Presentation is loading. Please wait.

Presentation is loading. Please wait.

Lecture 11 Overview Self-Reducibility. Overview on Greedy Algorithms.

Similar presentations


Presentation on theme: "Lecture 11 Overview Self-Reducibility. Overview on Greedy Algorithms."— Presentation transcript:

1 Lecture 11 Overview Self-Reducibility

2 Overview on Greedy Algorithms

3 Revisit Minimum Spanning Tree

4 Exchange Property

5 Self-Reducibility

6 Max Independent Set in Matroid

7 Exchange Property

8 Self-Reducibility

9

10 Overview on Greedy Algorithms Exchange Property Matroid Self-Reducibility

11 Local Ratio Method

12 Basic Idea Proof

13 Basic Idea

14 Minimum Spanning Tree

15 Activity Selection

16 Puzzle

17 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

18 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

19 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

20 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

21 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

22 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

23 23 Running Example P(I 1 ) = 5 -5 P(I 4 ) = P(I 3 ) = 5 -5 P(I 2 ) = 3 -5 P(I 6 ) = P(I 5 ) = Slide from

24 Minimum Weight Arborescence

25 Definition

26 Problem

27 Key Point 1

28 Key Point 2

29 Why?

30 Key Point 3 0

31 A Property of MST

32

33

34

35

36

37


Download ppt "Lecture 11 Overview Self-Reducibility. Overview on Greedy Algorithms."

Similar presentations


Ads by Google