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 Labiyengr@rpi.edu

http://networks.ecse.rpi.edu/~iyeng

Murat YukselUniversity of Nevada-Reno,

CSE Departmentyuksem@cse.unr.edu

http://www.cse.unr.edu/~yuksem

September 12, 2006

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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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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IEEE PIMRC 2006, Helsinki, Finland 8

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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IEEE PIMRC 2006, Helsinki, Finland 9

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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IEEE PIMRC 2006, Helsinki, Finland 15

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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IEEE PIMRC 2006, Helsinki, Finland 17

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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IEEE PIMRC 2006, Helsinki, Finland 18

Future Work Comparison of heuristics with the

optimal solution based on exhaustive search.

Better heuristics

Distributed solutions to the problem are needed

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IEEE PIMRC 2006, Helsinki, Finland 19

Thank you!

THE END