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Online Routing in Faulty Meshes with Sub-linear Comparative Time and Traffic Ratio

Online Routing in Faulty Meshes with Sub-linear Comparative Time and Traffic Ratio. Stefan Ruehrup Christian Schindelhauer Heinz Nixdorf Institute University of Paderborn Germany. Overview. Routing in faulty mesh networks Routing as an online problem

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Online Routing in Faulty Meshes with Sub-linear Comparative Time and Traffic Ratio

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  1. Online Routing in Faulty Mesheswith Sub-linear Comparative Time and Traffic Ratio Stefan Ruehrup Christian Schindelhauer Heinz Nixdorf Institute University of Paderborn Germany

  2. Overview • Routing in faulty mesh networks • Routing as an online problem • Basic strategies: single-path versus multi-path • Comparative performance measures • Our algorithm

  3. faulty node s source active node routing path t target Online Routing in Faulty Meshes • Mesh Network with Faulty Nodes: • Problem: Route a message from a source node to a target

  4. Offline versus Online Routing • Routing with global knowledge(offline) is easy • But if the faulty parts are not known in advance? • Online Routing: • no knowledge about the network • no routing tables • only neighboring nodes can identifyfaulty nodes s

  5. Why Online Routing is difficult barrier • Faulty nodes form barriers • barriers can be like mazes • Online routing in a faulty network = search a point in a maze • Related problems: navigation in an unknown terrain, maze traversal, graph exploration, position-based routing s perimeter t

  6. Basic Strategies: Single-path • Barrier Traversal • follow a straight line connecting source and target • traverse all barriers intersecting the line • leave at nearest intersection point • Time and traffic: h = optimal hop-distancep = sum of perimeters • no parallelism, traffic-efficient s t Problem: time consuming, if many barriers

  7. Basic Strategies: Multi-path • Expanding Ring Search: • start flooding with restricted search depth • if target is not in reach thenrepeat with double search depth • Time: Traffic:h = optimal hop-distance • asymptotically time optimal Problem: traffic overhead, if few barriers

  8. Competitive Time Ratio „ • competitive ratio: • competitive time ratio of a routing algorithm: • h = optimal hop-distance • algorithm needs T rounds to deliver a message “ solution of the algorithm optimal offline solution cf. [Borodin, El-Yanif, 1998] T h single-path

  9. h+p Comparative Traffic Ratio • optimal (offline) solution for traffic:h messages (length of shortest path) • this is unfair, because ... • offline algorithm knows all barriers • but every online algorithm has to pay exploration costs • exploration costs: sum of perimeters of all barriers (p) • comparative traffic ratio: M = # messages used h = length of shortest path p = sum of perimeters

  10. Comparative Ratios • measure for time efficiency: competitive time ratio • measure for traffic efficiency: comparative traffic ratio • Combined comparative ratiotime efficiency and traffic efficiency

  11. time ratio traffic ratio combined ratio maze Barrier Traversal (single-path) Expanding Ring Search (multi-path) open space Algorithms under Comparative Measures time traffic Barrier Traversal (single-path) Expanding Ring Search (multi-path) Is that good? on the scenario It depends ...

  12. How to beat the linear ratio • define a search area(including source and target) • subdivide the search area into squares (“frames”) • traverse the frames efficiently  decision: traversal or flooding? • enlarge the search area, if the target is not reached 4 3 2 1 s t barrier

  13. Frame Multicast Problem • Inform every node on the frame as fast as possible goal: constant competitive ratio • Traverse and Search: frame traversal stopped, start expanding ring search entry node starts frame traversal

  14. Performance of Traverse and Search • Traverse and Search in a mesh of size g x g • Time: constant competitive ratio • Traffic: • frame traversal • flooded area is quadratic in the number of barrier nodes... but also bounded by g2 • concurrent exploration costs a logarithmic factor 3 1 2

  15. Recursive Traverse and Search • Expanding ring search inside a frame: • Subdivide the flooded area in sub-frames • apply Traverse and Search on sub-frames • Traffic: 1st recursion: (g1g1-frame subdivided into g0g0-frames) 2nd recursion: 3rd recursion ... • Time: constant factor grows exponentially in #recursions replaced by toplevel frame

  16. Overall Asymptotic Performance • Toplevel frame = 1/4 search area, size = h2 • With an appropriate choice of g0, g1, ..., gl : • Time: • Traffic: • combined comparative ratio: • sub-linear, i.e. for all compared to

  17. Conclusion • Our algorithm is • nearly as fast as flooding ... and traffic efficient • approaches the online lower bound for traffic • Open question: Can time and traffic be optimized at the same time? ... or is there a trade-off?

  18. Thank you for your attention! Questions ... Stefan Ruehrup sr@upb.de Tel.: +49 5251 60-6722 Fax: +49 5251 60-6482 Algorithms and Complexity Heinz Nixdof Institute University of Paderborn Fuerstenallee 11 33102 Paderborn, Germany

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