Leader Election

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Leader Election

• Transform the system from configuration where
  every node is in the same state, to a
  configuration where one node is in state leader
  and all others are in state defeated.
• Motivation
• central coordinator often needed
• efficiency/simplicity (centralized protocols
  tend to be more efficient)
• fault tolerance recover from a crashed
  coordinator
• closely related to other problems spanning tree
  construction, minimum finding, cooperative
  merging of multiple computations

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Leader Election II

• Basic impossibility result
• leader cannot be elected in general in anonymous
  setting
• symmetry cannot be broken
• Ways to avoid the problem
• unique IDs (standard assumption)
• asymmetric network, configuration of non-unique
  IDs
• Results we already know
• in trees with unique IDs (saturation)
• multi-flooding (everybody broadcasts its ID,
  minimum wins)

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Leader Election III

• Leader election in rings
• very simple, but symmetrical network
• unidirectional, bidirectional (oriented,
  unoriented)
• all the way
• as far as it can, average cost analysis of it
• limited travel (controlled distance)
• Franklins algorithm, variants and improvements
• unidirectional protocols
• discussion

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Leader Election IV

• O(n) leader election in oriented hypercubes
• O(n) leader election in oriented complete
  networks
• Reducing the number of chords
• log n chords hypercube-like chordal rings
• O(1) chords O(n) leader election in torus
• How about lack of orientation?
• torus, hypercube, complete networks
• General approach to leader election in specific
  networks
• Leader election in arbitrary networks

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All the way (LeLann 1977)

• Basic idea
• ID of each entity travels all the way around the
  ring
• the leader is the node with minimal value
• solves also the problem of input collection
• Assumptions
• local orientation, distinct IDs
• can be unidirectional or bidirectional, also
  unoriented
• no FIFO on links, no knowledge of n
• Termination problem
• when receives own ID back (assumes FIFO)
• when received n distinct values (assumes n
  known)
• Solution
• compute n (have a counter in each message)

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As far as it can (Chang and Roberts 79)

• Main idea
• no need to forward IDs which cannot be the
  smallest one
• Overview
• each node maintains variable smallest seen
• only IDgtsmallest seen is forwarded
• only the smallest ID will travel all the way
  around
• notification phase is needed
• FIFO? knowledge of n?
• Complexity?
• worst case, best case, average case

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Limited travel (Hirschberg Sinclair 80)

• Main idea
• works in phases, the first phase is initiated by
  the spontaneously awaken processors (candidates)
• in phase i a candidate sends two explorers in
  opposite directions
• each explorer carriers the ID of its candidate
• each explorer travels distance 2i and either
  dies (if it encounters a node with lower ID) or
  returns back to its candidate
• when(if) a candidate receives back both its
  explorers, it is promoted to the next phase
• when a candidate receives its explorers from the
  opposite directions, it becomes the leader

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Limited travel (Hirschberg Sinclair 80)

• Correctness
• at least one candidate always survives phase i
  (the one with the smallest ID)
• a candidate with will eventually become a leader
  (when 2i n)
• at most one candidate will become leader (if a
  candidate receives both its explorers back from
  the opposite directions, they traveled the whole
  ring and there cannot be another active candidate
  left)
• Complexity
• at most n/2i candidates survive stage i (there
  is no candidate with lower ID within distance 2i
  )
• a candidate in stage i spends 4x2i messages
• therefore, the cost of one stage is 2n
• there are log n stages
• hence overall complexity is 8nlog n

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Electorial stages (Franklin 80)

• Main idea
• works in phases, the first phase is initiated by
  the spontaneously awaken processors (candidates)
• in phase i a candidate sends two explorers in
  opposite directions
• each explorer carriers the ID of its candidate
  and the stage number
• each explorer travels until it reaches another
  candidate of the same stage
• when(if) a candidate receives explorers from
  both sides and it is the smallest, it is promoted
  to the next phase
• when a candidate receives its explorers from the
  opposite directions, it becomes the leader
• Assumptions
• bidirectional, unoriented ring network, unique
  Ids
• FIFO (will be relaxed)

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Electorial stages (Franklin 80) II

• Correctness
• the candidate with the smallest ID will never be
  defeated
• the number of candidates in each stage is halved
  by at least 2
• show example when 2 stages are sufficient
• show the worst case for n8
• therefore the leader will eventually be elected
• Complexity
• at most log n1 phases
• 2n messages are sent in each phase
• total complexity 2n log n O(n)
• Removing FIFO requirement
• many ways, e.g. if a message of too high stage
  is received, queue it and wait for the message of
  the right stage
• higher stage beats lower stage

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Franklin with positive feedback

• Main idea
• when you receive messages from the two
  neighbouring candidates, send OK to the one with
  the smallest ID (if his ID is smaller then yours)
• a candidate proceeds to the next stage only
  after receiving OK from both sides
• Correctness/complexity
• the candidate with the smallest ID is never
  defeated
• in each stage the number of candidates is
  reduced by a factor of 3
• the number of messages per stage is 3n
• the overall complexity is 3nlog3n cca 1.89 n
  log n

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Franklin in unidirectional rings

• Main idea
• Cannot send left? Lets the whole ring move
  right.
• Implementation
• each message travels to the second candidate
• a candidate thus receives 2 messages from the
  left and plays the role of the candidate from
  which the first message arrived
• Correctness/complexity
• correctness follows from the bidirectional case
• complexity 4n messages per stage, 4n log n
  overall

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Alternate directions

• Main idea
• take Franklin and divide the sending to the
  right and to the left into two sub-stages
• only the candidate which survive the first
  sub-stage are active in the second sub-stage
• Correctness/complexity
• correctness similarly as before
• complexity there are at most F-1n (the first i
  such that the i-th Fibonacci number is at least
  n) sub-stages
• there are n messages in each sub- stage
• overall complexity is cca 1.44 n log n

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Summary