Download presentation

Presentation is loading. Please wait.

Published byJamel Garley Modified over 3 years ago

1
Chapter 4 Partition (3) Double Partition Ding-Zhu Du

2
Partition a big thing (hard to deal) into small ones (easy to deal). It is a natural idea.

3
Partition is also an important technique in design of approximation algorithms. Example: To find a dominating set, we may find a dominating set in each small area.

5
Weighted Dominating Set in unit disk graphs Given a unit disk graph G=(D,E) with node weight c:DR, find a dominating set with minimum total weight. + <1<1

6
Backgroud 72-approximation (Ambuhl, et al. 2006).

7
(6+ε)-approximation (Gao, et al. 2008).

8
Partition into big cells General Case

9
Partition No node lies on a cut-line.

10
Problem A(i,j) 1 Dominating area

11
Lemma Problem A(i,j) has 2-approximation. Theorem

12
16 ? 14 !

14
2-approximation for A(I,j) Case 1 Minimum weight of node in D ij

15
2-approximation for A(I,j) Case 2. nodes in N(D ij ) dominate nodes in D ij ? ALAM AR CL CR BL BM BR

16
A problem on strip: outside disks cover inside points p1p1 p2p2 pipi Ti(D,D) : minimum weight set with D, D, dominating p 1, …, p i such that (1) D (lowest intersection point on L) among disks above the strip (2) D(highest intersection point on L) among disks below the strip L

17
Dynamic Programming

19
p1p1 p2p2 p i-1 D 1 (lowest intersection point on L) among disks above the strip, in T i (D,D) D 2 (highest intersection point on L) among disks below the strip, in T i (D,D) L

20
pjpj pipi p i-1 D D1

21
2-approximation for A(I,j) Case 2. nodes in N(D ij ) dominate nodes in D ij ?

22
LM Lemma If p is dominated by u in LM area, then every point in is dominated by u.

23
p u v p u v

24
p p Lemma If p and p can be dominated by nodes in BM but not nodes in CL and CR, then every node in can be dominated in nodes in A and B. A B CLCR

25
is the leftmost one for p dominated by a node in BM, but not any node in CL and CR is the rightmost one for p dominated by a node in LM, but not any node in CL and CR contains all nodes dominated by nodes in BM but not nodes in CL and CR. pp Consider OPT

26
is the leftmost one for p dominated by a node in UM, but not any node in CL and CR is the rightmost one for p dominated by a node in UM, but not any node in CL and CR contains all nodes dominated by nodes in UM but not nodes in CL and CR. qq Consider OPT

27
L R U R

28
How do we find p, p, q, q? Try all possibilities. How many possibilities?

29
Idea: Combine cells into a strip Each strip contains m cells.

30
6-approximation for a special case: For every subset C of cells, 1.every cell e in C is in case 1; 2.every cell e not in C is in case 2.

31
6! 1 2 3 4 56

32
Partition into big cells General Case

33
(6+ε)-approximation in general case Shafting to minimize # of disks on boundaries

34
(9.875+ε)-approximation for minimum weight connected dominating set in unit disk graph. Connecting a dominating set into a cds needs to add at most 3.875 opt nodes. (Zou et al, 2008) (improved 17opt)

Similar presentations

OK

1 Data Structures CSCI 132, Spring 2014 Lecture 1 Big Ideas in Data Structures Course website:

1 Data Structures CSCI 132, Spring 2014 Lecture 1 Big Ideas in Data Structures Course website:

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on human chromosomes 23 Ppt on first conditional exercise Ppt on javascript events status Ppt on ramayana in hindi Ppt on network theory definition Ppt on medieval history of india Ppt on digital television technology Ppt on event driven programming ppt Ppt on weather and climate for grade 5 Ppt on 14 principles of management