Stochastic Event Capture Using Mobile Sensors Subject to a Quality Metric

1
Stochastic Event Capture Using Mobile Sensors
Subject to a Quality Metric

• N. Bisnik A. Abouzeid V. Isler
• MobiCom06
• Presented by Ray K. Lam
• Jan 26, 2007

2
Introducing Mobile Sensors

• Wireless sensor network applications
• Surveillance, environmental monitoring, health
  care
• Densely spread static sensors. Problems?
• Some points never covered
• Broken network due to sensor deaths
• Changes / new obstructions in environment

How about using mobile sensors?
3
Promises and Challenges

• The good of mobile sensors
• Robust a few sensors to cover entire area
• Connectivity send data when moved in ranges
• Challenges
• Mobile sensors cover large area over time
• But instantaneous coverage is the same
• Miss events happened at points not covered at the
  time

4
Focus Motion Planning and QoC

• Good motion planning is needed
• To ensure quality of coverage (QoC)
• QoC metrics
• Fraction of events captured
• Probability that an event is lost
• QoC depends on
• Sensor speed
• Event dynamics
• Number of sensors deployed

5
Outline

• Part I Fractions of Events Captured
• Single Sensor, Fixed Velocity
• Multiple Sensors, Fixed Velocity
• Single Sensor, Variable Velocity
• Part II Bounded Event Loss Probability
• Linear Case
• Simple Closed Curve Case
• General 2-D Case
• Conclusion

6
Events and PoIs

• Events temperature rises, target appears,
• Appear, and then disappear
• Be captured, or lost
• Point of Interest (PoI)
• A fixed point where events occur

r
Event detectable in distance r
PoIs
exp(?)
exp(µ)
state
1 (event present)
0 (no event)
time
7
Sensor Movement
PoI 1
PoI a

• Case 1
• Single sensor
• Fixed velocity v
• Case 2
• m sensors
• Fixed velocity v
• Equally separated
• Case 3
• Single sensor
• Velocity 0 vmax

PoI 2

r
PoI i

Mobile sensor
PoI 3
Closed curve C of length D
Q for Case 1 Critical velocity?
Q for Case 2 Gain of multiple sensors?
Q for Case 3 Optimal velocity pattern?
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Single Sensor, Fixed Velocity

• Expected fraction of events captured F1(v)
• Explicit, but convoluted expression
• Observations from plot
• F1(v) increases with v
• Stationary sensor gives 1/a
• Critical velocity lowestspeed to improve QoC
• Critical velocity increaseswith ? and µ

9
Multiple Sensors, Fixed Velocity

• Non-trivial case D/m gt 2r
• Otherwise, all PoIs covered at any time
• Observations
• QoC does not increase after certain m
• Gain of more sensors diminishes with high speed

10
Single Sensor, Variable Velocity

• Reasonable velocity pattern
• Move fast in futileregions at vmax
• Slow down to vc whenPoI visible
• Consider random spread of PoI
• Fv(vc) derived is complex
• No closed form for optimal vc

Futile regions
vmax
vc
vc
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Single Sensor, Variable Velocity

• Observations from plot
• Only for a2, vc lt vmax (vmax 40)
• Slow down at 1 PoI, missevents at a1 others
• Best policy in general
• Move with maximum possible speed

12
Outline

• Part I Fractions of Events Captured
• Single Sensor, Fixed Velocity
• Multiple Sensors, Fixed Velocity
• Single Sensor, Variable Velocity
• Part II Bounded Event Loss Probability
• Linear Case
• Simple Closed Curve Case
• General 2-D Case
• Conclusion

13
BELP Problem

• Number of PoIs a
• Each with event dynamics ?i and µi
• Ei an event occurs at PoI i not captured
• Bounded Event Loss Probability (BELP) problem
• Generate a motion plan such that
• PEi lt e for all 1 i a
• Stronger than bound on fraction of events
  captured
• No PoI can be ignored
• Implies fraction of events captured gt 1 e

14
Dissecting the Problem

• Problem nature
• PEi strictly increases with inter visit time of
  a PoI
• Unique Tcriti such that PEi e
• Find motion plan such that inter visit time lt
  Tcriti for all PoI i
• Minimum Velocity MV-BELP
• With one sensor, what is min velocity to satisfy
  QoC?
• Mimimum Sensor MS-BELP
• At fixed velocity v, what is min number of
  sensors needed to satisfy QoC?

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Both BELPs are NP-hard

• MV-BELP reduced to TSP
• MV-BELP finds optimal path for sensor motion such
  that inter visit time lt Tcriti for all PoI i
• TSP finds shortest path to visit a set of points

Tcrit
Tcrit
Minimum velocity Shortest path length / Tcrit
Tcrit
Tcrit
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Feasible Set

• Feasibility function
• Feasible for PoI i andj at velocity v if QoCcan
  be satisfied by 1sensor
• Feasible set
• S Set of all PoIs
• A subset N of S is a feasible set for velocity v
  if QoC can be satisfied by 1 sensor with velocity
  v
• Necessary condition A(i,j,v)1 for all i,j in N

PoI i
PoI j
Xi Xj
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MS-BELP is NP-hard

• MS-BELP reduced to min set cover problem
• Min set cover problem finds min number of input
  sets to cover all input elements
• Denote C(v) the collection of all feasible sets
  for velocity v
• MS-BELP finds C, a subset of C(v), with min
  cardinality, such that all PoIs covered by C

18
Restricted BELP

• Linear case
• All PoIs along a straight line
• Sensors move along the line
• Closed curve case
• All PoIs on a simple closed curve
• Sensors move along the curve
• General 2-D case
• PoIs on 2-D plane
• Sensors free to move in any way

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Linear Case MV-BELP

• Sensor moves back and forth
• Between x r and x Xa r
• Max inter visit time for PoI i max(2(Xi X1
  2r)/v, 2(Xa Xi 2r)/v, 0)
• Min velocity for PoI i max(2(Xi X1
  2r)/Tcriti, 2(Xa Xi 2r)/Tcriti, 0)
• Min velocity for all PoIs max1ia vmini
• Running time O(a)

r
r
X10
X2
X3
Xa-1
Xa
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Linear Case MS-BELP

• Restricted MS-BELP is still NP-hard
• Restriction on path eases finding feasible sets
• Finding min of sets to cover all PoIs is still
  SCP
• Consider approximation algorithm
• Restricted scope
• ? Each PoI covered by 1 sensor every Tcriti
• ? PoI covered by k sensors
• Each visits PoI every k Tcriti
• Requires synchronization

21
Linear Case MS-BELP Algorithm

• While not all PoIs assigned to sensors
• Start a new sensor group Gk
• Add the leftmost unassigned PoI m to Gk
• For i PoI m1 to PoI a
• If PoI i unassigned AND A(i,j,v) 1 for all j in
  Gk
• Add PoI i in Gk
• Return of sensor groups

G1
G2
G3
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Bound the Performance

• Running time O(a2)
• Denote by k the of sensors used by algorithm
• k 2 kOPT 1
• Some notations before proof

Sensor i1
Sensor i1
Sensor i
Sensor i1
Sensor i
Sensor i
si
ti
ei
ei1
si1
si
ti
ei
ei1
si1
si
ti
ei
ei1
si1
Disjoint
Complete overlap
Partial overlap
Gi
Gi1
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Some Properties for Proof

• Property 1
• For all i, 1ik, QoC of all PoIs in Gi is
  satisfied by using 1 sensor
• Why? Algorithm guarantees A(si,j,v) 1 and
  A(j,ei,v) 1 for all j in Gi
• Property 2
• si lt sj for all 1 i j k

24
Some Properties for Proof

• Property 3
• For all i, 1 i k1, there exists ti in Gi,
  such that si ti lt si1 and A(ti,si1,v) 0
• Why?
• From property 2, si1 gt si
• If no ti in Gi such that A(ti,si1,v) 0, si1
  would have been added to Gi

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Some Properties for Proof

• A(ti,si1,v) 0 implies
• (i) OR (ii)

Sensor i1
Sensor i1
Sensor i
Sensor i1
Sensor i
Sensor i
si
ti
ei
ei1
si1
si
ti
ei
ei1
si1
si
ti
ei
ei1
si1
Disjoint
Complete overlap
Partial overlap
(ii) is TRUE
(ii) is TRUE
(i) or (ii) or both is TRUE
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Some Properties for Proof

• If (i) is TRUE
• Sensor cannot sense ti, move to si1 or its
  further right, and return to ti within Tcritti
• A(ti,j,v) 0 for all j si1
• If (ii) is TRUE
• Sensor cannot sense si1, move to ti or its
  further left, and return to si1 within
  Tcritsi1
• A(j,si1,v) 0 for all j ti

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Proving the Performance Bound

• For all i, 1 i k1, there exists ti in Gi,
  such that si ti lt si1 and
• (i) A(ti,j,v) 0 for all j si1 OR
• (ii) A(j,si1,v) 0 for all j ti
• Construct sets H1 and H2
• For each 1 i k1, for the ti and si1 pair
• Add ti to H1 if (i) holds
• Add si1 to H2 if (ii) holds

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Proving the Performance Bound

• For each (i,i1) pair
• At least 1 PoI added to either H1 or H2
• H1 H2 k 1
• max(H1,H2) (k 1)/2
• For all m,n in H2
• m si tj-1 lt sj n
• Impossible to cover sj while covering tj-1 or its
  left
• A(m,n,v) 0

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Proving the Performance Bound

• A(m,n,v) 0 for all m,n in H2
• Every PoI in H2 must be covered by 1 sensor
• A(m,n,v) 0 for all m,n in H1 similarly
• Every PoI in H1 must be covered by 1 sensor
• kOPT max(H1,H2) (k 1)/2

k 2 kOPT 1
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Closed Curve Case MV-BELP

• Two types of paths
• Minimum velocity for type 1 D / mini Tcriti
• Minimum velocity for type 2
• Open the curve into straight line
• Obtain by linear case algorithm
• Minimum velocity minimum velocity of the a1
  possibilities
• Running time O(a2)

PoI 1
PoI 1
PoI a
PoI a
PoI 2
PoI 2


PoI i
PoI i


PoI 3
PoI 3
Type 1 1 possible path
Type 2 a possible paths
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Closed Curve Case MS-BELP

• Two types of PoIs
• Type 1 Tcriti lt D/v
• Type 2 Tcriti D/v
• Type 2 PoIs may be maintained by looping sensor
• If no type 2 PoIs
• All sensors move back and forth

32
To Open the Curve

• If no optimal sensor groups partially overlap
• Open the curve and solve the problem
• Claim If 2 sensor groups overlap
• Can reassign PoIs to sensors
• Such that no extra sensor is used, AND
• The 2 sensor groups do not overlap

33
How to Reassign PoIs?

• Suppose sensor k passes through sk
• Check if PoIs from sk to ek can be covered by
  sensor k
• If yes, reassign PoIs like this

Sensor k
Sensor k
sk
sk
ek
ek
Sensor k
Sensor k
sk
ek
ek
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How to Reassign PoIs?

• If no, there exists tk between sk and ek
• Such that tk cannot be covered by sensor k
• Implications

35
Closed Curve Case MS-BELP Algorithm

• Optimal solution exists
• In which no groups partially overlap
• But we dont know where to open the curve
• Try all possibilities open the curve in a ways
• If there are type 2 PoIs (looping sensor works)
• Seek solution when 1 sensor loops
• Seek solution when no sensor loops
• Running time O(a3)

36
2-D Case MV-BELP

• Main hurdle finding optimal path is NP-hard
• Heuristic algorithm
• Use approximation algorithm to find TSPN path
• Min velocity length of path / mini Tcriti
• Constant factor approximation

37
2-D Case MS-BELP

• Heuristic algorithm
• Use approximation algorithm to find TSPN path
• Apply Closed Curve Case algorithm
• Constant factor approximation

rmax max distance between 2 PoIs
38
Conclusion

• Strengths
• Algorithm design
• Proof on approximation factor
• Weaknesses
• PoIs fixed and known a priori
• Does not take full advantage of multiple sensors
• No simulation / experiment

Thank you and questions welcomed