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A Practical Algorithm for Constructing Oblivious Routing Schemes

A Practical Algorithm for Constructing Oblivious Routing Schemes. Marcin Bieńkowski Mirosław Korzeniowski Harald Räcke. Problem Definition. given a graph describing a computer network input sequence of routing requests

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A Practical Algorithm for Constructing Oblivious Routing Schemes

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  1. A Practical Algorithm for Constructing Oblivious Routing Schemes Marcin Bieńkowski Mirosław Korzeniowski Harald Räcke

  2. Problem Definition • given a graph describing a computer network • input sequence of routing requests • select a system of paths between and for each request • each request routed fractionally over the paths chosen for it • cost-measure: congestion, i.e., the maximum load of an edge Virtual Circuit Routing Problem:

  3. Problem Definition online algorithm: • online algorithm must specify each path-system without knowingfuture requests • compare congestion of the online-algorithm to the optimal offline congestion for oblivious algorithm: • the system of paths chosen for a request cannot depend on any other request

  4. Related Work algorithms for specific networks: • Maggs, Meyer auf der Heide, Vöcking and Westermann [1997]Bartal, Leonardi [1997] • lower bound of for the competitive ratio in the mesh • logarithmic upper bounds for many specific networks as e.g., hypercubic networks, meshes... non-polynomial algorithm for general networks: • Räcke [2002] • polylogarithmic () upper bound for general networks

  5. Related Work optimal algorithm • Azar, Cohen, Fiat, Kaplan, Räcke [2003] • optimal competitive ratio for oblivious algorithm • based on linear programming and the Ellipsoid algorithm with a separation oracle

  6. Results Improvement on Räcke’s result from 2002 • Simplified proof for the upper bound on the competitive ratio • competitive polynomial-time algorithm for general graphs • competitive polynomial-time algorithm for planar graphs

  7. Related Work a better result • Harrelson, Hildrum, Rao [2003] • a competitive polynomial-time algorithm based on decomposition

  8. Hierarchical Decomposition

  9. Hierarchical Routing Scheme ? if offline has congestion 1 weonly send messagesalong this virtual edge t s

  10. Questions Decomposition: • How is the partitioning done? Routing Scheme: • How are intermediate nodes chosen? • How are routing paths between intermediate nodes chosen?

  11. Choosing Intermediate Nodes (1) • Probability distribution for choosing blue intermediate node of cluster : • weight function is the bandwidth of edges connecting to nodes outside of

  12. Choosing Intermediate Nodes (2) • Probability distribution for choosing white intermediate node of cluster , whose subclustering is : • weight function is the bandwidth of edges connecting to nodes in other sub-clusters

  13. Choosing Routing Paths (1)

  14. Choosing Routing Paths (2)

  15. Choosing Routing Paths (3) The decomposition has to fulfill: • tree height is • throughput property for each cluster and its subclusters: for CMCF-problem with demands: the solution of the flow must satisfy each demand and produce a congestion of at most

  16. A Bad-Case Example • No clustering of the following example can fulfill the throughput property

  17. Precondition A set fulfills the precondition iff for each such that

  18. Decomposition Theorem Theorem We can partition any cluster that fulfills the precondition into subclusters such that • fulfills the throughput property • for each we have • each fulfills the precondition

  19. Illustration of the Algorithm • We divide the cluster into single-node subclusters • If there is no solution for the flow we find a witness for this fact • we merge it • .... and cut it to fulfill the precondition • We can round an „ugly” set losing logarithmic factor capacity of edges bet- ween different clusters decreases

  20. Precondition Lemma Lemma Acluster can be partitioned into subclusters such that • each fulfills the precondition A partitioning • exists for • can be computed in polynomial time for

  21. Future Work • Is there a class of networks for which adaptive online algorithms are asymptotically better than oblivious algorithms? • How low can we make the competitive ratio? ? • How to repair the structure of the tree quickly if the graph changes?

  22. A Practical Algorithm for Constructing Oblivious Routing Schemes Thank you for your attention

  23. Flows and Cuts Concurrent Multicommodity Flow and Sparsest Cut General graphs Existence Planar graphs CMCF SparsestCut Computed Cut

  24. CMCF, SparsestCut - definition • Concurrent MultiCommodity Flow problem • Deliver some fraction of each demand • Respect the edges capacities • To maximize: the smallest delivered fraction of a demand • If each demand satisfied with ratio then we can route with congestion Sparsity of a cut :

  25. Fulfilling the precondition • . • amortize the new created capacity against edges in • an edge is used at most times for amortization since

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