1. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland1 On the Packet Header Size and Network State Tradeoff for Trajectory-Based Routing in Wireless Networks Rajagopal Iyengar Rensselaer Polytechnic Institute, ECSE Dept., Networks Lab iyengr@rpi.edu http://networks.ecse.rpi.edu/~iyeng Murat Yuksel University of Nevada-Reno, CSE Department yuksem@cse.unr.edu http://www.cse.unr.edu/~yuksem

2. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland2 Talk Outline Trajectory-Based Routing (TBR) revisited Overview Long/complex trajectories – SINs Problem Definition Motivation Contributions Optimization Formulation Hardness Heuristics Future Work

3. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland3 Overview of TBR So, how does it work? What happens when a packet travels in the network? S D DATA Use parametric curves (e.g. Bezier, B-spline) for encoding.

4. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland4 Long/Complex Trajectories How to encode long/complex curves? longer curve  larger packet header IP 1 IP 2 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). D S I1I1 I2I2

5. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland5 Long/Complex Trajectories How to encode long/complex curves? longer curve  larger packet header IP 1 IP 2 D S I1I1 I2I2 SINs (i.e. I 1, I 2 ) 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 S D C5 C6 C3 C4 IP 2 S D

6. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland6 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]

7. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland7 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

8. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland8 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.

9. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland9 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.

10. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland10 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.

11. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland11 System Model K choices for representing each piece of the trajectory (r 1,…,r k ) (‘colors’ from a ‘palette’) Network state is maintained at points along the trajectory which divide the curve into portions which are represented using the r i 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 r i selected, then a subroutine to compute error e(Q i,j,r i ) is utilized. Deviation area, Normalized length Header overhead cost C p and network state C N associated with approximation selected for each portion of the curve.

12. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland12 Equivalent Graph Representation of Discretized Trajectory

13. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland13 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.

14. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland14 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.

15. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland15 Form of Objective Function Objective function captures packet header and network state costs, C p and C N. C p in reality could be a function of battery power at a node, since transmission of long packets causes greater power consumption. C N could be a function of available buffer space at a node, for example, sensor networks where simple, resource constrained nodes are used.

16. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland16 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

17. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland17 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

18. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland18 Future Work Comparison of heuristics with the optimal solution based on exhaustive search. Better heuristics Distributed solutions to the problem are needed

19. September 12, 2006IEEE PIMRC 2006, Helsinki, Finland19 Thank you! THE END