Friedhelm Meyer auf der Heide 1

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity

Welcome to the course

Algorithm Design

Summer Term 2011

Friedhelm Meyer auf der Heide

Lecture 4, 6.5.2011

Lecture today given by Ulf-Peter Schröder

Friedhelm Meyer auf der Heide 2

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Today:

Chapter 3:

Greedy Algorithms

Friedhelm Meyer auf der Heide 3

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity The Greedy Paradigm

Friedhelm Meyer auf der Heide 4

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Greedy is not always optimal

Friedhelm Meyer auf der Heide 5

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Greedy is not always optimal

Friedhelm Meyer auf der Heide 6

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity …but sometimes

Friedhelm Meyer auf der Heide 7

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity …but sometimes

Fundamental Questions

• When did a Greedy Algorithm succeeds in

solving a nontrivial problem optimally?

• How to prove that a Greedy Algorithm

produces an optimal solution to a problem?

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The Greedy Method • Greedy as Algorithmic technique can optimally solve an

optimization problem,

– if the greedy algorithm stays ahead. Show that after each step of the greedy algorithm, its solution is at least as good as any other algorithm’s or

– if we can find an exchange argument. Gradually transform any solution to the one found by the greedy algorithm without hurting its quality or

– if the problem has the greedy-choice property. Prove that a globally-optimal solution can always be found by a series of local improvements from a starting configuration or

– if we can define an adequate Matroid for the problem.

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Examples

• Activity Selection (or Intervall Scheduling) – Greedy stays ahead

• Scheduling to Minimize Lateness – Exchange Argument

• Theoretical Foundations of the Greedy Method – Matroid Theory

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Activity Selection

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If one measures the Greedy Algorithm’s progress in a step-by-step fashion, one sees that it does better than any other algorithm at each step (“greedy stays ahead”); it then follows that it produces an optimal solution!

What have we shown ?

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Examples

• Activity Selection (or Intervall Scheduling) – Greedy stays ahead

• Scheduling to Minimize Lateness – Exchange Argument

• Theoretical Foundations of the Greedy Method – Matroid Theory

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Any possible solution to the problem can be transformed (Exchange Property) into the solution found by the Greedy Algorithm without hurting its quality; it then follows that the greedy algorithm must have found a solution that is at least as good as any other solution!

What have we shown ?

Friedhelm Meyer auf der Heide 38

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity

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: fmadh@upb.de

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

Thank you for

your attention!