A PursuerEvader Game for Sensor Networks

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A Pursuer-Evader Game for Sensor Networks

• Sixth Symposium on Self-Stabilizing Systems, San
  Fransisco, June 2003

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Outline

• Introduction
• Problem definition
• Evader-centric Program
• Pursuer-centric Program
• A Hybrid Pursuer-Evader Program
• Efficient Version of Hybrid Program
• Discussion and Future Work

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Introduction

• Due to the importance in military contexts,
  pursuer-evader tracking has received significant
  attention
• Previous work on the pursuer-evader problem is
  not directly applicable to tracking in sensor
  networks
• Limited computing resource
• Centralized algorithms are not suitable
• Energy constrained
• Excessive communication burden on nodes are not
  acceptable
• Faulty
• Nodes may reach arbitrary states
• On-site maintenance is not feasible
• Must be self-stabilizing

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Problem Definition

• Sensor nodes communication with its neighbors is
  symmetric and form a connected graph
  (bi-connectivity)
• Each sensor node can immediately detect the
  pursuer/evader are resident at that node
• Both pursuer and evader are mobile, move
  atomically from one node to another
• Pursuer moves faster than evader
• The movement strategy
• Evader unknown, but can omnisciently inspecting
  the entire network to decide the next movement
• Pursuer based only on the stat of the node at
  which it resides
• Eventually the pursuer and evader reside at the
  same node (catch)

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Algorithm Design

• Tunability
• Constructing it to be a hybrid between two
  orthogonal programs
• Evader-centric program
• Pursuer-centric program
• Limit the tracking tree depth to a bounded R in
  the evader-centric program to save energy
• Tuning R for tracking speed or energy efficiency
• Self-stabilization
• Algorithm will be self-stabilized starting from
  any arbitrary states

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A Evader-centric Program

• Nodes action
• Pursuers action

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Steps of Evader-centric Program (1/3)
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Initially, ts.j 0 for all sensor j
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Steps of Evader-centric Program (2/3)
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Steps of Evader-centric Program (3/3)
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Properties of Evader-centric Program (1/2)

• The tracking tree is a spanning tree rooted at
  the mote where the evader resides
• The evader resides the only node which has
  highest timestamp
• The tracking tree is fully constructed in at most
  D steps
• The distance between the pursuer and evader does
  not increase once the constructed tree includes
  the mote where the pursuer resides
• The pursuer catches the evader in at most
• M2Ma/(1-a)

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Properties of Evader-centric Program (2/2)

• Stabilization
• The tracking tree stabilizes in at most D steps
• The pursuer catches the evader in at
  most D2Da/(1-a) steps when starting from an
  arbitrarily corrupted state
• Performance
• Not energy efficient ?N communications,
  effectively N when all communication treated as
  broadcast
• (? average degree of a node)
• But fast Pursuer catches evader when starting
  from an arbitrary stat at most D2Da/(1-a) steps

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A Pursuer-centric Program

• Nodes action
• Pursuers action

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Steps of Pursuer-centric Program
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Evader Action
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Pursuer Action
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Properties of Pursuer-centric Program (1/2)

• If the pursuer reaches a node j where ts.jgt0, the
  pursuer catches the evader in at most Na/(1-a)
• The pursuer reaches a mote j where ts.jgt0 within
  O(N2logN) steps
• Followed by recent research with a local topology
  information (degree of neighbor vertices)

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Properties ofPursuer-centric Program (2/2)

• Stabilization
• ts.j resets to zero upon a detection of the
  pursuer, arbitrary ts.j eventually disappear
• Performance metrics
• Energy efficient only ? communications to the
  neighbor that the pursuer resides at each step
  (? average degree of a node)
• But Slow O(N2logN) steps

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A Hybrid Pursuer-Evader Program (1/3)

• Modified evader-centric program

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A Hybrid Pursuer-Evader Program (2/3)

• Two stabilization actions for the bounded length
  tracking tree

For non-valid parents
For fake tree roots
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A Hybrid Pursuer-Evader Program (3/3)

• Modified pursuer-centric program

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Properties of Hybrid Pursuer-Evader Program

• The tracking tree is fully constructed in at most
  R steps
• The pursuer reaches the tracking tree within
  O((N-n)2log(N-n)) steps
• (n denotes the number of nodes in the tracking
  tree)
• Stabilization
• The tracking tree structure stabilizes in at most
  2R steps
• The pursuer catches the evader within
  O((N-n)2log(N-n)) steps starting from an
  arbitrary state
• Performance metrics
• n ? communications per step
• The random walk takes O((N-n)2log(N-n)) to find
  the tracking tree and from the point in takes
  Ra/(1-a) steps for the pursuer to catch the
  evader

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Efficient Version of the Hybrid Program (1/2)

• Efficient pursuer-centric program
• Concept replace the random talk action by
  searching a trace of evader using agents
• Agents can be implemented by constructing DFS or
  BFS trees rooted at the mote where pursuer
  resides
• If there is a node with ts.jgt0 in the pursuer
  tree, the pursuer is notified and move to j along
  the tree
• Then take Na/(1-a) steps to catch the evader
• Modified pursuer-centric program
• If pursuer resides a node k with no nbr.k has
  ts.(nbr.k)gt0, the node set next.j to ? and start
  a tree construction
• The tree will be stabilized in D steps

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Efficient Version of the Hybrid Program (2/2)

• Extended hybrid program
• The depth of pursuer tree should be set to D-R
  hops instead of D hops
• Performance metrics
• From O(N2logN) to 3DNa/(1-a) compared to
  pursuer-centric program
• (Two D for tree construction and information
  propagation,
• one D for pursuers movement)
• From O((N-n)2log(N-n)) to 3(D-R) for pursuer to
  reach the tracking tree compared to original
  hybrid program
• Extra overhead, but still energy efficiency

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Illustration of Extended Hybrid Program
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D5, R2
Tracking Tree
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Pursuer Tree
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Discussions (1/2)

• Energy consumption under special cases
• 1-pursuer 0-evader
• Evader-centric good
• Pursuer-centric bad
• Hybrid bad
• Extended Hybrid good with modified pursuer tree
  program which triggered if evader tree is
  encountered
• 0-pursuer 1-evader
• Achieve energy-efficiency with pursuers
  authenticate themselves when they join the
  network and notify the network when they leave

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Discussions (2/2)

• Shared memory model adopted instead of
  message-passing for simplicity
• Maximal parallelism v.s. event-based execution
• Asynchronous program
• Instead of underlying clock synchronization
  service
• Achieved by adding a variable which increases
  when detecting the evader
• Other energy issues
• The periodicity of soft-state updates for
  stabilization should be kept low if the faults
  are relatively rare in the network
• Maintain the tracking tree over a small number of
  nodes
• Ex directional

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Future Works

• The communication time between sensor nodes
• message broadcast v.s. shared memory
• Different number of pursuers and evaders
• Range of movement
• Computation interleaving instead of maximum
  parallelism
• General forms of the tracking problem