On the Packet Header Size and Network State Tradeoff for Trajectory-Based Routing in Wireless NetworksRajagopal 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

Talk OutlineTrajectory-Based Routing (TBR) revisitedOverviewLong/complex trajectories SINsProblem DefinitionMotivationContributionsOptimization FormulationHardnessHeuristicsFuture Work

Overview of TBRSo, how does it work?What happens when a packet travels in the network?SDUse parametric curves (e.g. Bezier, B-spline) for encoding.

Long/Complex TrajectoriesHow to encode long/complex curves? longer curve larger packet headerSplit the curve into simpler pieces:Each piece could be represented by a cubic Bezier curveThe 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).

Long/Complex TrajectoriesHow to encode long/complex curves? longer curve larger packet headerSINs (i.e. I1, I2) do special forwarding.Store the next Bezier curves control pointsUpdate the packet headers with that of the next Bezier curves control points

Problem DefinitionApplication-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 sizeRepresentation 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., INFOCOM03]

Problem DefinitionOverall problem: What can we say about the relationship between:Packet header sizeNetwork state sizeAccuracy of curve representation

Goal: accurate representation of a trajectory with the objective of minimizing the cost incurred due to header size and network state

Illustrative ExampleThe simplest representation of a trajectory: straight lineNegative: High error in representation accuracy.Positive: Small network state.Small packet header.These become higher when error needs to be bounded.

Illustrative Example

ContributionsProvide 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.

System ModelK 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 lengthHeader overhead cost Cp and network state CN associated with approximation selected for each portion of the curve.

Equivalent Graph Representation of Discretized Trajectory

Optimization Formulationk # of representation choicesm # of points where split can be done due to discretizationMaps to Constrained-Shortest Path Problem, i.e. NP-Complete.

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.

Form of Objective FunctionObjective 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.

Trajectory Partitioning HeuristicSplit the trajectory in halfStart with an error bound E.Try representing each piece within the leftover error tolerance E/2If so, deduct the error of this piece from EIf 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

Equal Error HeuristicSelect the number of SINs: mAllow each piece to use have a maximum error of: E/m

Positive:Better balanced approximation is likely

Negative: m can be significantly suboptimal

Future WorkComparison of heuristics with the optimal solution based on exhaustive search.

Better heuristics

Distributed solutions to the problem are needed

Thank you!THE END