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Tracking. Murat Demirbas SUNY Buffalo. A Pursuer-Evader Game for Sensor Networks. Murat Demirbas Anish Arora Mohamed Gouda. Pursuer-evader problem. Evader is omniscient; Strategy of evader is unknown

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tracking

Tracking

Murat Demirbas

SUNY Buffalo

a pursuer evader game for sensor networks

A Pursuer-Evader Game for Sensor Networks

Murat Demirbas

Anish Arora

Mohamed Gouda

pursuer evader problem
Pursuer-evader problem
  • Evader is omniscient; Strategy of evader is unknown
  • Pursuer can only see state of nearest node; Pursuer moves faster than evader ( ratio = f )
  • Required is to design a program for nodes and pursuer so that pursuer can catch evader (despite the occurrence of faults)
model
Model
  • Connected graph of sensor nodes
  • Transient faults; connectivity still maintained
  • Maximal parallelism in node actions
two approaches
Two approaches
  • Evader-centric program
    • move is costly, find is for free
    • sensor nodes communicate periodically with neighbors
    • stabilizes and tracks faster
  • Pursuer-centric program
    • find is costly, move is for free
    • sensor nodes communicate with neighbors only upon request
    • minimizes number of messages and energy efficient
  • Hybrid program
    • find & move are both tunable
outline
Outline
  • Evader-centric program
  • Pursuer-centric program
  • Hybrid program
  • Extensions
evader centric program
Evader-centric program
  • Nodes collectively maintain a tracking tree
    • ts.j : latest timestamp that j knows about detection of evader
    • p.j: parent of j in tree
    • d.j : distance of j from evader
    • root is the node where the evader resides

{Evader resides at j} ---> p.j := j; ts.j :=clock.j; d.j :=0

    • every node sets its parent to be the nbr with maximum ts

ts.k > ts.j ---> p.j :=k ; ts.j := ts.(p.j); d.j:= d.(p.j)+1

  • Find is for free: pursuer follows the tracking tree to its root
evader centric program cont
Evader-centric program (cont.)
  • Tracking tree is dynamically rooted at the evader
  • Parent of a node is closer to the evader
evader centric program cont1
Evader-centric program (cont.)
  • Tracking tree is dynamically rooted at the evader
  • Parent of a node is closer to the evader
evader centric program cont2
Evader-centric program (cont.)
  • Tracking tree is dynamically rooted at the evader
  • Parent of a node is closer to the evader
evader centric program proof of stabilization
Evader-centric program (proof of stabilization)
  • Tracking tree is rooted at the evader within D steps
    • soft-state stabilization
  • The distance between pursuer and evader does not increase once the constructed tree includes the pursuer
  • Starting from an arbitrary state pursuer catches evader in at most D+ 2D * f/(1-f) steps
pursuer centric program
Pursuer-centric program
  • Move is for free: ts.j is maintained locally

{evader detected at j} ---> ts.j := clock.j

  • Nodes communicate with nbrs only at the request of pursuer; pursuer is directed to nbring node with highest recorded time

{pursuer detected at j} ---> next.j :in {k | ts.k= max ts.nbr.j };

ts.j :=0

  • Pursuer action is to move to next.j
    • pursuer does a random walk until it reaches a node that evader has visited
pursuer centric program cont
Pursuer-centric program (cont.)
  • If the pursuer reaches a node j with ts.j>0, pursuer catches evader within N*f/(1-f) steps
pursuer centric program stabilization
Pursuer-centric program (stabilization)
  • Since ts.j is reset to 0 when pursuer visits j, bad values disappear
  • From random walk result, pursuer reaches a node j, ts.j >0 , within O( N2 * log N ) steps
  • Then, pursuer catches evader within N*f/(1-f) steps
hybrid program
Hybrid program

Tracking tree is bounded to a depth R

Pursuer-centric program is executed at nodes outside tracking tree

hybrid program stabilization
Hybrid program (stabilization)
  • Since tracking tree is bounded to a depth R, soft-state stabilization is not available for nodes outside tree
    • Cycles are detected and removed by counting to R
  • Starting from an arbitrary state pursuer finds the tracking tree in at most O( (N-n)2 * log (N-n) ) steps
    • n : number of nodes in tracking tree
  • Then, pursuer catches evader within R*f/(1-f) steps
  • Hybrid program is tunable by assigning R appropriately
outline1
Outline
  • Evader-centric program
  • Pursuer-centric program
  • Hybrid program
  • Extensions
extensions
Extensions
  • Asynchronous model :

Readily available

  • Faster convergence :

Extended hybrid program

  • Better scalability :

Hierarchical tracking program

asynchronous model
Asynchronous model
  • Evader-centric program
    • instead of ts.j maintain val.j
    • val.j denotes the number of detections of the evader that j is aware of
    • when j detects evader it increments val.j
    • tracking tree is rooted at evader in 2D steps
    • we have implemented the asynchronous version for June 2002 DARPA/NEST demo
  • Pursuer-centric program
    • readily available
extended hybrid program
Extended hybrid program
  • Pursuer-centric program can be modified query a radius Rp(instead of 1) s.t. R+ Rp = D
self stabilizing hierarchical tracking service for sensor networks

Self-Stabilizing Hierarchical Tracking Service for Sensor Networks

Murat Demirbas ,

Anish Arora ,

Tina Nolte ,

Nancy Lynch

stalk scalable tracking
STALK: Scalable tracking
  • Maintain tracking structure
    • over fewer number of nodes
    • with accuracy inversely proportional to the distance from evader
      • communication cost of msgj,k= distance(j,k), delay= δ*distance(j,k)
      • nearby nodes (cheap to update) have recent & accurate info
      • distant nodes (expensive to update) have stale & rough info
  • Local operations :
    • Cost of move proportional to the distance the evader moves
    • Cost of find proportional to the distance from the evader
    • Cost of healing proportional to the size of the initial perturbation
  • To this end we employ a hierarchical partitioning of the network

M. Demirbas, A. Arora, T. Nolte, and N. Lynch. A Hierarchy-based Fault-local Stabilizing Algorithm for Tracking in Sensor Networks. OPODIS, 2004.

hierarchical clustering
Hierarchical clustering

R: dilation factor of clustering to determine size at higher levels

Radius at level L is ≈ RL

M. Demirbas, A. Arora, V. Mittal, and V. Kulathumani. Design and Analysis of a Fast Local Clustering Service for Wireless Sensor Networks. BroadNets 2004.

slide24
Hierarchical tracking path

evader

evader

evader

  • Grow action for building a tracking path
  • Shrink action for cleaning unrooted paths
local find
Local find
  • Searching phase:
    • A find operation at j queries j’s neighbors & j’s clusterhead at increasingly higher levels to find the tracking path
  • Tracing phase:
    • Once path is found, operation follows the path to its root
slide26
Examples of find

A find for an evader d away incurs O(d) work/time cost

  • guaranteed to hit the tracking path at level logRd of hierarchy

evader

find

find

find

a problem for move
A problem for move

evader dithering between cluster boundaries may lead to nonlocal updates

evader

evader

evader

local move
Local move
  • Laterallinks to avoid nonlocal updates
  • When evader moves to new location j:
    • a new path is started from j
    • the new path checks neighbors at each level to see whether insertion of a lateral link is possible
  • Restricts lateral links to 1 per level in order not to deteriorate the tracking path
    • otherwise find would not be local since it could not hit the path at level logRd for an evader d away
slide29
Examples of move

A move to distance d away incurs O(d*logRd) work/time cost

  • a level L pointer is updated at every iL-1Ridistance; level L is updated d/iL-1Ritimes
  • update at L incurs O(RL) cost

evader

evader

evader

evader

evader

evader

evader

local healing
Local healing means work/time for recovery proportional to perturbation size & not the network size

In the presence of faults

a grow can be mistakenly initiated; shrink should contain grow

a shrink can be mistakenly initiated; grow should contain shrink

Local healing
fault containment
Fault-containment
  • Give more priority to the action that has more recent info regarding the validity of the path
  • A shrink or grow action is delayed for longer periods as the level of the node executing the action gets higher
      • j.grow-timer = g * R lvl(j)
      • j.shrink-timer = s * R lvl(j)
  • Catching occurs within a constant number of levels
    • For g=5δ, s=11δ, b=11δR
      • grow catches shrink in 2 levels:

logR ((bR–b+sR2–gR-δR)/(sR-gR-3δ))

      • shrink catches grow in 4 levels:

logR ((bR–b+sR+gR-2s+3δR)/(gR-s-δ))

seamless tracking
Seamless tracking
  • Fault-containment does not affect responsiveness
    • Total delaying up to l is a constant factor of communication delay up to l, δR l
  • Concurrent move operations
    • move occurs before tracking path is updated
    • a complete path is no longer possible; discontinuity in the path
    • give a bound on evader speed to maintain a reachable path
  • Concurrent find operations
    • when find reaches a dead-end, search phase is re-executed
    • reachability condition guarantees that new path is nearby
  • Cost of find & move unaffected

find

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