Download presentation

Presentation is loading. Please wait.

Published byJesse Lockhart Modified over 3 years ago

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

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 http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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 http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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 http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

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 http://www.cs.technion.ac.il/~reuven/STOC2000.ppt

23
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

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

Similar presentations

Presentation is loading. Please wait....

OK

Forward looking statement

Forward looking statement

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on 2nd world war dates Free ppt on brain machine interface software Ppt on different types of reactions and their applications Ppt on video teleconferencing standards Ppt on airbag working principle of air Training ppt on team building Ppt on product life cycle in marketing Ppt on computer languages basic Ppt on 4g mobile technology free download Ppt on team building process