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: [email protected]
http://www.upb.de/cs/ag-madh
Thank you for
your attention!
