chapter 6 relaxation (1) cds in unit disk graph ding-zhu du
TRANSCRIPT
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Sensor NetworksSensor NetworksA sensor network is an ad hoc A sensor network is an ad hoc wireless networkwireless network which consists of a which consists of a huge amount of static or mobile huge amount of static or mobile sensorssensors. The sensors collaborate to . The sensors collaborate to sense, collect, and process the raw information of the sense, collect, and process the raw information of the phenomenonphenomenon in in the sensing area (in-network), and transmit the processed information the sensing area (in-network), and transmit the processed information to the to the observersobservers..
Sensing AreaSensing Area
phenomenonphenomenon
User1User1
SinkSink
Internet / Internet / SatelliteSatellite
Sensor networkSensor network
User2User2
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Sensor Networks (Cont.)
Sensor Node
• Sensing + Computation + Communication
• Small size• Limited power
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Military applications
Applications
Example 1
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Environmental Monitoring
Example 2
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Biological Systems
Example 3
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Example 4
Traffic Control
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Virtual Backbone Flooding
Reduction of communication
overhead
Redundancy
Contention
Collision
Reliability Unreliability
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
B
A
C D
A B ?A: B:C:D:
A B ?A:B: C: D:
A B
Only nodes in CDS maintain routing tables Routing information localized Save storage space
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Unit Disk Graph
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Unit Ball Graph
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Connected Dominating Set
connected. is by inducedsubgraph if
set dominating connected a called is e,Furthermor
. oadjacent tor in either is
nodeevery ifset dominating a is subset nodeA
).,(graph aConsider
C
C
CC
C
EVG
Dominating set
Connected dominating set
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CDS in unit disk graphs
y.cardinalit minimumset with dominating
connected a find graph,disk unit aGiven
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CDS in unit ball graphs
y.cardinalit minimumset with dominating
connected a find graph, ballunit aGiven
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Two Stage Algorithm
Dominating set
Connected dominating set
Stage 1. Compute a dominating set D.
Stage 2. Connect D into a connected dominating set.
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Stage 1
set. dominating a isset t independen maximalEvery
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MCDS (opt)
MIS
23|mis| opt
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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 < 1?
8!(Wu et al, 2006)
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How many independent points can be packed Into four disks that one contains centers of other three?
< 15!
(Yao et al, 2008)
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In unit disk graph
set. dominating
connected minimum theof size theis and
sett independen maximal a is mis where
14|mis|
opt
opt
2.18.3|mis| opt
9748.3|mis| opt
3
11
3
23|mis| opt
(Wu et al. 2006)
(Funke et al. 2006)
(Yao et al. 2008)
(Wan et al, 2002)
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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
>1
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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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1.5
.5
1
Relationship between problems 1, 2 and 3?
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For balls not touched each other, 12!! (Hoppe, 1874)
Allow balls to touch, 12!!
icosahedron
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11!
How many independent points can be contained In a ball subtracting another ball?
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How many independent points can be contained by two balls with radius 1 and center distance < 1?
1
>1
22!
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How many unit balls can kiss two intersecting unit balls?
20?!
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In unit ball graph
111|mis| opt
12
11
11
(Butenko, et al, 2007)
??|mis| opt (Zhang, et al, 2008)
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Connect all nodes in an MIS with a spanning tree
Stage 2
122 optfor unit ball graphs
(Butenko, 2007)
3
22
3
120 opt for unit disk graphs
(Wan-Yao)
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Stage 2: Connect all nodes in an MIS D.
Consider a greedy method.
)(}){()(
by inducedsubgraph
of components connected of #)(
.return
};{
and )( maximize to node a choose
do 2)( while
;
CfxCfCf
C
Cf
C
xCC
Cfx
Cf
DC
x
x
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Connect all nodes in an MIS with greedy algorithm
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Theorem
.10ln13ln2 and 11 space,In
.3ln6)1ln(2 and 4 plane,In
.1))1ln(2(
mostat size with CDS
a produces algorithmgreedy then the
,1 |mis| If
opt
opt
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Proof
.1)()*(
Then
}.,...,{* and }{, Denote
subgraph. connected a induces },...,{ ,any for and
set dominating connected minimum a be }{Let
algorithm.Greedy by in turn selected are ,..., Suppose
1
1110
1
1
1
iyjiy
jjiii
i
opt
g
CfCCf
yyCxCCDC
yyi
, ...,yy
xx
jj
1y1jy jy
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)1
1(
Then ).(1 Denote
.)(1
)()(
)()(
/))((
/))(*)(1(
/))*(1(
/))(()( Thus,
1 allfor
)()( So,
1
1
11
1
1
1
optaa
Cfopta
opt
CfoptCf
opt
CfoptCfCf
optCfopt
optCfCCfopt
optCCfyopt
optCfyCfx
optj
CfCfx
ii
ii
i
i
iii
i
ii
optjji
optjii
iyi
j
ji
ji
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.
2 Then
.)12 hence and 2||
then, i.e., exist,t doesn' such If(
.
satisfying onelargest thebe to Choose
)/11(
/0
0
/00
opti
i
optiii
eaopt
optig
optgoptD
optai
aopt
i
eaoptaa
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.1))1ln(2(|mis|
Therefore,
.)1 |mis| :(
.1 /|)mis|1(/
)/(ln 2
2
Thus,
)/(ln
0
0
0
/0
optg
optnote
optoptopta
optaoptopt
ioptg
optaopti
eaopt opti
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mathematics
Operations Research
Computer Science
Packing
Dominating
Wireless Networking
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Thanks, End