Download presentation

Presentation is loading. Please wait.

Published byJesse Lockhart Modified over 2 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

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

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google