1. 1 Routing in Error-Correcting Networks Edwin Soedarmadji May 10, 2006 California Institute of Technology Department of Electrical Engineering Pasadena, CA 91125, USA

2. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Introduction Start from an unrelated network problem  Route planning under fuel capacity and refueling constraints Especially relevant  Increasing energy cost  Vehicles with alternative fuel  Vehicles exploring remote areas

3. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 The Gas Station Problem  Shortest Path Problem in  Vehicle has limited a fuel capacity  Refueling nodes in the network  Edge weights expressed in fuel units  Vehicle starts at with fuel units 

4. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 The Gas Station Problem  Feasible paths are paths where the vehicle always carries a non- negative amount of fuel on the path nodes  Is {all feasible paths} an empty set ?  If not, what is the path that minimizes the travel distance? 

5. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Theorem

6. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Example  Remove infeasible edges in E and vertices in V  Compute all-pairs shortest path between nodes in V’  Remove infeasible edges in E’ and vertices in V’  Calculate the shortest path from s to t  Solution produced in = = 12

7. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Error-Correcting Network Possible Generalization to Communication Networks  The Gas-Station algorithm works for transportation networks  Is it applicable to communication networks? There are many similarities  Vehicle  information packet  Gas tank capacity  error budget  Gas station  error correction node  Gas consumption  packet error

8. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Error Budget  MURFLES MARBLES  Suppose each packet contains seven symbols, and the error-correction scheme employed in the network can correct up to (but not more than) three errors. Then the error budget is three units for a given alphabet size. 

9. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Edge model: Symmetric Channel U A

10. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 The Cascaded SC UA

11. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 UA The ary Erasure Channel

12. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Generalized Dijkstra Algorithm x + min( y, z ) = min ( x + y, x + z )

13. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 The Error Distribution MUFFLER MARBLES

14. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Edge Weight: Worst-Case Error p = 0.10 p = 0.25 p = 0.98 p = 0.50

15. May 10, 2006Maximum Capacity QoS MetricLee Center Workshop 06 Routing in Error-Correcting Networks  WCE x is a non-decreasing function of p  The algorithm used in the Gas Station Problem can be used to find the feasible path with minimum WCE from s to t.  Questions? 