algorithm design - heinz nixdorf institut · friedhelm meyer auf der heide 5 heinz nixdorf...

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




Algorithms and Complexity Topics

- Divide & conquer

- Dynamic programming

- Greedy Algorithms

- Approximation Algorithms

- Randomized Algorithms

- Online Algorithms





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.)





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




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.




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.




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.




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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Optimal Schedule

List schedule

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Friedhelm Meyer auf der Heide

Heinz Nixdorf Institute

& Computer Science Department


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!

algorithm design - heinz nixdorf institut · friedhelm meyer auf der heide 5 heinz nixdorf...

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