chapter 4 partition (3) double partition ding-zhu du
TRANSCRIPT
Partition a big thing (hard to deal) into small ones (easy to deal).
It is a natural idea.
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.
)/1( size cell
in timeion approximat-)1( )/1( 2
O
nO
timerunning ratio eperformanc
ratio? eperformanc improve to
how grow,cannot size cellWhen
Weighted Dominating Set in unit disk graphs
Given a unit disk graph G=(D,E) with node weightc:D→R, find a dominating set with minimum total weight.+
<1
Backgroud
• 72-approximation (Ambuhl, et al. 2006).
known. ision approximat-constant) (small no
,2/2 size edge-cellFor
exists.ion approximat-2
,2/2 size edge-cellFor
• 72-approximation (Ambuhl, et al. 2006).
• (6+ε)-approximation (Gao, et al. 2008).
Partition into big cells
ijB),( jmim
2
2m
General Case
Partition
ijS),( ji
2
2
No node lies on a cut-line.
.in nodes ofset thedenote ijij SD
.in node aleast at dominatingeach
,in nodes ofsubset thedenotes )(
ij
ijij
D
DDDN
.in nodes dominating
)(in subset -nodeweight -minimum a Find
ij
ijij
D
DND
Problem A(i,j)
1
Dominating area
Lemma Problem A(i,j) has 2-approximation.
for WDS.ion approximat-28 is Then
j).A(i, Problemfor ion approximat-2 be Let
ijij
ij
U
U
Theorem
?)(in becan node a ,many howFor ijij SNS
16 ?
14 !
,282)()(
Then
.14
Then j).A(i, Problemfor
solution optimalan of weight total thebe Let
optoptUcUc
optopt
opt
ij ijij ijijij
ij ij
ij
j).A(i, Problemfor ion approximat-2 be where ijU
2-approximation for A(I,j)
Case 1
Minimum weight of node in Dij
2-approximation for A(I,j)
Case 2. nodes in N(Dij) dominate nodes in Dij
?
AL AM AR
CL CR
BL BM BR
A problem on strip: outside disks cover inside points
p1
p2
pi
Ti(D,D’) : minimum weight set with D, D’, dominating p1, …, pi 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
otherwise. 0
, if 1][
, passing line with ' ofn that higher thanot point on intersecti
havingeach disks allover is , passing line with ofthat
lower thannot point on intersecti havingeach disk allover is
where
)}(][
)(][
)({min)(
11
2
1
2
1
211, 21
DDDD
pD
DpD
D
D'cD'D+
DcDD+
,DDT = D,D'T
i
i
iDDi
Dynamic Programming
otherwise. 0
, if 1][
, passing line with ' ofn that higher thanot point on intersecti
havingeach disks allover is , passing line with ofthat
lower thannot point on intersecti havingeach disk allover is
where
)}(][
)(][
))(({min))((
11
2
1
2
1
211, 21
DDDD
pD
DpD
D
D'cD'D+
DcDD+
,DDTw = D,D'Tw
i
i
iDDi
Dynamic Programming
p1
p2
pi-1
D1 (lowest intersection point on L’) among disks above the strip, in Ti(D,D’)
D2 (highest intersection point on L’) among disks below the strip, in Ti(D,D’)
L’
pj
pi
pi-1
D
. does so ),1( covers If 1DijpD j
D1
2-approximation for A(I,j)
Case 2. nodes in N(Dij) dominate nodes in Dij
?
)( plow
p
LMu
Lemma If p is dominated by u in LM area, then every point in is dominated by u.)( plow
p
u
v
22
p
u
v
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
CL CR
)',( pplow
)',( pplow
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 BMbut not nodes in CL and CR.
)',( pplow
p p’
Consider OPT
)'( plow
)( plow
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 UMbut not nodes in CL and CR.
)',( qqup
q q’
Consider OPT
)'(qup
)(qup
R Lin nodesby dominated are
)',()',(in not node All
L.in U nodesby dominated
are )',()',(in nodes All
qqpp
qqpp
uplow
uplow
Consider OPT
L R
U
R
How do we find p, p’, q, q’?
Try all possibilities.
4441 )14( nnn t
How many possibilities?
Idea: Combine cells into a strip
Each strip contains m cells.
nscombinatio )( 2mOn
6-approximation for a special case:
constant a is m
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.
s22
Cm
nscombinatio )( 2mOn
)()( 222
2 :time mOmOm nn
?innot but dominate,can node a ,strips howFor
6!
1
2
3
4 5 6
Partition into big cells
ijB),( jmim
2
2m
General Case
(6+ε)-approximation in general case
Shafting to minimize # of disks on boundaries
optm
6)8
1(
/48m
(9.875+ε)-approximation for minimum weight connected dominating set in unit disk graph.
Connecting a dominating set into a cdsneeds to add at most 3.875 opt nodes. (Zou et al, 2008)
(improved 17opt)
