Rajagopal Iyengar Rensselaer Polytechnic Institute, ECSE Dept., Networks Lab iyengr@rpi

1. 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 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 IEEE PIMRC 2006, Helsinki, Finland

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

4. IP1 IP2 I2 I1 D S Long/Complex Trajectories • How to encode long/complex curves? • longer curve  larger packet header • 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). IEEE PIMRC 2006, Helsinki, Finland

5. IP1 IP2 C5 C2 I2 I1 D IP2 S S C1 D C3 C4 S C5 C3 C6 D C6 C4 Long/Complex Trajectories • How to encode long/complex curves? • longer curve  larger packet header • 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 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] 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 IEEE PIMRC 2006, Helsinki, Finland

8. Illustrative Example Negative: High error in representation accuracy. The simplest representation of a trajectory: straight line Positive: Small network state. Small packet header. These become higher when “error” needs to be bounded. IEEE PIMRC 2006, Helsinki, Finland

9. 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. Illustrative Example 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. 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 errore(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. IEEE PIMRC 2006, Helsinki, Finland

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

13. Network state cost. Packet header cost. Application-specific error bound. Max of 1 representation per splitting point. Optimization Formulation k – # of representation choices m – # of points where split can be done due to discretization Maps to Constrained-Shortest Path Problem, i.e. NP-Complete. IEEE PIMRC 2006, Helsinki, Finland

14. Why different than “curve compression”? Curve Compression:minimize the number of line segments matching a target error requirement. Packet header cost – a new dimension to the curve compression algorithms. One might ask:Why is this different than the curve compression algorithms in computational geometry? 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. IEEE PIMRC 2006, Helsinki, Finland

16. 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 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 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 IEEE PIMRC 2006, Helsinki, Finland

19. THE END Thank you! IEEE PIMRC 2006, Helsinki, Finland