LEADER ELECTION

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LEADER ELECTION
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Leader election

• all processes are in the same state at the
  initial configuration, and arrive at a
  configuration where exactly one process is in
  state leader and all other processes are in the
  state lost.
• an initial phase of centralized algorithms
• assumptions
• reliable processes and channels
• asynchronous system
• processes are distinguished by unique identities

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Leader election algorithms

• algorithms are decentralized
• the length of messages is O(log N)
• process identities are used to break symmetry
  between processes
• extrema-finding algorithms
• tree algorithms with wake-up phase
• the phase algorithm O (D E) messages, O (D)
  time
• MST algorithm O ( E N log N) messages

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CHANG-ROBERTS RING ALGORITHM
find the node with the maximum id and make it the
leader
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CHANG-ROBERTS Ring Algorithm
?
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Ring networks

• Chang - Roberts
• O(N2) messages, O(N) time
• time complexity O(N)
• all processes start at the same time N
• the process to the left from Pmax starts ( N -
  1) N
• communication complexity (all processes start)
  O(N2 )
• processes are in increasing order 2N - 1
• processes are in decreasing order ?N i1 1/2
  N(N1)

Pmax
9
2
14
8
11
3
send
6
10
4
7
9

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Hirschberg - Sinclaire

• bidiretional ring
• sendLR
• pass
• respond
• complexity
• O(N logN) messages, O(N) time

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Hirschberg - Sinclaire
?
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Hirschberg Sinclaire complexity
length processes
2 . 1 1 2. 20
phase 0
2 . 21 1 2. 20
phase 1
22
22
2 . 22 1 2. 20
phase 2
N/12 . 22 processes in 3rd phase
N/12.2i-1 processes in i-th phase length of i-th
phase 2i1
4 messages
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Hirschberg - Sinclaire

• phase i - 1 leaves N / (2i 1) candidates
• the length of the phase i 2i 1
• 4 messages are sent by each node
• lt 4N (2 2 log N)
• O (N log N)
• Complexity
• O(N logN) messages, O(N) time

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Anonymous networks

• a probabilistic algorithm is modeled as a process
  that tosses a coin with every step it executes.
• ? (?1 , ?2 , . . . ) is an infinite sequence
  of coin flips, i.e., of zeroes and ones in a
  process p.
• ? -computation is a computation in which the i-th
  transition is specified by the i-th element of ? .

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Definitions

• A probabilistic algorithm is ? -terminating if it
  terminates in every ? -computation.
• A probabilistic algorithm is ? -partially correct
  if each terminal configuration of a ?
  -computation satisfies a condition ? .
• A probabilistic algorithm is ? -correct if it is
  both ? -terminating and ?
  -partially correct.

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. . . contd

• A probabilistic algorithm is terminating if it is
  ? -terminating for every ?
  .
• A probabilistic algorithm is partially correct
  if it is ? - partially correct for
  every ? .
• A probabilistic algorithm is correct if it is ?
  -correct for every ? .

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. . . contd

• A probabilistic algorithm is terminating with
  probability P if the probability that the
  algorithm is ? -terminating is at
  least P.
• A probabilistic algorithm is partially correct
  with probability P
  
  if the probability that the
  algorithm is ? - partially correct is at least
  P.
• A probabilistic algorithm is correct with
  probability P if the probability that
  the algorithm is ? -correct
  is at least P.

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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.