© 2006–2011 Antonio Carzaniga 1

Algorithms and Data StructuresCourse Introduction

Antonio Carzaniga

Faculty of InformaticsUniversity of Lugano

September 20, 2011

© 2006–2011 Antonio Carzaniga 1

General Information

On-line course information

ñ on Moodle: ‘INFO.M053’http://www.icorsi.ch/course/view.php?id=3777

ñ and on my web page:http://www.inf.usi.ch/carzaniga/edu/algo/

ñ last edition also on-line:http://www.inf.usi.ch/carzaniga/edu/algo10f/

Announcements

ñ you are responsible for reading the announcements page orthe messages sent through Moodle

Office hours

ñ Antonio Carzaniga: by appointmentñ Koorosh Khazaei: by appointment


Textbook

Introduction to Algorithms

Third Edition

Thomas H. CormenCharles E. LeisersonRonald L. RivestClifford Stein

The MIT Press


Additional Readings (Textbook 2)

Algorithms

Sanjoy DasguptaChristos H. PapadimitriouUmesh V. Vazirani

Mc Graw-Hill


Evaluation

+30% projectsñ 3–5 assignmentsñ grades added together, thus resulting in a weighted average

+30% midterm exam

+40% final exam

±10% instructor’s discretionary evaluation

ñ participationñ extra creditsñ trajectoryñ . . .

−100% plagiarism penalties


Plagiarism

A student should never take someone else’s material and present itas his or her own. Doing so means committing plagiarism.

“material” means ideas, words, code, suggestions, correctionson one’s work, etc.

Using someone else’s material may be appropriateñ e.g., software librariesñ always clearly identify the external material, and acknowledge

its source. Failing to do so means committing plagiarism.ñ the work will be evaluated based on its added value

Plagiarism on an assignment or an exam will result in

ñ failing that assignment or that examñ loosing one or more points in the final note!

Penalties may be escalated in accordance with the regulationsof the Faculty of Informatics


Deadlines

Deadlines are firm.

Exceptions may be granted

ñ at the instructor’s discretion

ñ only for documented medical conditions or other documentedemergencies

Each late day will reduce the assignment’s grade by one thirdof the total value of that assignment

ñ Corollary 1. The grade of an assignment turned in more thantwo days late is 0

ñ The proof of Corollary 1 is left as an exercise


Now let’s move on to the realinteresting and fun stuff. . .


Fundamental Ideas

Johannes Gutenberg invents movable type and the printing pressin Mainz, circa 1450 AD (already known in China, circa 1200 AD)


Maybe More Fundamental Ideas

The decimal numbering system(India, circa 600 AD)

Persian mathematician Khwarizmıwrites a book (Baghdad, circa 830 AD)

ñ methods for adding, multiplying,and dividing numbers (and more)

ñ these procedures were precise,unambiguous, mechanical, efficient,and correct

ñ they were algorithms! Muhammad ibn Musaal-Khwarizmı


Example

A sequence of numbers

0,1,1,2,3,5,8,13,21,34, . . .

The well-known Fibonacci sequence

Leonardo da Pisa (ca. 1170–ca. 1250)son of Guglielmo “Bonaccio”a.k.a. Leonardo Fibonacci


The Fibonacci Sequence

Mathematical definition: Fn =

0 if n = 0

1 if n = 1

Fn−1 + Fn−2 if n > 1

Implementation on a computer:

“pseudo-code”

Fibonacci(n)1 if n == 02 return 03 elseif n == 14 return 15 else return Fibonacci(n− 1)+ Fibonacci(n− 2)


Questions on Our First Algorithm


1. Is the algorithm correct?ñ for every valid input, does it terminate?ñ if so, does it do the right thing?

2. How much time does it take to complete?

3. Can we do better?© 2006–2011 Antonio Carzaniga 13

Correctness


Fn =

0 if n = 0

1 if n = 1

Fn−1 + Fn−2 if n > 1

The algorithm is clearly correctñ assuming n ≥ 0


Performance

How long does it take?

Let’s try it out. . .


Results

20 25 30 35 40 45 500

20

40

60

n

runnin

gti

me

(sec

onds)

RubySchemePython

CC-wizJava

C-gcc


Comments

Different implementations perform differently

ñ it is better to let the compiler do the optimization

ñ simple language tricks don’t seem to pay off

However, the differences do not seem to be substantial

ñ all implementations sooner or later seem to hit a wall. . .

Conclusion: the problem is with the algorithm


Complexity of Our First Algorithm

We need a mathematical characterization of the performanceof the algorithm

We’ll call it the algorithm’s computational complexity

Let T (n) be the number of basic steps needed to computeFibonacci(n)


T (0) = 2;T (1) = 3

T (n) = T (n− 1)+ T (n− 2)+ 3 ⇒ T (n) ≥ Fn


Complexity of Our First Algorithm (2)

So, let’s try to understand how Fn grows with n

T (n) ≥ Fn = Fn−1 + Fn−2

Now, since Fn ≥ Fn−1 ≥ Fn−2 ≥ Fn−3 ≥ . . .

Fn ≥ 2Fn−2 ≥ 2(2Fn−4) ≥ 2(2(2Fn−6)) ≥ . . . ≥ 2n2

This means that

T (n) ≥ (√

2)n ≈ (1.4)n

T (n) grows exponentially with n

Can we do better?


A Better Algorithm

Again, the sequence is 0,1,1,2,3,5,8,13,21,34, . . .

Idea: we can build Fn from the ground up!

SmartFibonacci(n)1 if n == 02 return 03 elseif n == 14 return 15 else pprev = 06 prev = 17 for i = 2 to n8 f = prev + pprev9 pprev = prev

10 prev = f11 return f


Results

20 40 60 80 100 120 140 160 180 2000

20

40

60

n

runnin

gti

me

(sec

onds)

RubySchemePython

CC-wizJava

C-gcc(Python) SmartFibonacci


Complexity of SmartFibonacci

SmartFibonacci(n)1 if n == 02 return 03 elseif n == 14 return 15 else prev = 06 pprev = 17 for i = 2 to n8 f = prev + pprev9 pprev = prev

10 prev = f11 return f

T (n) = 6+ 6(n− 1) = 6n

The complexity of SmartFibonacci(n) is linear in n© 2006–2011 Antonio Carzaniga 22

Documents

© 2006–2011 Antonio Carzaniga 1