Embed Size (px)
Citation preview
Intersection Disk Graph
Consider n points in the Euclidean plane, each is associated with a disk. An edgeexists between two points if and only if their associated disks have nonempty intersection.
Maximum Independent Set in Intersection Disk Graph
Given a intersection disk graph D, find a maximumIndependent set opt(D).
Multi-layer
Suppose the largest disk has diameter 1-ε. Let dmin be The diameter of smallest disk. Fix an integer k > 0.Let
.)/1(log min1 dm k
Put all disks into m+1 layers. For 0 < j < m, layer j consists of all disks with diameter di,
)1()1()1( ji
j kdk
Partition P(0,0) in layer j
jkk )1(
jk )1(
(0,0)
Partition P(0) in layer j and layer j+1
jkk )1()1()1( jkk
Partition P(a,b) in layer j
jkk )1(
jk )1(
))1( ,)1(( jj kbka
)1()1( jk
jka )1( jkika )1)((
)1()1)(( jkika)1()1( jka
)1()1)()(()1)(( jj kkikaiakika
Layer j
Layer j+1
A disk hits a cut line.
At each layer, a disk can hit at most one among Parallel lines apart each other with distance .
jk )1(
In partition P(a,b), delete all disks each hitting a cut line in the same layer. The remaining disks form a collection D(a,b).
D(a,b)
Maximum Independent set opt(D(a,b)) can be computed in time
)( 4kOnby dynamic programming. Why use it?
Dynamic Programming
j-cell is a cell in layer j.
For any j-cell S and a set I of independent disks in layers < j, intersecting S,
Table(S,I) = maximum independent set of disks layers > j, contained in S, and disjoint from I.
opt(D(a, b)) = US Table(S, Ǿ)where S is over all cells in layer 0.
Recursive Relation
For j-cell S and I,
}.' hits | {'for
)','(Table),'Table(
and in cells-)1( all
over is ' , fromdisjoint and ngintersecti
,layer in disks of subsets allover is where
)),'(Table(y cardinalit-max),(Table'
SdiskJIdiskJ
JSJIS
Sj
SIS
jJ
IJSJISsJ
# of Table(S,I)
# of S = too large
How do we overcome this difficulty?
Relevant cell:
A j-cell is relevant if it contains a disk in layer j.
Dynamic Programming
j-cell is a cell in layer j.
For any relevant j-cell S and a set I of independent disks in layers < j, intersecting S,
Table(S,I) = maximum independent set of disks layers > j, contained in S, and disjoint from I.
opt(D(a, b)) = USTable(S, Ǿ)where S is over all maximal relevant cells.
Children of a relevant cell
).(
'''such that '' cell-relevant
another no is thereand ' if cell-
relevant a of child a is ' cell-relevant A
jhi
SSSSh
SSj
Si
SS’’S’
Maximal relevant cell
A relevant cell is maximal if it is not contained byAnother relevant cell.
Recursive Relation
For j-cell S,
}.' hits | {'for
)','(Table),'Table(
and in children all
over is ' , fromdisjoint and ngintersecti
,layer in disks of subsets allover is where
)),'(Table (y cardinalit-max),(Table'
SdiskJIdiskJ
JSJIS
S
SIS
jJ
IJSJISsJ
# of Table(S,I)
# of relevant S = n.
# of I = )( 2kOn
# of Table(S,I) = )( 2kOn
# of I
S
jkk )1(jkk )1)(2(
jk )1(
22
2
)2(4
)2/)1((
))1)(2((
kk
kkj
j
# of I’s =)( 2kOn
Computation Time of Recursion
# of S’ =
# of J =
)(nO
)( 4kOn
Time = )()( 44
)( kOkO nnnO
Running Time of dynamic programming
)()()( 442 kOkOkO nnn
# of J
S
jkk )1(
22
2)1(
2
)1(4
)2/)1((
))1(( ||
kk
k
kkJ
j
j
)( 4
of # kOnJ
(1+ε)-Approximation
Compute opt(D(0,0)), opt(D(1,1)), …, opt(D(k-1,k-1).
Choose k = ?.
Choose maximum one among them.
Analysis
• Consider an optimal solution D*.
• For each partition P(a,b), let H(a,b) be the collection of all disks hitting cut line in the same layer.
• Estimate |H(0,0)|+|H(1,1)|+···+|H(k-1,k-1)|.
|H(0,0)|+|H(1,1)|+···+|H(k-1,k-1)|
Each disk appears in at most two terms in this sum.
There exists i such that |H(2i,2i)| < 2|D*|/k.
Performance ratio
Opt/approx =1/(1-2/k) = 1 + 2/(k-4)
Choose
1
12k
We obtain a (1+ε)-approximationWith time
)/1( 4On
Thanks, End
