Welcome to the course Algorithm Design

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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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Topics

- Divide & conquer

-

Dynamic programming

-

Greedy Algorithms

- Approximation Algorithms

- Randomized Algorithms

- Online Algorithms

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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) = §_e_{2 B} w(e).

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Subset-Systems and the Canonical Greedy-Algorithm

Canonical Greedy ((E,U))

(1) Sort E such that w(e

₁

)

¸

…

¸

w(e

_n

).

(2) B

?

.

(3) For k=1 to n

if B

[

{e

_k

}

2 U then B B

[

{e

_k

}

(4) Return B

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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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Matroids and the canonical Greedy Algorithm

Proof:

“(”: Consider a matroid (E,U) and a weight function w. Let w(e₁)¸ …¸ w(e_n).

- Consider an optimal solution T’={e_i1, …, e_ik}.

- Assume the solution T={e_j1, …, e_jk} found by Canonical Greedy were not optimal, i.e. w(T’)> w(T).

- Then there is an index p with w(e_ip) > w(e_jp). Let p be minimal with this property. Note that i_p < j_p, because the items are sorted w.r.t weight. - Apply the exchange property to A = {e_j1, …,e_jp-1} and B ={e_i1, …, e_ip} : As |A| < |B|, there is e_iq 2 B-A with A [ {e_iq} 2 U.

- As w(e_iq ) ¸ w(e_ip) > w(e_jp), Canonical Greedy would have chosen e_iq before

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Matroids and the canonical Greedy Algorithm

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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Topics

- Divide & conquer

-

Dynamic programming

-

Greedy Algorithms

-

Approximation Algorithms

- Randomized Algorithms

- Online Algorithms

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23.05.2011 10

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

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Load Balancing: List Scheduling A D F B C E Machine 3 Machine 2 Time 0 Machine 1 I H J G

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Machine 3 Machine 2 Machine 1

Load Balancing: List Scheduling

A D F B C E Time 0 I H J G

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A F B C E Time 0 I H J G D

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G

A F B C E Time 0 I H J D

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A F B C G E Time 0 I H J G D

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A F B C G E Time 0 I H J D

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A D F B C G E 0 I H J Machine 3 Machine 2 Machine 1 A D F B C G E 0 I H J Optimal Schedule List schedule

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1. 2.

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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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23.05.2011 40

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 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

Welcome to the course Algorithm Design

Welcome to the course

Algorithm Design

- Divide & conquer

-

Dynamic programming

-

Greedy Algorithms

- Approximation Algorithms

- Randomized Algorithms

- Online Algorithms

Canonical Greedy ((E,U))

(1) Sort E such that w(e

)

¸

…

¸

w(e

).

(2) B

?

.

(3) For k=1 to n

if B

[

{e

}

2

U then B B

[

{e

}

(4) Return B

- Divide & conquer

-

Dynamic programming

-

Greedy Algorithms

-

Approximation Algorithms

- Randomized Algorithms

- Online Algorithms

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

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

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

Thank you for

your attention!