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Vertical strips Each disk appears once.
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)
O(a 2)
O(a 2)
O(a 2)
O(a 2)
n1 + n2 + ···+ nk < n
Approximation Algorithm
For x=0, -2, …, -(a-2), compute S(x). Choose minimum one from S(0), S(-2), …, S(-a+2).
component
1
Construct PTAS
For each partition P(a,a), construct C(a) as follows: 1. In each cell, construct MCDS for each connected
component in the inner area. 2. Connect those minimum connected dominating sets with a part of 8-approximation lying in boundary area. Choose smallest C(a) for a = 0, h+1, 2(h+1), ….
Theorem There is a PTAS for connected dominating set in unit disk graph.
central area
h+1 h
Boundary area
Why overlapping?
cds for G
cds for each
connected
a
(x,x)
Partition P(x)
Construct Minimum Unit Disk Cover in Each Cell
Each square with edge length
1/√2 can be covered by a unit
disk.
1/√2
Hence, each cell can be covered
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 modificationmost a 2 2disks.
Suppose a cell contains ni points. Then there are ni(ni-1) possible positions for each disk.
Minimum cover can be computed In time nOi (a2 )
add some disks and estimate how many added disks.
Added Disks
Count twice
Count four times
2
Shifting
2
Estimate # of added disks
Shifting
Estimate # of added disks
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.
< 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 .
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
• 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.
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.
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.
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)
O(a 2)
O(a 2)
O(a 2)
O(a 2)
n1 + n2 + ···+ nk < n
Approximation Algorithm
For x=0, -2, …, -(a-2), compute S(x). Choose minimum one from S(0), S(-2), …, S(-a+2).
component
1
Construct PTAS
For each partition P(a,a), construct C(a) as follows: 1. In each cell, construct MCDS for each connected
component in the inner area. 2. Connect those minimum connected dominating sets with a part of 8-approximation lying in boundary area. Choose smallest C(a) for a = 0, h+1, 2(h+1), ….
Theorem There is a PTAS for connected dominating set in unit disk graph.
central area
h+1 h
Boundary area
Why overlapping?
cds for G
cds for each
connected
a
(x,x)
Partition P(x)
Construct Minimum Unit Disk Cover in Each Cell
Each square with edge length
1/√2 can be covered by a unit
disk.
1/√2
Hence, each cell can be covered
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 modificationmost a 2 2disks.
Suppose a cell contains ni points. Then there are ni(ni-1) possible positions for each disk.
Minimum cover can be computed In time nOi (a2 )
add some disks and estimate how many added disks.
Added Disks
Count twice
Count four times
2
Shifting
2
Estimate # of added disks
Shifting
Estimate # of added disks
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.
< 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 .
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
• 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.
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.
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.