Chapter 6 Relaxation 1 CDS in unit disk graph

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Chapter 6 Relaxation(1) CDS in unit disk graph

• Ding-Zhu Du

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Sensor Networks
A sensor network is an ad hoc wireless network
which consists of a huge amount of static or
mobile sensors. The sensors collaborate to sense,
collect, and process the raw information of the
phenomenon in the sensing area (in-network), and
transmit the processed information to the
observers.
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Sensor Networks (Cont.)

• Sensor Node
• Sensing Computation Communication
• Small size
• Limited power

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Applications
Example 1
Military applications
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Environmental Monitoring
Example 2
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Example 3
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Example 4
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Applications of CDS Virtual backbone
CDS is used as a virtual backbone in wireless
networks.
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Applications of CDS Broadcast

• Only nodes in CDS relay messages
• Reduce communication cost
• Reduce redundant traffic

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Applications of CDS Unicast

• Only nodes in CDS maintain routing tables
• Routing information localized
• Save storage space

C
D
B
A ? B
A
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Unit Disk Graph
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Unit Ball Graph
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Connected Dominating Set
Dominating set
Connected dominating set
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CDS in unit disk graphs
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CDS in unit ball graphs
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Two Stage Algorithm
Stage 1. Compute a dominating set D. Stage 2.
Connect D into a connected
dominating set.
Dominating set
Connected dominating set
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Stage 1
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MCDS (opt)
MIS
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Disk Packing
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How many independent points can be contained by
a disk with radius 1?
5!
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How many independent points can be contained
by two disks with radius 1 and center distance
lt 1?
(Wu et al, 2006)
8!
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How many independent points can be packed Into
four disks that one contains centers of other
three?
lt 15!
(Yao et al, 2008)
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In unit disk graph
(Wan et al, 2002)
(Wu et al. 2006)
(Funke et al. 2006)
(Yao et al. 2008)
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Sphere Packing
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1. How many independent points can be packed
by a ball with radius 1?
1
gt1
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2. How many (untouched) unit balls can be packed
into a ball with radius 1.5?
0.5
1.5
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3. Gregory-Newton Problem (1694)
How many unit balls (not touch each other) can
kiss a unit ball?
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Relationship between problems 1, 2 and 3?
1.5
1
.5
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For balls not touched each other,
12!! (Hoppe, 1874)
icosahedron
Allow balls to touch,
12!!
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How many independent points can be contained In
a ball subtracting another ball?

11!
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How many independent points can be contained
by two balls with radius 1 and center distance lt
1?
22!
1
gt1
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How many unit balls can kiss two intersecting
unit balls?
20?!
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In unit ball graph
(Butenko, et al, 2007)
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12
11
(Zhang, et al, 2008)
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Stage 2
Connect all nodes in an MIS with a spanning tree
for unit disk graphs
(Wan-Yao)
for unit ball graphs
(Butenko, 2007)
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Stage 2 Connect all nodes in an MIS D.
Consider a greedy method.
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Connect all nodes in an MIS with greedy algorithm
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Theorem
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Proof
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Operations Research
Dominating
Packing
Wireless Networking
mathematics
Computer Science
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Thanks, End