A PursuerEvader Game for Sensor Networks

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A Pursuer-Evader Game for Sensor Networks

• By Murat Demirbas, Anish Arora, Mohamed Gouda
• Presented 4/17/03 by John Oleszkiewicz

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Roadmap

• Problem Description
• System Description
• Evader-Centric Algorithm
• Pursuer-Centric Algorithm
• Hybrid Algorithm
• Extended Hybrid Algorithm

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• Problem Description

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The Pursuer-Evader Problem

• Goal For each computation, the pursuer must
  eventually catch an evader
• Catching being in the same location in a graph

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Evader

• May know state of entire graph
• e.g. May be omniscient
• Can choose any evasion strategy it likes

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Pursuer

• May only use local information
• Is faster than the evader

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Pursuer-Evader in Sensor Networks

• Previous work does not present solutions
  applicable to sensor networks
• Nodes are too energy constrained
• Limited computational resources are available
• Sensors are fault-prone
• No on-site maintenance is available
• Plus...
• Purser-Evader in Sensor Networks money

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• System Description

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Notation

• ltguardgt --gt statement
• j, k, l represent motes
• var.j A variable at mote j
• x e A x is assigned to an element of set A
• Evader resides at j true for all states where
  evader is at j
• Evader detected at j true in any step where
  the evader moves to j false any other time

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Motes

• Each mote consists of
• A system clock
• A timestamp variable
• A pointer to another mote
• Each mote can
• Sense the evader
• Sense the pursuer
• Change its own variables
• Read the variables of its neighbors

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Graph

• Undirected, connected
• N number of motes
• D diameter
• M distance between pursuer and evader
• a ratio of speed of evader to pursuer
• ? average degree of each mote
• Transient faults only
• Faults do not cause loss of connectivity

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Pursuer and Evader Actions
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System Graph
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Performance Metrics

• How many communications are performed?
• How fast is the evader caught?

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Evader-Centric Algorithm
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Mote Actions
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Tracking Tree
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Performance Metrics

• Each mote communicates with each neighbor at
  every time step
• Number of communications at each step N?
• Not very energy efficient
• Evader will be captured quickly
• At most in M 2M a / (1-a) steps
• M steps for tracking tree construction, 2M for
  maximum distance between evader and pursuer after
  construction

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Stabilization

• Assuming
• ts.j lt clock.j
• p.j is either a nbr.j, j or NULL
• A fresh tracking tree is reconstructed in at most
  D steps
• Starting from an arbitrarily corrupted state, the
  pursuer catches the evader in at most
• D 2D a / (1 a)

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Pursuer-Centric Algorithm
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Mote Actions
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Pursuer-Centric Graph
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Performance Metrics

• At each time step, only one node communicates
  with all of its neighbors
• Number of communications at each step ?
• Energy efficient
• Evader will not be captured quickly
• Random walk to pick up the trail - O(n3)
• Can be improved to O(N2logN) with topology info
• Once pursuer finds the trail, evader will be
  captured in N a / (1 a) steps

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Stabilization

• Timestamps are reset to 0 every time the pursuer
  visits
• Eventually corrupted timestamps disappear.

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Hybrid Algorithm
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Mote Actions
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Mote Actions (con't)
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Hybrid Graph
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Performance Metrics

• Metrics are tuneable to the application needs
• Tracking tree communication and pursuer-centric
  communication required.
• n ? ?
• Evader will be caught faster than pursuer-centric
• O(N-n)2 log (N-n) steps to find the tracking
  tree
• R (a / 1 a) steps to catch pursuer

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Stabilization

• Tracking tree ok...
• Explicit stabilizing actions need to be added for
  motes outside of tracking tree.
• Detect invalid parents
• Correct fake tree roots

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Mote Stabilization Actions
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Extended Hybrid Algorithm
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Extended Pursuer-Centric

• The pursuer uses agents to search for a trace of
  the evader.
• Previous method random walk
• If an agent finds a timestamp greater than 0, it
  reports back to the pursuer
• Does not use mote communications
• Provides path from pursuer to mote

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Extended Hybrid
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Performance Metrics

• Communication
• Each step n ? communications occur
• One-time cost of 2(D-R)? for pursuer tree
• Evader caught more quickly than hybrid version
• 3(D-R) R a / (1 a) steps
• (2D steps for pursuer tree construction D steps
  to follow path through pursuer tree)

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Other Issues

• 1-pursuer 0-evader scenario
• Evader-centric no energy expended
• Pursuer-centric ? communications per step
• Hybrid ? communications per step
• Extended Hybrid Path returned only if evader
  tree encountered. No energy expended otherwise.
• 0-pursuer 1-evader scenario
• Force pursuers to authenticate themselves when
  they join the network and notify the network when
  they leave.

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Thanks for Listening!