September 12, 2006
IEEE PIMRC 2006, Helsinki, Finland 1
On the Packet Header Size and Network State Tradeoff for Trajectory-Based Routing in Wireless Networks
Rajagopal IyengarRensselaer Polytechnic Institute,
ECSE Dept., Networks [email protected]
http://networks.ecse.rpi.edu/~iyeng
Murat YukselUniversity of Nevada-Reno,
http://www.cse.unr.edu/~yuksem
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IEEE PIMRC 2006, Helsinki, Finland 2
Talk Outline Trajectory-Based Routing (TBR) revisited
Overview Long/complex trajectories – SINs
Problem Definition Motivation Contributions
Optimization Formulation Hardness
Heuristics Future Work
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IEEE PIMRC 2006, Helsinki, Finland 3
Overview of TBR So, how does it work?
What happens when a packet travels in the network?
S
DDATADATA
Use parametric curves (e.g. Bezier, B-spline) for encoding.
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Long/Complex Trajectories How to encode long/complex curves?
longer curve larger packet header
IP1
IP2
Split the curve into simpler pieces: Each piece could be represented by a
cubic Bezier curve The complete trajectory is
concatenation of the pieces.
Source performs signaling and sends a control packet that include:
end locations of the cubic Bezier curves, i.e. Intermediate Point (IP)
all the control points
The nodes closest to the IPs will be the Special Intermediate Nodes (SINs).
DS
I1
I2
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IEEE PIMRC 2006, Helsinki, Finland 5
Long/Complex Trajectories How to encode long/complex curves?
longer curve larger packet header
IP1
IP2D
S
I1
I2
SINs (i.e. I1, I2) do special forwarding. Store the next Bezier curve’s control points Update the packet headers with that of the
next Bezier curve’s control points
C1
C2
C3 C4
C5
C6
SD
C5C6C3
C4IP2
SD
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IEEE PIMRC 2006, Helsinki, Finland 6
Problem Definition Application-specific goals may require different levels of
accuracy in trajectory
Accuracy is affected by the selection of: # of SINs – network state size # of bits to encode each trajectory piece – packet header size Representation accuracy of each piece – error in the encoded
trajectory
Tradeoff: Packet header size vs. network state vs. representation accuracy
A similar tradeoff was studied between MPLS stack depth and label sizes [Gupta et. Al., INFOCOM’03]
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IEEE PIMRC 2006, Helsinki, Finland 7
Problem Definition Overall problem: What can we say
about the relationship between: Packet header size Network state size Accuracy of curve representation
Goal: accurate representation of a trajectory with
the objective of minimizing the cost incurred due to header size and network state
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Illustrative Example
The simplest representation of a trajectory: straight line
Negative: High error in
representation accuracy.
Positive: Small network
state.Small packet
header.
These become higher when
“error” needs to be bounded.
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Illustrative Example
Each piece can be represented by different choice of basic representation units, each causing different amount of
error to the representation of the complete trajectory.
Each piece can be represented by different choice of basic representation units, each causing different amount of
error to the representation of the complete trajectory.
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IEEE PIMRC 2006, Helsinki, Finland 10
Contributions Provide insight into the relationship between
packet header size, network state and accuracy of representation
Generic optimization formulation for trajectory optimization when provided with a set of encoding/decoding options.
Show hardness of problem and provide initial heuristics which can work well for certain classes of problem instances.
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IEEE PIMRC 2006, Helsinki, Finland 11
System Model K choices for representing each piece of the trajectory (r1,…,rk)
(‘colors’ from a ‘palette’)
Network state is maintained at points along the trajectory which divide the curve into portions which are represented using the ri
Trajectory is discretized using m equally spaced points
Binary valued matrix Q(m,n) used to represent which color from the palette is used on a given portion of the curve
If some ri selected, then a subroutine to compute error e(Qi,j,ri) is utilized.
Deviation area, Normalized length Header overhead cost Cp and network state CN associated with
approximation selected for each portion of the curve.
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IEEE PIMRC 2006, Helsinki, Finland 12
Equivalent Graph Representation of Discretized Trajectory
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Optimization Formulation
k – # of representation choices
m – # of points where split can be done due to discretization
Network state cost.
Application-specific
error bound.
Max of 1 representation
per splitting point.
Maps to Constrained-Shortest Path Problem, i.e. NP-Complete.
Packet header cost.
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IEEE PIMRC 2006, Helsinki, Finland 14
Why different than “curve compression”?
One might ask: Why is this different than the curve compression algorithms in
computational geometry?
Packet header cost – a new dimension to the
curve compression algorithms.
Curve Compression:
minimize the number of line
segments matching a target error requirement.
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Form of Objective Function Objective function captures packet header and
network state costs, Cp and CN.
Cp in reality could be a function of battery power at a node, since transmission of long packets causes greater power consumption.
CN could be a function of available buffer space at a node, for example, sensor networks where simple, resource constrained nodes are used.
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Trajectory Partitioning Heuristic Split the trajectory in half Start with an error bound E. Try representing each piece within the leftover error
tolerance E/2 If so, deduct the error of this piece from E If not, keep halving each piece until it is possible to
represent within the leftover error tolerance
Positive: Uses the error budget as much as possible
Negative: The later (or earlier depending on design) pieces will have
higher error in representation
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Equal Error Heuristic Select the number of SINs: m’ Allow each piece to use have a maximum error
of: E/m’
Positive: Better balanced approximation is likely
Negative: m’ can be significantly suboptimal
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Future Work Comparison of heuristics with the
optimal solution based on exhaustive search.
Better heuristics
Distributed solutions to the problem are needed