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Lecture 11 Overview Self-Reducibility

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Overview on Greedy Algorithms

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Revisit Minimum Spanning Tree

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Exchange Property

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Self-Reducibility

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Max Independent Set in Matroid

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Exchange Property

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Self-Reducibility

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Overview on Greedy Algorithms Exchange Property Matroid Self-Reducibility

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Local Ratio Method

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Basic Idea Proof

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Basic Idea

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Minimum Spanning Tree

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Activity Selection

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Puzzle

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17 Independent Set in Interval Graphs Activity 9 Activity 8 Activity 7 Activity 6 Activity 5 Activity 4 Activity 3 Activity 2 Activity 1 We must schedule jobs on a single processor with no preemption. Each job may be scheduled in one interval only. The problem is to select a maximum weight subset of non-conflicting jobs. time

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18 Independent Set in Interval Graphs Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1 Maximize s.t.For each instance I For each time t time Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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19 Maximal Solutions We say that a feasible schedule is I-maximal if either it contains instance I, or it does not contain I but adding I to it will render it infeasible. Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1 time I2I2 I1I1 The schedule above is I 1 -maximal and also I 2 -maximal

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20 An effective profit function P 1 = P( Î) P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1 Let Î be an interval that ends first; Î P 1 = P( Î) Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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21 An effective profit function P 1 = P( Î) P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1 Î P 1 = P( Î) For every feasible solution x: p 1 ·x p(Î) For every Î-maximal solution x: p 1 ·x p(Î) Every Î-maximal is optimal. Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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22 Independent Set in Interval Graphs: An Optimization Algorithm Algorithm MaxIS( S, p ) 1.If S = Φ then return Φ ; 2.If I S p(I) 0 then return MaxIS( S - {I}, p); 3.Let Î S that ends first; 4. I S define: p1 (I) = p(Î) (I in conflict with Î) ; 5.IS = MaxIS( S, p- p1 ) ; 6.If IS is Î-maximal then return IS else return IS {Î}; Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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23 Running Example P(I 1 ) = 5 -5 P(I 4 ) = 9 -5 -4 P(I 3 ) = 5 -5 P(I 2 ) = 3 -5 P(I 6 ) = 6 -4 -2 P(I 5 ) = 3 -4 -5 -4 -2 Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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Minimum Weight Arborescence

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Definition

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Problem

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Key Point 1

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Key Point 2

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Why?

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Key Point 3 0

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A Property of MST

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