Chapter 4 Partition

I. Covering and Dominating

Unit 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

Partition as a restriction

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 for all cells.

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

Added DisksCount twice

Count four times

2

So, we have a 4-approximation.

2

Shifting

Estimate # of added disks

Shifting

Estimate # of added disks

In vertical strips,

each disk appearsonce.

Estimate # of added disks

In 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 exists an x such that # of added disks for P(x) < (6/a) opt.

(each disk can be addedonly to one P(a).)

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.

• Note: This is just the unit disk covering problem with the restriction that each disk must be centered at an input point.

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.

Existence of 4-approximation

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

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

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

Partition

But, how do we combine solutionsin each cell together?

Boundary area

central area

h

h+1

Add extra vertices around the Boundary area

Why overlapping?

cds for G

cds for eachconnectedcomponent 1

1. In each cell, construct MCDS for each conn. component in the central area.

Constructing a PTAS

2. Find a 4-approximation D of MCDS of the whole graph, and add Dbound to the solution

Step 1 is a restriction:

In each central area of a cell e, the feasible solution C[e] must satisfy:

Each conn. component H of G[e] is dominatedby a single conn. component of C[e].

The resulting set of nodes must be a dominating set.

This set is also connected:

1. Two conn. components of Dbound can be connected in D through a conn. component in a central area A. The end points of these components must be dominated by MCDS of A. So, the two components together with MCDS of A are connected together.

2. Every conn. component C of MCDS of a central area is connected to Dbound.

• A point x in C must be dominated by some point y in D.

• y is connected to a point z in Dbound, with all points in the path lying in central area.

• This path and C are in the same conn. component, and so is dominated by C.

• So, C is connected to z.

MCDS (time)

2/2

2)2(a

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

a

2/2

2/2

2. The total size of MCDSs for connected components in a central area is at most .

a

3)2(3 a

nnaO

ii

aO

i

i

n

n

aO

)()(

22

)2(

is time total thecells, allOver

.

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.

• D*: MCDS for G

• D*[e]: MCDS in a cell e

• D*[e] may not satisfy the restriction; i.e., D*[e] may contain k > 1 components that are in the same component of G[e].

Estimate P.R.

For an MCDS D*, modify it as follows:

(1) In each central area, connect all

conn. components of D*[e] that are in

the same component of G[e].

(2) Add Dbound to it.

Use Charging Method to count the extra from (1).

Use Shifting Technique to reduce it.

Charge the extra vertices toa boundary point

charging

Charged to

Rule 1: Each componentIs charged at most twice.

Rule 2: In each component,charge to the point just outsidethe central area.

Multiple Charges

charge

How many possiblecharges for each Boundary node?

How many componentscan each node be adjacent to?

How many independent points can be packed in a half disk with radius 1?

1

>1

3

Each node can be charged at most 6 times!!!

Each node can connect to at most 3 components.

Each component makes at most 2 charges to a node.

Therefore, each node can be charged at most 6 times.

Shifting

• Shifting the

partition with

distance 1.

• Each vertex can

appear in the

boundary area of

at most 4(h+1)

partitions.

Extra nodes in a fixed partition P(a)

• Each boundary point of D* may be charged 6 times

• Each boundary point of D is used once.

• Together, we get 6 |D*bound| + |Dbound|

By shifting, the total extra nodes in all partitions:

(6 x 4(h+1) + 4 x 4(h+1)) |D*|

Charging and Shifting

3

p.r.= 1+40(h+1)/a

Time=nO(a )2

Set h=3

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?