Simultaneous Placement and Scheduling of Sensors

1
Simultaneous Placement andScheduling of Sensors

• Andreas Krause, Ram Rajagopal,Anupam Gupta,
  Carlos Guestrin

rsrg_at_caltech
..where theory and practice collide
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Traffic monitoring

• CalTrans wants to deploy wireless sensors under
  highways and arterial roads
• Deploying sensors is expensive(need to close and
  open up roads etc.)
• ? Where should we place the sensors?
• Battery lifetime ¼ 3 years
• Need 10 years lifetime for feasible deployment ?
• Solution Sensor scheduling (e.g., activate every
  4 days)
• ? When should we activate each sensor?

3
Monitoring water networks

• Contamination of drinking watercould affect
  millions of people

Contamination
Sensors
Simulator from EPA

• Place sensors to detect contaminations
• Battle of the Water Sensor Networks competition
• ? Where and when should we sense to
  detect contamination?

4
Traditional approach
1.) Sensor PlacementFind most informative
locations
2.) Sensor SchedulingFind most informative
activation times (e.g., assign to groups round
robin)

• If we know that we need to schedule, why not
  take that into account during placement?

5
Our approach
1.) Sensor PlacementFind most informative
locations
Simultaneously optimize overplacement and
schedule
2.) Sensor SchedulingFind most informative
activation times (e.g., assign to groups round
robin)

• If we know that we need to schedule, why not
  take that into account during placement?

6
Model-based sensing

• Utility 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)

Model predicts High impact
Low impactlocation
Contamination
Medium impactlocation
S3
S1
S2
S3
S4
S2
Sensor reducesimpact throughearly detection!
S1
S4
Set V of all network junctions
S1
Low sensing quality F(A)0.01
High sensing quality F(A) 0.9
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Problem formulation

• Sensor Placement
• Given finite set V of locations, sensing quality
  F
• Want Aµ 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 Problem

• Simultaneous placement and scheduling (SPASS)
• Given finite set V of locations, sensing quality
  F
• Want 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)
Greedily choose s sensor location t time step
to add s to
Start with A1,,Ak For i 1 to m (s,t)
argmax(s,t) F(At s) F(At) At At s
s8
3
2
1
s1
4
1
s10
4
s13
s5
s12
s6
2
s2
1
s11
s9
s7
2
3
1
How well can this simple heuristic do?
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Key property Diminishing returns
Placement A S1, S2
Placement B S1, S2, S3, S4
S2
S2
S3
S1
S1
S4
Theorem Krause et al., J Wat Res Mgt
08 Sensing quality F(A) in water networks is
submodular!
Adding S will help a lot!
Adding S doesnt help much
S
New sensor S

S
B
Large improvement
A
Submodularity

S
Small improvement
For A µ B, F(A S) F(A) F(B S) F(B)
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Performance guarantee

• Theorem
• GAPS provides constant factor approximation
• ?t F(AGAPS,t) 1/2 ?t F(At)
• Proof Sketch
• SPASS requires maximization of a monotonic
  submodular function over a truncated partition
  matroid
• Theorem 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 unfair

• Consider V s1,,sn, k 4, m 10
• ? Want to ensure balanced coverage

?t F(At) high!
mint F(At) low ?
s1
s6
Score F(Ai)
s13
s10
s5
s8
s12
s11
s2
s7
s2
A2
A3
A4
A1
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Balanced SPASS

• Want 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-case

• Suppose we learn the value attained by optimal
  solution
• c mint F(At) 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 Truncation

• Need to find a feasible solution such that
• F(At) c for all t
• For Fc(A) minF(A), c
• F(At) c for all t ? ?t Fc(At) k c
• Truncation preserves submodularity! ?
• Hence, to check whether OPT mint F(At) c,
  we need to solve average-case problem

F(A)
Fc(A)
A
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Challenge Use of approximation

• Only have an ½-approximation algorithm (GAPS) for
  average case problem

nocoverage!
? Can lead to unbalanced solution! mint F(At) 0
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Remedy Can rebalance solution

• Can attempt to rebalance the solution, to obtain
  uniformly high score for all buckets

c
Score Fc(Ai)
s6
s12
s5
s11
s2
A2
A3
A4
A1
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Is rebalancing always possible?

• If there are elements s where F(s) is large,
  rebalancing may not be possible

Rebalanced solutionstill has mint F(At)
0
c
s7
s3
Score Fc(Ai)
s2
A2
A3
A4
A1
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Distinguishing big and small elements

• Element s2 V is big if F(s) ? c for some fixed
  0lt?lt1
• If we can ensure that F(At) ? c for all tthen
  we get ? approximation guarantee!
• Can remove big elements from problem instance!

Will find out howto choose ? later!
c
big elements
Score Fc(s)
? c
s1
s2
s3
s4
sn

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How large should ? be?

GAPS solutionon small elements
c
s9
s8
s7
Score Fc(Ai)
rebalanced solution
s6
s5
? c
s10
s11
s2
s12
s4
A1
A2
A3

Ak
satisfied time steps
Lemma If ? 1/6, can always successfully
rebalance (i.e., ensure all time steps are
satisfied)
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eSPASS Algorithm

• eSPASS
• Efficient Simultaneous Placement and Scheduling
  of Sensors
• Initialize cmin0, cmax F(V)
• Do binary search c (cmincmax)/2
• Allocate 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
  solution
• If mint F(At) c/6 increase cmin
• If mint F(At) lt c/6 decrease cmax
• until convergence

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Performance guarantee

• Theorem
• eSPASS provides constant factor 6 approximation
• mint F(AeSPASS,t) 1/6 mint F(At)

Can also obtain data-dependent bounds which are
often much tighter
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Experimental studies

• Questions 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 networks
• Traffic monitoring
• Community sensing
• Selecting informative blogs on the web

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Traffic monitoring

• Goal Predict normalized road speeds on
  unobserved road segments from sensor data
• Approach
• Learn probabilistic model (Gaussian process) from
  data
• Use eSPASS to optimize sensing quality
• F(A) Expected reduction in MSE when 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
OP Optimized Placement OS Optimized
Schedule RP Random Placement RS Random Schedule
Higher is better
Traffic data
Lifetime improvement (k groups)

• ¼ 30 lifetime improvement for same accuracy!
• For large k, random scheduling hurts more than
  random placement

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Average-case vs. Balanced Score
Higher is better
Traffic data

• Optimizing for balanced score leads to good
  average-case performance, but not vice versa

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Data-dependent bounds
Higher is better
Traffic data

• Our data-dependent bounds show that eSPASS
  solutions are typically much closer to optimal
  than 1/6

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Water network monitoring

• Real metropolitan area network (12,527 nodes)
• Water flow simulator provided by EPA
• 3.6 million contamination events
• Multiple objectives Detection time, affected
  population,
• Place sensors that detect well on average

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Benefit of simultaneous optimization
OP Optimized Placement OS Optimized
Schedule RP Random Placement RS Random Schedule
Higher balanced score
Water networks
More sensors

• Simultaneous optimization significantly
  outperforms traditional approaches

E.g., 3x reduction in affected population when m
24, k 3
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Comparison with existing techniques

• Comparison 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 sensors
• Goal Minimize expected MSE

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Comparison with existing techniques
Lower error (MSE)
Worst-case error

• eSPASS outperforms existing approaches for sensor
  scheduling

Temperature data
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Trading off power and accuracy

• Suppose that we sometimes activate all
  sensors(e.g., determine boundary of traffic jam,
  localize source of contamination)
• Want to simultaneously optimize mint F(At)
  and F(A1 Ak)
• Scalarization for some 0 lt l lt 1, we want to
  optimize l mint F(At) (1-l) F(A1 Ak)
• Theorem Our algorithm, mcSPASS (multicriterion
  SPASS) guarantees factor 8 approximation!

Balanced performance
High-density performance
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Tradeoff results
max l mint F(At) (1-l) F(A1 Ak)
Stage-wise (? 0)
eSPASS (? 1)
mcSPASS (? .25)
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Tradeoff results
1
l mint F(At) (1-l) F(A1 Ak)
0.99
0.98
High-density performance
0.97
0.96
0.95
0.94
0.82
0.84
0.86
0.88
0.9
0.92
Water networks
Scheduled performance

• Can simultaneously obtain high performancein
  scheduled and high-density mode

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Conclusions

• Introduced simultaneous placement and scheduling
  (SPASS) problem
• Developed efficient algorithms with strong
  guarantees
• GAPS 1/2 approximation for average
  performance
• eSPASS 1/6 approximation for balanced
  performance
• mcSPASS 1/8 approximation for trading off
  high-density and balanced performance
• Data-dependent bounds show solutions close to
  optimal
• Presented results on several real-world sensing
  tasks