chapter 4 partition
DESCRIPTION
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. a. (x,x). Partition P(x). Construct Minimum Unit Disk Cover in Each Cell. Each square with edge length - PowerPoint PPT PresentationTRANSCRIPT
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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.
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(x,x)
Partition P(x)
a
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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
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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
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Approximation Algorithm
For x=0, -2, …, -(a-2), compute S(x).
Choose minimum one from S(0), S(-2), …, S(-a+2).
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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.
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Added DisksCount twice
Count four times
2
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2
Shifting
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Estimate # of added disks
Shifting
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Estimate # of added disks
Vertical strips
Each disk appearsonce.
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Estimate # of added disks
Horizontal strips
Each disk appears once.
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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.
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Performance Ratio
P.R. < 1 + 6/a < 1 + ε when we choose a = 6 ⌠1/ε .
Running time is n.O(1/ε )2
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Unit disk graph
< 1
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Dominating set in unit disk graph
• Given a unit disk graph, find a dominating set with the minimum cardinality.
• Theorem This problem has PTAS.
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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.
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Boundary area
central area
h
h+1
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Why overlapping?
cds for G
cds for eachconnectedcomponent 1
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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), ….
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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.
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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.
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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
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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
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is time total thecells, allOver
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findingThen nodes. cotains cell a Suppose
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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
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Connect & Charge
charge
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Multiple Charge
charge
How many possiblecharges for each node?
How many componentscan each node be adjacent to?
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1. How many independent points can be packed by a disk with radius 1?
1
>1
5!
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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
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Shifting
3
a/(2(h+1)) = integer
Time=nO(a )2
h=2
dimesion.any in in timeion approximat-)1( )/1( 2 On
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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?
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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?
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Thanks, End