Worst and BestCase Coverage in Sensor Networks

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Worst and Best-Case Coveragein Sensor Networks

• Seapahn Meguerdichian, Farinaz Koushanfar,
• Miodrag Potkonjak, and Mani Srivastava

IEEE TRANSACTIONS ON MOBILE COMPUTING, 2005
Presented by Cheng-Ta Lee 11/17/2009
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Outlines

• Introduction
• Preliminaries
• Stochastic Coverage
• Worst-case Coverage and Maximal Breach Path
• Best-case Coverage and Maximal Support Path
• Experimental Results
• Conclusion
• Future Works

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Introduction

• In general, coverage can be considered as a
  measure of the quality of service of a sensor
  network.
• Furthermore, coverage formulations can try to
  find weak points in a sensor field and suggest
  future deployment or reconfiguration schemes for
  improving the overall quality of service.
• By using best and worst-case coverage information
  as heuristics to deploy sensors to improve
  coverage.

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Preliminaries

• Computational Geometry
• Voronoi Diagram
• Delaunay Triangulation

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Stochastic Coverage

• In the simulation studies for this paper, authors
  have generally assumed uniform sensor
  distribution.
• Given
• A field A.
• Sensors S, where for each sensor si?S, the
  location (xi,yi) is known.
• Areas I and F corresponding to initial (I) and
  final (F) locations of an agent.

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Worst-case Coverage and Maximal Breach Path
(maxmin) (1/6)

• Definition Breach.
• Given a path P connecting areas I and F, breach
  is defined as the minimum Euclidean distance from
  P to any sensor in S.
• Problem Maximal Breach Path.
• PB is defined as a path through the field A, with
  end- points I and F and with the property that
  for any point p on the path PB, the distance from
  p to the closest sensor is maximized, thus the PB
  must lie on the line segments of the Voronoi
  diagram.
• Theorem 1.
• At least one Maximal Breach Path must lie on the
  line segments of the bounded Voronoi diagram
  formed by the locations of the sensors in S.

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Worst-case Coverage and Maximal Breach Path (2/6)

• The following steps outline the algorithm for
  finding PB
• Generate Voronoi diagram D for S.
• Apply graph theoretic abstraction by transforming
  D to a weighted graph.
• Find PB using binary-search and
  breadth-first-search.

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Worst-case Coverage and Maximal Breach Path (4/6)
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Worst-case Coverage and Maximal Breach Path (5/6)
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Worst-case Coverage and Maximal Breach Path (6/6)

• The complexities of the subalgorithms
• For generating the Voronoi diagram, O(n log(n)),
  where n is the number of vertex.
• For BFS O(log(m)) where m is the number of edges.
• For binary search O(log(range)).

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Best-case Coverage and Maximal Support Path
(minmax) (1/3)

• Definition Support.
• Given a path P connecting areas I and F, support
  is defined as the maximum Euclidean distance from
  the path P to the closest sensor in S.
• Problem. Maximal Support Path .
• PS is defined as a path through the field A, with
  end- points I and F and with the property that
  for any point p on the path PS, the distance from
  p to the closest sensor is minimized.
• Theorem 2.
• At least one Maximal Support Path must lie on the
  edges of the Delaunay triangulation (with the
  exceptions of the start and end points connecting
  PS to I and F).

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Best-case Coverage and Maximal Support Path (2/3)

• The algorithm for finding PS is very similar to
  the breach algorithm above, with the following
  exceptions
• The Voronoi diagram is replaced by the Delaunay
  triangulation as the underlying geometric
  structure.
• Each edge in graph G is assigned a weight equal
  to the largest distance from the corresponding
  line segment in the Delaunay triangulation to the
  closest sensor.
• The search parameter breach_weight is replaced by
  the new parameter support_weight and the search
  is conducted in such a way that support_weight is
  minimized.

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Best-case Coverage and Maximal Support Path (3/3)
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Experimental Results (1/3)
If new sensors can be deployed or existing
sensors moved such that this breach_weight is
decreased, then the worst-case coverage is
improved.
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Experimental Results (2/3)
If additional sensors can be deployed or existing
sensors moved such that support_weight is
decreased, then the best-case coverage is
improved.
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Experimental Results (3/3)
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Conclusion

• Authors presented best and worst-case
  formulations for sensor coverage in wireless ad
  hoc sensor networks.
• An optimal polynomial time algorithm that uses
  graph theoretic and computational geometry
  constructs was proposed for solving for best and
  worst-case coverages
• Maximal Breach Path (worst-case coverage)
• Maximal Support Path (best-case coverage)
• Additional sensor deployment heuristics to
  improve coverage.

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Future Works

• In practice, other factors influence coverage
  such as
• Obstacles
• nonhomogeneous sensors
• Authors have introduced heuristics based on this
  coverage model that may perform well for
  single-sensor deployment, it is interesting to
  investigate methods of optimally deploying
  multiple sensors at a time.

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References

• SeapahnMeguerdichian, Farinaz Koushanfar, Miodrag
  Potkonjak, and Mani B. Srivastava,Coverage
  Problems in Wireless Ad-hoc Sensor Networks,
  IEEE INFOCOM 2001.
• Laura Kneckt, Summary of Coverage Problems in
  Wireless Ad-hoc Sensor Networks, 2005.

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