机动车安全技术检验机构检验
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路漫漫其悠远
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.
路漫漫其悠远
a
路漫漫其悠远
(x,x)
Partition P(x)
路漫漫其悠远
Construct Minimum Unit Disk Cover in Each Cell
Each square with edge length
1/√2 can be covered by a unit
disk.
1/√2
路漫漫其悠远
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
<1
路漫漫其悠远
Dominating set in unit disk graph
• Given a unit disk graph, find a dominating set with the minimum cardinality.
• Theorem This problem has PTAS.
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.
路漫漫其悠远
Performance Ratio
P.R. < 1 + 6/a < 1 + ε when we choose a = 6 ⌠1/ε .
Running time is n.O(1/ε 2 )
路漫漫其悠远
Unit disk graph
Hence, each cell can be covered
By at most
disks.
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 )
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), ….
add some disks and estimate how many added disks.
路漫漫其悠远
Added Disks
Count twice
Count four times
2
路漫漫其悠远
Shifting
2
路漫漫其悠远
Estimate # of added disks
Shifting
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
路漫漫其悠远
路漫漫其悠远
central area
h+1 h
Boundary area
பைடு நூலகம்
路漫漫其悠远
Why overlapping?
cds for G
cds for each
connected
component
1
路漫漫其悠远
Construct PTAS
For each partition P(a,a), construct C(a) as follows: 1. In each cell, construct MCDS for each connected
路漫漫其悠远
Existence of 8-approximation
O(a 2)
O(a 2)
O(a 2)
O(a 2)
n1 + n2 + ···+ nk < n
路漫漫其悠远
Approximation Algorithm
For x=0, -2, …, -(a-2), compute S(x). Choose minimum one from S(0), S(-2), …, S(-a+2).
机动车安全技术检验机 构检验
路漫漫其悠远
2020/7/16
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.
路漫漫其悠远
Estimate # of added disks
Vertical strips Each disk appears once.
路漫漫其悠远
Estimate # of added disks
Horizontal strips Each disk appears once.
路漫漫其悠远
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.
路漫漫其悠远
a
路漫漫其悠远
(x,x)
Partition P(x)
路漫漫其悠远
Construct Minimum Unit Disk Cover in Each Cell
Each square with edge length
1/√2 can be covered by a unit
disk.
1/√2
路漫漫其悠远
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
<1
路漫漫其悠远
Dominating set in unit disk graph
• Given a unit disk graph, find a dominating set with the minimum cardinality.
• Theorem This problem has PTAS.
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.
路漫漫其悠远
Performance Ratio
P.R. < 1 + 6/a < 1 + ε when we choose a = 6 ⌠1/ε .
Running time is n.O(1/ε 2 )
路漫漫其悠远
Unit disk graph
Hence, each cell can be covered
By at most
disks.
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 )
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), ….
add some disks and estimate how many added disks.
路漫漫其悠远
Added Disks
Count twice
Count four times
2
路漫漫其悠远
Shifting
2
路漫漫其悠远
Estimate # of added disks
Shifting
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
路漫漫其悠远
路漫漫其悠远
central area
h+1 h
Boundary area
பைடு நூலகம்
路漫漫其悠远
Why overlapping?
cds for G
cds for each
connected
component
1
路漫漫其悠远
Construct PTAS
For each partition P(a,a), construct C(a) as follows: 1. In each cell, construct MCDS for each connected
路漫漫其悠远
Existence of 8-approximation
O(a 2)
O(a 2)
O(a 2)
O(a 2)
n1 + n2 + ···+ nk < n
路漫漫其悠远
Approximation Algorithm
For x=0, -2, …, -(a-2), compute S(x). Choose minimum one from S(0), S(-2), …, S(-a+2).
机动车安全技术检验机 构检验
路漫漫其悠远
2020/7/16
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.
路漫漫其悠远
Estimate # of added disks
Vertical strips Each disk appears once.
路漫漫其悠远
Estimate # of added disks
Horizontal strips Each disk appears once.
路漫漫其悠远