ANONYMOUS NETWORKS

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ANONYMOUS NETWORKS
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Las Vegas and Monte Carlo

• A Las Vegas algorithm is a probabilistic
  algorithm that
• terminates with positive probability
• is partially correct.
• A Monte Carlo algorithm is a probabilistic
  algorithm that
• terminates
• is partially correct with positive probability.

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The Itai-Rodeh election algorithm

• Las Vegas algorithm for rings of known size
• Chang-Roberts algorithm with random IDs ? 1 ..
  N
• Selecting the minimal ID node
• Ring size N detects the return of the token to
  the initiator
• un(seen) variable detects the same ID node
• If un false another round is started with new
  IDs

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Itai-Rodeh
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Itai-Rodeh
my token
pass
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Partial correctness

• If at least one process generates a token at
  level l, either one process becomes elected at
  that level and no process generates a token at
  level l 1, or at least one process generates a
  token at level l 1.
• a process generates a token on level l 1 iff it
  has received its own token from level l
• 1. the token is not minimal on level l -gt is
  purged by a process on level l 1 or by a
  process having with smaller identity
• the process with the smaller identity is elected
  or it will generate a token on level l 1
• 2. the token is minimal on level l it will
  either be purged by a process on level l 1 or
  will return to the initiator
• this token is minimal on level l, un is true and
  the process is elected
• un is false gt the process generates a token on
  level l 1

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Itai-Rodeh correctness

• The Itai-Rodeh algorithm terminates with
  probability one.
• Proof
• let k be a number of processes that generated a
  token on level l
• let Pk be the probability that one of k processes
  selected a minimal value (algorithm ends) let P
  mink Pk
• the probability that a token is generated on
  level l 1 is (1 - P)
• the probability that the algorithm does not stop
  after l levels is (1 - P)l
• (1 - P)l for a big value of l is 0

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A probabilistic algorithm for the ring size

• Monte-Carlo
• terminates
• when it terminates, estp N with probability
  depending on a parameter R (R is a
  range of random numbers)
• Process p increases its estimate when
• est gt estp a token is received that contains an
  estimate est gt estp.
• est estp a token with estimate est estp is
  received,
  this token has made est hops, but the included
  label differs from ps label.
• Process p purges the token
• est lt estp
• Otherwise pass the token

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Ring size algorithm
1
a
2
2
1
b
2
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Complexity

• The algorithm message-terminates after exchanging
  O(N3 ) messages. In the terminal configuration
  all estp are equal, and their common value is
  bounded by N.
• Proof
• the estimate remains conservative (never
  decreases), lbl changes only when est increases
• Assume, by induction that when the estimate is
  received in a token it has been conservative so
  far
• 1. estp est and est ?? N because est is
  conservative estimate (by induction) (case
  a2)
• 2. estp esp 1 when
• (a1) h est estp lt est gt est ? N (not
  my token) and est ?? N (conservative)
• (b2) estp est h , lbl ? lblp gt est
  ? N (not my token) and est ?? N
  (conservative)

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. . . contd

• every successor on a ring either keeps the value
  est or increases it
• estNextp ? estp and estp ? N gt
• est has to be the same for all processes
• time complexity O(N3 )
• N processes
• each sends at most N tokens
• each token is at most N - times forwarded

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Termination

• The algorithm terminates,
• and upon termination est N for all p
• with probability at least 1 - (N - 2)
  (1/R)N/2
• Proof
• the algorithm terminates in the configuration
  when all estimates are equal and less or equal to
  N

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• Consider the probability of termination with a
  wrong result (est lt N)
• Probability that processes make a low estimate
  est lt N and it will not be improved
• a token lttest, e, lblp , hgt is forwarded e hops
• e-th process does not increase its estimate
  becasuse lblp lbl
• lets divide processes to f groups f gcd (N,
  e)
• processes in each group have the same estimate

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. . . contd

• processes in a distance const f have the
  same label
• labels are chosen randomly from 1, . . . , R
• probability that all processes in one group have
  the same label R (1/R) N/f R (1/R)
  (N/f ) - 1
• this holds for all groups ((1/R) N/f-1) f
  (1/R) N/f (1/R) N/2
• because f lt N/2
• all possible wrong estimates 2 ... N-1
• probability of a wrong answer (N - 2) (1/R)
  N/f
• probability of a correct answer 1 - (N - 2)
  (1/R) N/f