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Self Stabilizing Smoothing and Counting

This paper discusses the concepts of smoothing and counting networks in self-stabilizing systems and analyzes their behavior with improper initialization. It also explores techniques for recovering from illegal states and presents results on the self-stabilization of balancing networks.

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Self Stabilizing Smoothing and Counting

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  1. Self Stabilizing Smoothing and Counting Maurice Herlihy, Brown University Srikanta Tirthapura, Iowa State University

  2. Overview • Smoothing and Counting Networks • Analysis of behavior without proper initialization- upper and lower bounds • Self stabilization of smoothing networks

  3. Smoothing Networks 2-smoothing network In a k-smoothing Network, the numbers of Tokens on different output wires differ by at most 2

  4. Counting Networks • 1-smoothing networks with other additional properties • Aspnes, Herlihy and Shavit in 1991 • Since then, scalable Construction and Properties well studied • Bitonic and Periodic networks are two popular counting networks

  5. Balancer

  6. Balancer

  7. Balancer

  8. Counting Network Depth = 4 Width = 4 Initial State: All balancers pointing up

  9. 1-Smoothing Property

  10. Questions • How do counting networks perform when initialized incorrectly (or by an adversary)? • How to recover from illegal states reached during execution?

  11. Motivation • Initializing to a “correct” global state is hard or may be impossible • global reconfiguration expensive • network switches reboot • Step towards building fault tolerant and dynamic smoothing networks

  12. Our Results(1) Periodic and Bitonic Counting Networks: • When started from an arbitrary state, output is log w smooth (w = width of network) • Tight lower bound: We demonstrate inputs such that the output is not log k smooth for any k < w

  13. Our Results (2) Self-stabilization of Balancing Networks • Add extra state and actions • If network begins in illegal state, will eventually return to a legal state • Upper bound on the time till stabilization, and extra space required

  14. Block[w] Block[w] Block[w] Periodic[w] Counting Network

  15. 000 001 Block[4] 010 011 Block[2] 100 101 Block[4] 110 111 Block Network:Inductive Definition

  16. Definitions • Sequence is k-smooth if for all • Matching layer of balancers for sequences X and Y joins and in a one-to-one correspondence

  17. Matching Lemma If X and Y are each k-smooth then result of matching X and Y is (k+1)-smooth Holds irrespective of the orientations of balancers 3 54 2 X 1 53 53 104 52 Y 103 52 51 102

  18. Block[w] is (log w)-smooth • Proof by Induction • Assume Output of Block[n] is log n smooth • Show that output of Block[2n] is log (2n) smooth A Block[n] B Block[n] Block[2n]

  19. Lower Bound • Worst Case bound: There exist input sequences and initial states such that output of Block[w] is not k-smooth for any k < log w • Show a fixed-point sequence for Block[w]which is not k-smooth for any k < log w

  20. Block[2] Fixed Point Sequence 5 5 5 5 5 6 6 6 Block[4] 6 4 4 5 5 7 4 4 6 6 5 7 Sequence not k-smooth for anyk < log (width) 7 6 Block[4] 4 6 6 5 5 5

  21. Bitonic Counting Network Starting from an arbitrary initial state • Output is always log w smooth, where w=width • Matching worst case lower bound on smoothness

  22. Self Stabilization • Extra state and actions added to the network • Self-stabilizing Actions enabled only if network in illegal state otherwise, normal execution

  23. Self Stabilization • Definition:Legal State can be reached in an execution starting from the Correct Initial State • Natural definition, but hard to use directly, so need alternate characterization • Local state can be observed easily • Strategy: Characterize legality in terms of local states

  24. Global vs Local States

  25. Additional State Top In Top Out Bot In Bot Out These counters can be bounded – details in paper

  26. Local States • Balancer is Legal if (1)Top In + Bot In = Top Out + Bot Out(2)Toggle State is correct • Wire is Legal ifTokens entering the wire = Tokens leaving the wire + Tokens in Transit

  27. Global Legality in terms of Local Theorem: Iff (every wire and every balancer is in legal local state), then (the network is in a legal global state) Now focus on stabilizing the local states- simpler problem

  28. Space and Time Complexity • Time to Stabilization = d parallel timesteps where d = depth of network • Total additional space =w = width of network

  29. Issues • Lazy versus pro-active stabilization • Transient Behavior till stabilization might differ from “legal” behavior • Tokens might be unevenly distributed till then

  30. Summary • Even if bitonic and periodic networks are not initialized, they are log smooth • If only approximate smoothing is needed, then use (log w) depth uninitialized block network • Can be converted into 1-smooth behavior by self-stabilization - overhead is small and analytically bounded

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