Simultaneous Placement andScheduling of Sensors

Andreas Krause, Ram Rajagopal,Anupam Gupta, Carlos Guestrin

rsrg@caltech..where theory and practice collide

2

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?

3

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

4

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)

5

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

6

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

7

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!

8

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

9

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)

132

4

1

1

14

3

2

2

s10

10

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!

11

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

12

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

s13

s7 s2

t F(At) high!

mint F(At) high!

s5

s10

Poor coverage at t=4!

13

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!

14

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?

15

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)

16

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

s5

s7

s8

s9

s10

s1

s13

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

s7

s8

s9

s19

s18

s31

s32

s45

s49

s27

s16

nocoverage!

17

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

s7

s8

s9

18

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

19

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!

20

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)

21

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

22

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

23

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

24

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

25

Benefit of simultaneous optimization

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

5 10 15 200

20

40

60

80

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

RP/RS

5 10 15 200

20

40

60

80

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

OP/RS

RP/RS

5 10 15 200

20

40

60

80

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

RP/OS

OP/RS

RP/RS

5 10 15 200

20

40

60

80

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

OP/OS

RP/OS

OP/RS

RP/RS

5 10 15 200

20

40

60

80

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

26

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

2 4 6 8 1042

44

46

48

50

Lifetime improvement (5 sensors / time slot)

Var

ianc

e re

duct

ion

Avg. scoreGAPS

2 4 6 8 1042

44

46

48

50

Lifetime improvement (5 sensors / time slot)

Var

ianc

e re

duct

ion

Avg. scoreGAPS

Balanced scoreGAPS

2 4 6 8 1042

44

46

48

50

Lifetime improvement (5 sensors / time slot)

Var

ianc

e re

duct

ion

Avg. scoreGAPS

Balanced scoreGAPS

Balanced scoreeSPASS

2 4 6 8 1042

44

46

48

50

Lifetime improvement (5 sensors / time slot)

Var

ianc

e re

duct

ion

Avg. scoreGAPS

Avg. scoreeSPASS

Balanced scoreGAPS

Balanced scoreeSPASS

27

Data-dependent bounds

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

Traffic data

High

er is

bet

ter

5 10 15 200

100

200

300

400

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

eSPASS

5 10 15 200

100

200

300

400

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

Bound fromTheorem 4.1

eSPASS

5 10 15 200

100

200

300

400

Lifetime improvement (#time slots k)

Min

imum

var

ianc

e re

duct

ion

Bound fromTheorem 4.1

Data-dependentbound

eSPASS

28

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”

29

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

0.6

0.8

1

Number m of sensors (k=3)

Min

. pop

ulat

ion

prot

ecte

d

RP/RS

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Number m of sensors (k=3)

Min

. pop

ulat

ion

prot

ecte

d

RP/OS

RP/RS

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Number m of sensors (k=3)

Min

. pop

ulat

ion

prot

ecte

d

OP/RS

RP/OS

RP/RS

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Number m of sensors (k=3)

Min

. pop

ulat

ion

prot

ecte

dOP/OS

OP/RS

RP/OS

RP/RS

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

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

30

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”

33

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

0.96

0.97

0.98

0.99

1

= 1

= 0 = 0.25

High

-den

sity

perf

orm

ance

Scheduled performance Water networks

mint F(At) + (1-) F(A1[ … [ Ak)

35

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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