HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Online Routing in Faulty Meshes with Sub-linear Comparative Time and Traffic Ratio Stefan Ruehrup Christian Schindelhauer Heinz Nixdorf Institute University of Paderborn Germany

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 2 Overview Routing in faulty mesh networks Routing as an online problem Basic strategies: single-path versus multi-path Comparative performance measures Our algorithm

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 3 Online Routing in Faulty Meshes Mesh Network with Faulty Nodes: Problem: Route a message from a source node to a target active node faulty node s t target source routing path

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 identify faulty nodes s

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 5 Why Online Routing is difficult 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 perimeter barrier s t

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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-distance p = sum of perimeters no parallelism, traffic-efficient Problem: time consuming, if many barriers s t

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 7 Basic Strategies: Multi-path Expanding Ring Search: –start flooding with restricted search depth –if target is not in reach then repeat with double search depth Time:Traffic: h = optimal hop-distance asymptotically time optimal Problem: traffic overhead, if few barriers

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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] „ “ h T single-path

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 9 M = # messages used h = length of shortest path p = sum of perimeters 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: h+p Comparative Traffic Ratio

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 10 Comparative Ratios measure for time efficiency: competitive time ratio measure for traffic efficiency: comparative traffic ratio Combined comparative ratio time efficiency and traffic efficiency

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 11 Algorithms under Comparative Measures Barrier Traversal (single-path) Expanding Ring Search (multi-path) traffictime scenario maze open space Barrier Traversal (single-path) Expanding Ring Search (multi-path) time ratio traffic ratio combined ratio Is that good? It depends... on the

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

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 13 Frame Multicast Problem Inform every node on the frame as fast as possible goal: constant competitive ratio Traverse and Search: frame entry node starts frame traversal entry node starts frame traversal traversal stopped, start expanding ring search traversal stopped, start expanding ring search

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 14 Performance of Traverse and Search Traverse and Search in a mesh of size g x g –Time: constant competitive ratio –Traffic: 1.frame traversal 2.flooded area is quadratic in the number of barrier nodes... but also bounded by g 2 3.concurrent exploration costs a logarithmic factor 1 2 3

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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: (g 1  g 1 -frame subdivided into g 0  g 0 -frames) 2nd recursion: 3rd recursion... Time: constant factor grows exponentially in #recursions replaced by toplevel frame

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 16 Overall Asymptotic Performance Toplevel frame = 1/4 search area, size = h 2 With an appropriate choice of g 0, g 1,..., g l : Time: Traffic: combined comparative ratio: sub-linear, i.e. for all compared to

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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?

Stefan Ruehrup HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 18 Thank you for your attention! Questions... Thank you for your attention! Questions... Stefan Ruehrup Tel.: Fax: Algorithms and Complexity Heinz Nixdof Institute University of Paderborn Fuerstenallee Paderborn, Germany