algorithm design - heinz nixdorf institut · friedhelm meyer auf der heide 5 heinz nixdorf...
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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 6, 20.5.2011
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Friedhelm Meyer auf der Heide 2
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity Topics
- Divide & conquer
- Dynamic programming
- Greedy Algorithms
- Approximation Algorithms
- Randomized Algorithms
- Online Algorithms
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Friedhelm Meyer auf der Heide 3
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
For which problems are Greedy-Algorithms
optimal?
Consider a finite set E and a system U of subsets of E. (E,U) is called a
subset-system, if the following holds:
(i) ? 2 U
(ii) For each B 2 U, also each subset of B is in U.
B2 U is called maximal, if no proper superset of B is in U.
The Optimization problem corresponding to (E,U) is :
Given a weight function w:E Q+ ,compute a maximal set B 2 U with
maximizes w(B) = §e2 B w(e).
(Minimization problems are described analogously.)
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Friedhelm Meyer auf der Heide 4
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Subset-Systems and the
Canonical Greedy-Algorithm
Canonical Greedy ((E,U))
(1) Sort E such that w(e1) ¸ … ¸ w(en).
(2) B ?.
(3) For k=1 to n
if B [ {ek} 2 U then B B [ {ek}
(4) Return B
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Friedhelm Meyer auf der Heide 5
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity Greedy-Algorithms und Matroids
A system of subsets (E,U) is a matroid, if in addition, the following
exchange property holds:
(iii) For all A,B 2 U with |A|<|B|, there is x2 B-A such that A[{x} 2 U.
Remark: All maximal sets of a matroid have the same size. (homework)
Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal
for (E,U) for every weight function w, if and only if (E,U) is a matroid.
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Friedhelm Meyer auf der Heide 6
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity Matroids and the canonical Greedy Algorithm
Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal
for (E,U) for every weight function w, if and only if (E,U) is a matroid.
Proof:
“(”: Consider a matroid (E,U) and a weight function w.
Let w(e1)¸ …¸ w(en).
- Consider an optimal solution T’={ei1, …, eik
}.
- Assume the solution T={ej1, …, ejk
} found by Canonical Greedy were not
optimal, i.e. w(T’)> w(T).
- Then there is an index p with w(eip) > w(ejp
). Let p be minimal with this
property. Note that ip < jp, because the items are sorted w.r.t weight.
- Apply the exchange property to A = {ej1, …,ejp-1
} and B ={ei1, …, eip
} :
As |A| < |B|, there is eiq 2 B-A with A [ {eiq
} 2 U.
- As w(eiq ) ¸ w(eip
) > w(ejp), Canonical Greedy would have chosen eiq
before
ejp , a contradiction.
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Friedhelm Meyer auf der Heide 7
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity Matroids and the canonical Greedy Algorithm
Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal for (E,U) for every weight function w, if and only if (E,U) is a matroid.
Proof:
“)”: Assume that, for some A,B 2 U with |A|<|B|, it holds that A[{e} is not in U, for every e2B-A.
Let |B|=r and consider the weight function w with
- w(e) = r+1 for e2A
- w(e) = r for e2B-A
- w(e) = 0 else
Then, Canonical Greedy would select a solution T with T µ A and T Å (B-A) =?.
As w(T)=|A|(r+1) · (r-1)(r+1)· r2 -1, T is not optimal, because B is a solution with larger weight |B|r = r2.
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Friedhelm Meyer auf der Heide 8
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity Topics
- Divide & conquer
- Dynamic programming
- Greedy Algorithms
- Approximation Algorithms
- Randomized Algorithms
- Online Algorithms
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Approximation Algorithms
• Greedy Techniques – Load-Balancing Problem
– Center Selection Problem
• Pricing Method – Vertex Cover Problem
• Linear Programming and Rounding – Vertex Cover Problem
– Generalized Load-Balancing Problem
• Polynomial Time Approximation Scheme – Knapsack Problem
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Load Balancing: List Scheduling
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Approximation Algorithms
• Greedy Techniques – Load-Balancing Problem
– Center Selection Problem
• Pricing Method – Vertex Cover Problem
• Linear Programming and Rounding – Vertex Cover Problem
– Generalized Load-Balancing Problem
• Polynomial Time Approximation Scheme – Knapsack Problem
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Approximation Algorithms
• Greedy Techniques – Load-Balancing Problem
– Center Selection Problem
• Pricing Method – Vertex Cover Problem
• Linear Programming and Rounding – Vertex Cover Problem
– Generalized Load-Balancing Problem
• Polynomial Time Approximation Scheme – Knapsack Problem
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Friedhelm Meyer auf der Heide 46
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!