++) p = p * x;cout << p << endl;

C:p = 1;for (i=1; i<=n; i++) p = p * x;printf("%d\n", p);

Java:p = 1;for (i=1; i<=n; i++) p = p * x;System.out.println(p);

Pascal:p := 1;for i := 1 to n do p := p * x;writeln(p);

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Pseudo Codeiteration i p

start 1

1 1 3

2 2 9

3 3 27

4 4 81

end 5

suppose n=4, x=3

trace table

p = 1for i = 1 to n do p = p * xoutput p

Another way to describe algorithm is by pseudo code

similar to programming language

more like EnglishCombination of

both


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Pseudo Code: conditionalConditional statement

if condition then statement

if condition then statementelse statement

if a > 0 then b = aelse b = -a output b

if a < 0 then a = -ab = aoutput b

What is computed?absolute

value


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Pseudo Code: iterative (loop)Iterative statement

for var = start_value to end_value do statement

while condition do statement

repeat statementuntil condition

condition to CONTINUE the loop

condition to STOP the loop

condition for while loop is NEGATION of condition for repeat-until loop


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for loop

i=1

i <= n?

sum = sum+i

No

Yes

Sum of 1st n nos.:

input: nsum = 0for i = 1 to n dobegin sum = sum + iendoutput sum

i=i+1

sum=0


the loop is executed n times


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for loop

iteration i sum

start 0

1 1 1

2 2 3

3 3 6

4 4 10

end 5

suppose n=4

Sum of 1st n nos.:

input: nsum = 0for i = 1 to n dobegin sum = sum + iendoutput sum


the loop is executed n times


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while loopwhile condition do

statement

Sum of 1st n numbers:

input: nsum = 0i = 1 while i <= n dobegin sum = sum + i i = i + 1endoutput sum

do the same as for-loop in previous slides

condition to CONTINUE the loop


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Sum of all input numbers:

sum = 0while (user wants to continue) dobegin ask for a number sum = sum + numberendoutput sum

while loop – example 2execute undetermined number of times

continue?

ask for numbersum = sum+number

No

Yes


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repeat-untilrepeat statementuntil condition

condition to STOP the loop

Sum of all input numbers:

sum = 0repeat ask for a number sum = sum + numberuntil (user wants to stop)output sum

Stop?

ask for numbersum = sum+number

Yes

No

also execute undetermined number of times

How it differs from while-loop?

execute at least

once


More Example 1

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input: m, nr = mq = 0while r = n dobegin r = r − n q = q + 1endoutput r and q

(@ end of) iteration r q

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0

1 10

1

2 6 2

3 2 3

suppose m=14, n=4

(@ end of) iteration r q

1 9 1

2 4 2

suppose m=14, n=5

(@ end of ) iteration r q

1 7 1

2 0 2

suppose m=14, n=7

What is computed?

remainder & quotient


input: m, nif m < n then swap m & ni = nwhile i = 1 dobegin if m%i==0 && n%i==0 then output i i = i-1end

suppose m=15, n=6

More Example 2

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(@ end of ) iteration

output (this

iteration)

i

4

1 4 3

2 2

3 2 1

4 1 0

suppose m=12, n=4

6

1 5

2 4

3 3

4 3 2

5 1

6 1 0

What values are output?

a%b remainder ofa divided b

all common factors


More Example 3input: m, nif m < n then swap m & ni = nfound = falsewhile i = 1 && !found dobegin if m%i==0 && n%i==0 then found = true else i = i-1endoutput i

What value is output?

Questions:what value of found makes the loop stop?when does found change to such value?

Highest Common Factor (HCF)

Greatest Common Divisor (GCD)

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Pseudo Code: ExerciseWrite a while-loop to

1.Find the product of all integers in interval [x, y]

e.g., if x is 2 & y is 5, then output is 2*3*4*5 = 120

assuming x and y are both integers

product = ??i = ??while ?? dobegin ?? i = ??endoutput ??


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Pseudo Code: ExerciseWrite a while-loop to

1.Find the product of all integers in interval [x, y]


product = 1i = xwhile i <= y dobegin product = product * i i = i+1endoutput product


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Pseudo Code: Exercise 2Write a while-loop for this:

2.Given two positive integers x and y, list all factors of x which are not factors of y

if x is 30 & y is 9, output is 2, 5, 6, 10, 15, 30 (not 1 or 3)

i = ??

while ?? do

begin

if ?? then

output ??

i = ??

end


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Pseudo Code: Exercise 2Write a while-loop for this:

2.Given two positive integers x and y, list all factors of x which are not factors of y

i = 1

while i <= x do

begin

if x%i == 0 && y%i != 0 then

output i

i = i+1

end


Challenges …

Convert while-loops to for-loops & repeat-loop


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Convert to for loops Find the product of all integers in interval [x, y]

assuming x and y are both integersproduct = ??

for i = ?? to ?? do

begin

??

end

output ??


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Convert to for loops Find the product of all integers in interval [x, y]

assuming x and y are both integersproduct = 1

for i = x to y do

begin

product = product * i

end

output product


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Convert to repeat loopsFind the product of all integers in interval [x, y]


product = 1

i = x

repeat


i = i+1

until i > y

output product What’s wrong with this?

If given x is larger than y?


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(Basics)

Convert to repeat loopsFind the product of all integers in interval [x, y]


product = 1

if x <= y then begin

i = x

repeat


i = i+1

until i > y

end

output product


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Convert to for loops (2)Given two positive integers x and y, list all

factors of x which are not factors of yfor i = ?? to ?? do

begin

if ?? then

??

end


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Convert to for loops (2)Given two positive integers x and y, list all

factors of x which are not factors of yfor i = 1 to x do

begin

if x%i == 0 && y%i != 0 then

output i

end


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Convert to repeat loops (2)Given two positive integers x and y, list all

factors of x which are not factors of yi = ??

repeat

if ?? then

??

i = ??

until ??


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Convert to repeat loops (2)Given two positive integers x and y, list all

factors of x which are not factors of yi = 1

repeat

if x%i==0 && y%i!=0 then

output i

i = i+1

until i > x


Searching …


Searching Input: n numbers a1, a2, …, an; and a number X

Output: determine if X is in the sequence or not

Algorithm (Sequential search):

1. From i=1, compare X with ai one by one as long as i

<= n.

2. Stop and report "Found!" when X = ai .

3. Repeat and report "Not Found!" when i > n.

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Sequential Search 12 34 2 9 7 5 six numbers

7 number X 12 34 2 9 7 5

7 12 34 2 9 7 5

7 12 34 2 9 7 5

7 12 34 2 9 7 5

7 found!

To find 7

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Sequential Search (2) 12 34 2 9 7 510

12 34 2 9 7 510

12 34 2 9 7 510

12 34 2 9 7 510

12 34 2 9 7 510

12 34 2 9 7 510 not found!

To find 10

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Sequential Search – Pseudo Codei = 1

while i <= n do

begin

if X == a[i] then

report "Found!" and stop

else

i = i+1

end

report "Not Found!"

Challenge: Modify it to include

stopping conditions in the

while loop

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Number of comparisons?i = 1

while i <= n do

begin

if X == a[i] then

report "Found!" & stop

else

i = i+1

end

report "Not Found!"

Best case: X is 1st no. 1 comparison

Worst case: X is last OR X is not found n comparisons

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How many comparisons this algorithm requires?


Finding maximum / minimum...

2nd max / min…


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Finding max from n +ve numbersinput: a[1], a[2], ..., a[n]M = 0i = 0while (i < n) dobegin i = i + 1 M = max(M, a[i])endoutput M

What about minimum?


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Finding min from n +ve numbersinput: a[1], a[2], ..., a[n]M = a[1]i = 1while (i < n) dobegin i = i + 1 M = min(M, a[i])endoutput M

How many comparisons?

n-1


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Finding 1st and 2nd mininput: a[1], a[2], ..., a[n]M1 = min(a[1], a[2])M2 = max(a[1], a[2])i = 2while (i < n) dobegin i = i + 1 if (a[i] < M1) then M2 = M1, M1 = a[i] else if (a[i] < M2) then M2 = a[i]endoutput M1, M2

Two variables: M1, M2

How to update M1, M2?


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Finding location of minimuminput: a[1], a[2], ..., a[n]loc = 1 // location of the min numberi = 1while (i < n) dobegin i = i + 1 if (a[i] < a[loc]) then loc = iendoutput a[loc]

(@ end of)

Iterationloc a[loc] i

1234

Example

a[]={50,30,40,20,10}

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algorithmic foundations comp108 comp108 algorithmic foundations basics prudence wong...

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