Simultaneous Placement and Scheduling of Sensors
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Transcript of Simultaneous Placement and Scheduling of Sensors
Simultaneous Placement andScheduling of Sensors
Andreas Krause, Ram Rajagopal,Anupam Gupta, Carlos Guestrin
[email protected] theory and practice collide
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Traffic monitoringCalTrans wants to deploy wireless sensors under highways and arterial roadsDeploying sensors is expensive(need to close and open up roads etc.)
Where should we place the sensors?
Battery lifetime ¼ 3 yearsNeed 10+ years lifetime for feasible deployment Solution: Sensor scheduling (e.g., activate every 4 days)
When should we activate each sensor?
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Monitoring water networksContamination of drinking watercould affect millions of people
Contamination
Place sensors to detect contaminations“Battle of the Water Sensor Networks” competition Where and when should we sense to detect contamination?
Sensors
Simulator from EPAYSI 6600 Sonde
~$7K75 days
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Traditional approach
If we know that we need to schedule, why not take that into account during placement?
1.) Sensor Placement:Find most informative locations
2.) Sensor Scheduling:Find most informative activation times(e.g., assign to groups + round robin)
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Our approach
If we know that we need to schedule, why not take that into account during placement?
1.) Sensor Placement:Find most informative locations
2.) Sensor Scheduling:Find most informative activation times(e.g., assign to groups + round robin)
Simultaneously optimize overplacement and schedule
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Model-based sensingUtility of sensing based on model of the world
For traffic monitoring: Learn probabilistic models from data (later)For water networks: Water flow simulator from EPA
For each subset A µ V compute “sensing quality” F(A)
S2
S3
S4S1 S2
S3
S4
S1
High sensing quality F(A) = 0.9 Low sensing quality F(A)=0.01
Model predictsHigh impact
Medium impactlocation
Low impactlocation
Sensor reducesimpact throughearly detection!
S1
Contamination
Set V of all network junctions
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Problem formulationSensor Placement:
Given: finite set V of locations, sensing quality FWant: A*µ V such that
Sensor Scheduling:Given: sensor placement A* µ V
Want:Partition A* = A1* [ A2
* [ … [ Ak* s.t.
Ak* = sensors activated
at time k
Want to maximize average performance over time!
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The SPASS ProblemSimultaneous placement and scheduling (SPASS):
Given: finite set V of locations, sensing quality FWant:Disjoint sets A1
*, …, Ak* such that
| A1* [ … [ Ak
*| · m and
Typically NP-hard!
At = sensors activated at time t
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Greedy average-case placement and scheduling (GAPS)
Start with A1,…,Ak = ;For i = 1 to m
(s*,t*) := argmax(s,t) F(At [ {s}) – F(At)
At* := At* [ {s*}
How well can this simple heuristic do?
Greedily choose:s: sensor locationt: time step to add s to
Sco
re F
(Ai)
A1 A2 A3 A4
s11
s12
s9
s5 s6
s8
s1
s13
s7s2
Contribution of s2 to F(A4)F(A4 [ {s2}) – F(A4)
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s10
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S’
S2S3
S4S1
Key property: Diminishing returns
S2
S1
S’
Placement A = {S1, S2} Placement B = {S1, S2, S3, S4}
Adding S’ will help a lot!
Adding S’ doesn’t help muchNew
sensor S’B A
S’
+
+
Large improvement
Small improvement
For A µ B, F(A [ {S’}) – F(A) ¸ F(B [ {S’}) – F(B)
Submodularity:
Theorem [Krause et al., J Wat Res Mgt ’08]:Sensing quality F(A) in water networks is submodular!
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Performance guarantee
TheoremGAPS provides constant factor approximation
t F(AGAPS,t) ¸ 1/2 t F(A*t)
Proof Sketch:SPASS requires maximization of a monotonic submodular function over a truncated partition matroidTheorem then follows from result by Fisher et al ’78
Generalizes analysis of k-cover problem (Abrams et al., IPSN ’04) Can also get slightly better guarantee (¼ 0.63) using
more involved algorithm by Vondrak et al. ‘08
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Average-case scheduling can be unfairConsider V = {s1,…,sn}, k = 4, m = 10
Want to ensure balanced coverage
Sco
re F
(Ai)
A1 A2 A3 A4
s11
s12
s2
s6
s8
s1s13
s7 s2
t F(At) high!mint F(At) low
Sco
re F
(Ai)
A1 A2 A3 A4
s11
s12
s2
s5s6
s8s10
s1
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s7 s2
t F(At) high!
mint F(At) high!
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s10
Poor coverage at t=4!
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Balanced SPASSWant: A1
*, …, Ak* disjoint sets s.t. |A1
* [ … [ Ak *| · m and
Greedy algorithm performs arbitrarily badly
We now develop an approximation algorithm for this balanced SPASS problem!
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Key idea: Reduce worst-case to average-caseSuppose we learn the value attained by optimal solution:
c* = mint F(A*t) = OPT
Then we need to find a feasible solution A1,…,Ak such that
F(At) ¸ c* for all t
If we can check feasibility for any c, we can find optimal c* using binary search!
How can we find such a feasible solution?
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Trick: TruncationNeed to find a feasible solution such that
F(At) ¸ c for all t
For Fc(A) = min{F(A), c}:
F(At) ¸ c for all t t Fc(At) = k c
Truncation preserves submodularity!
Hence, to check whether OPT = mint F(A*t) ¸ c,
we need to solve average-case problem
c
|A|
F(A)Fc(A)
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Challenge: Use of approximationOnly have an ½-approximation algorithm (GAPS) for average case problem
Can lead to unbalanced solution! mint F(At) = 0
c
Sco
re F
c(Ai)
A1 A2 A3 A4
s3
s12
s2
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Approximate solutionguarantees only 2c
Optimal solutionhas value 4c
c
Sco
re F
c(Ai)
A1 A2 A3 A4
s11
s12
s2
s5
s6
s14
s15
s20
s10
s1
s13
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s8
s9
s19
s18
s31
s32
s45
s49
s27
s16
nocoverage!
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Remedy: Can rebalance solutionCan attempt to rebalance the solution, to obtain uniformly high score for all buckets
c
Sco
re F
c(Ai)
A1 A2 A3 A4
s11
s12
s2
s5
s6 s10
s1
s13
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Is rebalancing always possible?If there are elements s where F({s}) is large, rebalancing may not be possible:
c
Sco
re F
c(Ai)
A1 A2 A3
s2
A4
s3
s7
Rebalanced solutionstill has
mint F(At) = 0
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s1 s2 s3 s4 sn
Distinguishing big and small elementsElement s2 V is big if F({s})¸ c for some fixed 0<<1
If we can ensure that F(At) ¸ c for all tthen we get approximation guarantee!Can remove big elements from problem instance!
c
c
Sco
re F
c({s}
)
…
“big” elements
Will find out howto choose later!
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How large should be?
c
c
Sco
re F
c(Ai)
s11 s12 s4
A1 A2 A3 Ak’…
“satisfied” time steps
s2
s5
s6
s7
s8
s9
s10
GAPS solutionon small elements
rebalanced solution
Lemma: If = 1/6, can always successfully rebalance (i.e., ensure all time steps are satisfied)
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eSPASS AlgorithmeSPASS:Efficient Simultaneous Placement and Scheduling of Sensors
Initialize cmin=0, cmax = F(V)
Do binary search: c = (cmin+cmax)/2Allocate big elements to separate time steps (and remove) Run GAPS with Fc to find A1,…,Ak’, where k’ = k - #big elements
Reallocate small elements to obtain balanced solutionIf mint F(At) ¸ c/6: increase cmin
If mint F(At) < c/6: decrease cmax
until convergence
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Performance guarantee
TheoremeSPASS provides constant factor 6 approximation
mint F(AeSPASS,t) ¸ 1/6 mint F(A*t)
Can also obtain data-dependent bounds which are often much tighter
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Experimental studiesQuestions we ask:
How much does simultaneous optimization help?Is optimizing the balanced performance a good idea?How does eSPASS compare to existing algorithms (for the special case of sensor scheduling)?
Case studies:Contamination detection in water networksTraffic monitoringCommunity sensingSelecting informative blogs on the web
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Traffic monitoringGoal: Predict normalized road speeds on unobserved
road segments from sensor dataApproach:
Learn probabilistic model (Gaussian process) from dataUse eSPASS to optimize sensing quality
F(A) = Expected reduction in MSEwhen sensing at locations A
Data: from 357 sensors deployed on highway I-880 South (PeMS)Sampled between 6am and 11am during work days
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Benefit of simultaneous optimization
¼ 30% lifetime improvement for same accuracy!For large k, random scheduling hurts more than random placement
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Lifetime improvement (#time slots k)
Min
imum
var
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RP/RS
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Lifetime improvement (#time slots k)
Min
imum
var
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OP/RS
RP/RS
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Min
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RP/OS
OP/RS
RP/RS
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Min
imum
var
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e re
duct
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OP/OS
RP/OS
OP/RS
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Lifetime improvement (#time slots k)
Min
imum
var
ianc
e re
duct
ion eSPASS
OP/OS
RP/OS
OP/RS
RP/RSHigh
er is
bet
ter
Lifetime improvement (k groups)Traffic data
OP: Optimized PlacementOS: Optimized ScheduleRP: Random PlacementRS: Random Schedule
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Average-case vs. Balanced Score
Optimizing for balanced score leads to good average-case performance, but not vice versa
Traffic data
High
er is
bet
ter
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Lifetime improvement (5 sensors / time slot)
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ianc
e re
duct
ion
Avg. scoreGAPS
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Lifetime improvement (5 sensors / time slot)
Var
ianc
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duct
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Avg. scoreGAPS
Balanced scoreGAPS
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Lifetime improvement (5 sensors / time slot)
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Avg. scoreGAPS
Balanced scoreGAPS
Balanced scoreeSPASS
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Lifetime improvement (5 sensors / time slot)
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Avg. scoreGAPS
Avg. scoreeSPASS
Balanced scoreGAPS
Balanced scoreeSPASS
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Data-dependent bounds
Our data-dependent bounds show that eSPASS solutions are typically much closer to optimal than 1/6
Traffic data
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Lifetime improvement (#time slots k)
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eSPASS
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Lifetime improvement (#time slots k)
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Bound fromTheorem 4.1
eSPASS
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Lifetime improvement (#time slots k)
Min
imum
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Bound fromTheorem 4.1
Data-dependentbound
eSPASS
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Water network monitoringReal metropolitan area network (12,527 nodes)Water flow simulator provided by EPA3.6 million contamination eventsMultiple objectives: Detection time, affected population, …Place sensors that detect well “on average”
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Benefit of simultaneous optimization
Simultaneous optimization significantly outperforms traditional approaches
OP: Optimized PlacementOS: Optimized ScheduleRP: Random PlacementRS: Random Schedule
High
er b
alan
ced
scor
e
More sensors Water networks
5 10 15 20 25 30 35 400
0.2
0.4
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Number m of sensors (k=3)
Min
. pop
ulat
ion
prot
ecte
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RP/RS
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Number m of sensors (k=3)
Min
. pop
ulat
ion
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ecte
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RP/RS
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Number m of sensors (k=3)
Min
. pop
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ion
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ecte
d
OP/RS
RP/OS
RP/RS
5 10 15 20 25 30 35 400
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Number m of sensors (k=3)
Min
. pop
ulat
ion
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ecte
dOP/OS
OP/RS
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Number m of sensors (k=3)
Min
. pop
ulat
ion
prot
ecte
deSPASS
OP/OS
OP/RS
RP/OS
RP/RS
E.g., ~3x reduction in affected population when m = 24, k = 3
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Comparison with existing techniquesComparison of eSPASS with existing algorithms for scheduling (m = |V|):
MIP: Mixed integer program for domatic partitioning with accuracy requirements (Koushanfary et al. 06)SDP: Approximation algorithm for domatic partitioning (Deshpande et al. 08)
Results on temperature monitoring (Intel Berkeley) data set with 46 sensorsGoal: Minimize expected MSE
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Comparison with existing techniques
eSPASS outperforms existing approaches for sensor scheduling
Low
er e
rror
(MSE
)
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Temperature data
Worst-case error Average-case error
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Trading off power and accuracySuppose that we sometimes activate all sensors(e.g., determine boundary of traffic jam,
localize source of contamination)
Want to simultaneously optimizemint F(At) and F(A1 [ … [ Ak)
Scalarization: for some 0 < < 1, we want to optimize:
mint F(At) + (1-) F(A1[ … [ Ak)
Theorem: Our algorithm, mcSPASS (multicriterion SPASS) guarantees factor 8 approximation!
“Balanced performance” “High-density performance”
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Tradeoff results
Stage-wise ( = 0) eSPASS ( = 1) mcSPASS ( = .25)
max mint F(At) + (1-) F(A1[ … [ Ak)
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Tradeoff results
Can simultaneously obtain high performancein scheduled and high-density mode
0.82 0.84 0.86 0.88 0.9 0.920.94
0.95
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1
= 1
= 0 = 0.25
High
-den
sity
perf
orm
ance
Scheduled performance Water networks
mint F(At) + (1-) F(A1[ … [ Ak)
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ConclusionsIntroduced simultaneous placement and scheduling (SPASS) problemDeveloped efficient algorithms with strong guarantees:
GAPS: 1/2 approximation for average performanceeSPASS: 1/6 approximation for balanced performancemcSPASS: 1/8 approximation for trading off high-density and
balanced performance
Data-dependent bounds show solutions close to optimalPresented results on several real-world sensing tasks
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LAB
KITCHEN
COPYELEC
PHONEQUIET
STORAGE
CONFERENCE
OFFICEOFFICE5051
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