Chapter 4 Partition

(1) Shifting

Ding-Zhu Du

Disk Covering

• Given a set of n points in the Euclidean plane, find the minimum number of unit disks to cover the n given points.

(x,x)

Partition P(x)

a

Construct Minimum Unit Disk Cover in Each Cell

1/√2

Each square with edge length1/√2 can be covered by a unitdisk.Hence, each cell can be coveredBy at most disks.

Suppose a cell contains ni points.Then there are ni(ni-1) possiblepositions for each disk.

Minimum cover can be computed In time ni

O(a )2

22a

Solution S(x) associated with P(x)

For each cell, construct minimum cover.S(x) is the union of those minimum covers.

Suppose n points are distributed into k cells containing n1, …, nk points, respectively.Then computing S(x) takes time

n1 + n2 + ··· + nk < nO(a ) O(a ) O(a ) O(a )

2 2 2 2

Approximation Algorithm

For x=0, -2, …, -(a-2), compute S(x).

Choose minimum one from S(0), S(-2), …, S(-a+2).

Analysis

• Consider a minimum cover.

• Modify it to satisfy the restriction, i.e.,

a union of disk covers each for a cell.

• To do such a modification, we need to add some disks and estimate how many added disks.

Added DisksCount twice

Count four times

2

2

Shifting

Estimate # of added disks

Shifting

Estimate # of added disks

Vertical strips

Each disk appearsonce.

Estimate # of added disks

Horizontal strips

Each disk appears once.

Estimate # of added disks

# of added disks for P(0)

+ # of added disks for P(-2)+ ···+ # of added disks for P(-a+2)

< 3 opt

where opt is # of disk in a minimum cover.

There is a x such that # of added disks for P(x) < (6/a) opt.

Performance Ratio

P.R. < 1 + 6/a < 1 + ε when we choose a = 6 ⌠1/ε .

Running time is n.O(1/ε )2

Unit disk graph

< 1

Dominating set in unit disk graph

• Given a unit disk graph, find a dominating set with the minimum cardinality.

• Theorem This problem has PTAS.

Connected Dominating Set in Unit Disk Graph

• Given a unit disk graph G, find a minimum connected dominating set in G.

Theorem There is a PTAS for connected dominating set in unit disk graph.

Boundary area

central area

h

h+1

Why overlapping?

cds for G

cds for eachconnectedcomponent 1

1. In each cell, construct MCDS for each connected component in the inner area.

Construct PTAS

2. Connect those minimum connected dominating setswith a part of 8-approximation lying in boundary area.

For each partition P(a,a), construct C(a) as follows:

Choose smallest C(a) for a = 0, h+1, 2(h+1), ….

Existence of 8-approximation

1. There exists (1+ε)-approximation for minimum dominating set in unit disk graph.

2. We can reduce one connected component with two nodes.

Therefore, there exists 3(1+ε)-approximation for mcds.

8-approximation

1. A maximal independent set has size at most 4 mcds +1.

2. There exists a maximal independent set, connecting it into cds need at most 4mcds nodes.

MCDS (Time)

2/2

2)2(a

1. In a square of edge length , any node can dominate every bode in the square. Therefore, minimum dominating set has size at most .

a

2/2

MCDS (Time)

2/2

2. The total size of MCDSs for connected components in an inner square area is at most .

a

3)2(3 a

nnaO

ii

aO

i

i

n

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is time total thecells, allOver

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time takesareainner in the components

connected allfor cell in the MCDSs all

findingThen nodes. cotains cell a Suppose

MCDS (Size)

• Modify a mcds for G into MCDSs in each cell.

• mcds(G): mcds for G

• mcdscell(inner): MCDS in a cell for connected components in inner area

Connect & Charge

charge

Multiple Charge

charge

How many possiblecharges for each node?

How many componentscan each node be adjacent to?

1. How many independent points can be packed by a disk with radius 1?

1

>1

5!

Each node can be charged at most 10 times!!!

charges. 10most at receives nodeEach

nodes. on mde be willchanges 10most At

nodes. 2 tocharge a makecomponent Each

.components 5most at connect tocan nodes

kk

kk

Shifting

3

a/(2(h+1)) = integer

Time=nO(a )2

h=2

dimesion.any in in timeion approximat-)1( )/1( 2 On

Weighted Dominating Set

• Given a unit disk graph with vertex weight, find a dominating set with minimum total weight.

• Can the partition technique be used for the weighted dominating set problem?

Dominating Set in Intersection Disk Graph

• An intersection disk graph is given by a set of points (vertices) in the Euclidean plane, each associated with a disk and an edge exists between two points iff two disks associated with them intersects.

• Can the partition technique be used for dominating set in intersection disk graph?

Thanks, End