Algorithm Design · Welcome to the course Algorithm Design Friedhelm Meyer auf der Heide 1 Summer Term 2011 Friedhelm Meyer auf der Heide Lecture 1, 8.4.2011. HEINZ NIXDORF INSTITUTE

HEINZ NIXDORF INSTITUTE

University of PaderbornAlgorithms and Complexity

Welcome to the course

Algorithm Design

Friedhelm Meyer auf der Heide 1

Summer Term 2011

Friedhelm Meyer auf der Heide

Lecture 1, 8.4.2011



Organization

Familiarity with some basic algorithms and data structures and their analyses as presented in the course "Datenstrukturen und Algorithmen" is assumed.

The lecture and tutorials will be given in English. (The German title of this course is “Methoden des Algorithmenentwurfs”.)


Lecture Friday 11:15 - 13:00 F0.530 Tutorial I Friday 10:15 - 11:00 F2.211 (Ralf Petring)Tutorial II Friday 13:15 - 14:00 F2.211 (Ralf Petring)

Exercises will be available on Wednesday afternoon. Return deadline is next weeks Thursday at 10 am. The solutions can be sent via e-mail (only PDF documents are accepted) to Ralf Petring or dropped at F1.203.



Examination

Oral examinations of 20 minutes duration. The exams will cover the lecture and also the tutorials and homework exercises.

You need at least 30% of the total number of points in the homework exercises for the admission to exam.

Bonus Scheme

You can improve your grade by actively participating in the weekly tutorials and


You can improve your grade by actively participating in the weekly tutorials andsuccessfully solving the weekly homework exercises. For achieving at least 50 % of the total number of points in the homework exercises, your grade will improve by 1/3, for achieving at least 75 %, your grade will improve by 2/3. This only applies if you pass the examination.

Module Information

Module II.2.1 Modelle und Algorithmen (Models and algorithms)V2 + Ü1 SWS4 ECTS credits (workload)



Topics

- Divide & conquer

- Dynamic programming

- Greedy Algorithms

- Randomized Algorithms

- Approximation Algorithms


- Approximation Algorithms

- Online Algorithms



Asymptotic Runtime Analysis




The O-Notation




The O-Notation




The O-Notation




Chapter 1:

Divide & Conquer


Divide & Conquer



Divide & Conquer

Divide the problem in a number of smaller subproblems.

Conquer the subproblems by solving them recursively.

Combine the results of the subproblem into a solution of the original problem.


of the original problem.



Divide & Conquer: Integer Multiplication

The school method needs Θ(n2) operations.

Divide the problem in a number of smaller subproblems.Conquer the subproblems by solving them recursively.





Divide & Conquer: Integer Multiplication

Divide the problem in a number of smaller subproblems.Conquer the subproblems by solving them recursively.



If n=1: c = a0×b0

If n>1: c = A0×B0 + (A0×B1 + B0×A1)×2k + A1×B1×22k

Cost measure: # bit operations

T(1) = 1, For n>1, n a power of 2: T(n)=4 T(n/2) + αααα n

Thus T(n) = O(n²) � no improvement

Multiplications with powers of 2 (shifts) and additions.



The Karazuba-Offman algorithm for integer multiplication, 1961


Cost measure: # bit operations

T(1) = 1For n>1, n a power of 2: T(n) ≤ 3 T(n/2 + 1) + αααα n

Thus T(n) = O(nlog2(3) ) ~ O(n1.58 ) � improvement !!!

Multiplications with powers of 2 (shifts) and additions.



A master theorem

Theorem: Let a, c > 0, b>1, be integer constants. Let T: N → N be recursively defined by

T(1) ≤ c, and for n > 1, n a power of b: T(n) ≤ a T(n/b) + cn

Then, for- a < b: T(n) ≤ cb/(b-a) n

a b c


- a = b: T(n) ≤ cn (logb(n) + 1) = O(n log(n))

- a > b: T(n) ≤ ca/(a-b) nlogb(a)

Proof: Homework



Divide & Conquer: Closest Pair




















































Thank you for your attention!Thank you for your attention!


Friedhelm Meyer auf der Heide

Heinz Nixdorf Institute

& Computer Science Department

University of Paderborn

Fürstenallee 11

33102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 80

Fax: +49 (0) 52 51/62 64 82

E-Mail: [email protected]

http://www.upb.de/cs/ag-madh

your attention!your attention!

Documents

Algorithm Design · Welcome to the course Algorithm Design Friedhelm Meyer auf der Heide 1 Summer Term 2011 Friedhelm Meyer auf der Heide Lecture 1, 8.4.2011. HEINZ NIXDORF INSTITUTE