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Stabilization in the Small World

Stabilization in the Small World. Andrew Berns Sukumar Ghosh University of Iowa. Intro to small world. Small world topology guarantees the existence of a “short path” between any pair of nodes. Watts-Strogatz model Kleinberg’s model. Our problem. 0. N-1. Elect a leader. j. j+1.

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Stabilization in the Small World

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  1. Stabilization in the Small World Andrew Berns Sukumar Ghosh University of Iowa

  2. Intro to small world Small world topology guarantees the existence of a “short path” between any pair of nodes. Watts-Strogatz model Kleinberg’s model

  3. Our problem 0 N-1 Elect a leader j j+1 Unidirectional ring: Large N Each process j has two neighbors: (j+1) and k Probability (j chooses k as a neighbor = 1/|k-j| k

  4. Our problem 0 Initially, each process i has an arbitrary value of L(i) N-1 ? j j+1 • i,j: L(i) = L(j) = k, where k is a “real” process. Usually it is the real process • with the largest id k

  5. The major steps Ping to check if L is real or fake Propagate the largest real id as the leader’s id to all. We will use controlled broadcast CB (j, k) j<k: process j sends k’s id to all processes in (j+1, .., k-1, k) j>k : process j sends k’s id to all processes in (j+1, .., N,0,1, ..k)

  6. Results from Kleinberg’s model Using a greedy routing, any node j can send a message of k in an expected number of O(log2N) hops. We hope the largest id will be sent to virtually all processes with almost similar efficiency, making the system “almost stabilizing” in polylog rounds (which is good). Fake leaders will fail to respond to the ping (again in polylog time) and will be discarded after a timeout period

  7. The proposed steps Execution takes place in rounds. Each round costs unit time Start: every process j initiates CB (j, L(j)); Thereafter, when process i receives j, L(j): R1: L(j) ≤ L(i) -> skip R2: L(j) > L(i) -> L(i):= L(j); CB (i, L(i)) R3: L(i) does not exist -> skip (how is this detected?) R4: timeout -> L(i) := I What is the final configuration?

  8. Analysis To be done. Identify the chain of events. We hope to show that almost the entire system stabilizes with high probability in O(log3N) rounds, and the remainder of the system asymptotically attains stability.

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