Multi-Valued Logic Synthesis

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Routing in Wireless Ad Hoc Networks by Analogy to
Electrostatic Theory Mehdi Kalantari Mark
Shayman University of Maryland, College
Park mehkalan,shayman_at_eng.umd.edu
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Electrostatic Routing
Motivation Natural phenomena with similarity in
structure Fluid dynamics How the flow forms
among the sources and the sinks Heat exchange
How heat is exchanged between hot and cold
environments Electrostatic How the electric
field is propagated in a dielectric with a
density of positive and negative charges Common
features Transfer something from sources to
sinks, Gauss Law
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Assumptions

• Basic assumptions
• A network of N wireless nodes that can
  communicate through radio frequency wireless
  links. Nodes are densely distributed in the
  network.
• Nodes have no mobility.
• There are M source-destination pairs in the
  network, the
• communication demand for each of them is known.

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Definition of Load Vector Field
Given a set of source-destination pairs and a
set of paths between each pair, the load vector
field represents the actual load or demand at
every place of network. Its direction shows the
desired direction of relaying packets.
Example Line of sight routing
The weight of the demand of source-destination
pair i The unit vector aiming from the
source to the destination The line of sight of
the source and destination
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One important property of
Gauss' law The amount of flow that crosses the
boundary of a closed contour is the amount of net
sources inside that contour In the other
words
C
A differential element normal to the boundary
of C
The net sum of sources--positive for sources,
negative for destinations.
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PDE form of the Gauss Law
The density of sources
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One major problem of definition of
Cancellation The value of may not
represent the actual load density at a place of
the network. See example below. However, there
is one special case in which represents
the actual demand and load of the network Many
to one communication There is a single
destination in the network.
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Many to One Communication

• Many sources in the network want to communicate
  to a single node.
• Nodes can use multi-path and multi-hop routes to
  communicate.
• In this case routes uniquely specify , and
  uniquely specifies routes.
• The value of at each place of the network
  specifies the actual value of the communication
  demand at that point.

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Abstract Definition of
In general is any vector field that
satisfies the following equations
The normal component of on the boundary
of the network
Routes are generated from by following the
direction of this vector field by relaying a
packet in the direction of . We assume the
nodes are distributed densely in the network.
Routes
Observation The above equation does NOT specify
uniquely.
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Spreading the Load As Evenly As Possible
Intuition spreading the load as evenly as
possible. -The number of collisions in the
network decreases. -Network resources are
utilized better. Mathematical formulation of
even load distribution
s.t.
and
Solution
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The Analogy to the Maxwells Equations
Maxwells Equations
The above equations with boundary condition give
uniquely.
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A Useful Observation
This means is a conservative vector
field, so it can be written as the gradient of a
scalar field.
is known as a potential function. The
routing is done in direction of the gradient of
. This potential function satisfies This
equation is analogous to the Poisson equation in
electrostatic theory.
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Simulation Experiment

• Consider a wireless network of 1500 nodes
  randomly
• distributed in a 1000m x 1000m square.
• The transmission range of the nodes is 85m.
• 40 source nodes want to communicate to a node in
  the
• center of the network.
• Each source node uses multi-path routing with 8
  paths
• to the destination.

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A Simulation Experiment (contd)
Legend
X Sources Destination Other Nodes
The placement of the nodes
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A Simulation Experiment (contd)
The direction of optimal
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A Simulation Experiment (contd)
The value of
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A Simulation Experiment (contd)
Electrostatic Routes
Shortest Path Routes
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Performance Comparison
Throughput (bits/s)
Shortest path
Electrostatic
70 improvement
Other experiments with random seeds
Average improvement 34
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Conclusion
Electrostatic routing is inspired by many
phenomena in nature, and it can improve network
performance.

• Possible Future Work
• Making the approach decentralized
• Trying to extend the results to the case in which
  we have partial location information
• Extending the results to many to many
  communication case
• Adding mobility to the model