DISTRIBUTED ALGORITHMS

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• DISTRIBUTED ALGORITHMS
• By
• Nancy.A.Lynch
• Chapter 18
• LOGICAL TIME
• By
• Sudha Elavarti

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Introduction

• Logical time is a notion used for simplifying
  the job of programming an asynchronous network
• Logical time is an element of some fixed totally
  ordered set Tt1,t2.., which is to be assigned
  for every execution of an asynchronous network A
• Logical time of all events should respect all the
  dependencies among the events within a system A.

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Logical Time for Send/Receive Systems

• If a is an execution of an asynchronous
  send/receive network, then logical time for a is
  assigning a value of Tt1.t2 to every event
  in a.
• Assumptions
• Only universal reliable FIFO send/receive
  channels are considered.
• No internal events are considered

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Properties for assigning Logical time to S/R

• 1)No two events get assigned the same logical
  time.
• 2)The logical times of events at each process are
  strictly increasing, according to their order of
  occurrence in a.
• 3)The logical time of any send event is strictly
  smaller than that of the corresponding receive
  event.
• 4)For any t belongs to T, there are only finite
  many events that get assigned logical times
  smaller than t.

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Logical time for Broadcast Systems

• To define logical time for broadcast systems all
  the events are considered.
• For any execution a ,logical time assignment is
  assigning a value T to very event in a , in such
  a way as to satisfy same properties as for S/R
  systems except property 3 i.e
• The logical time of any bcast event is strictly
  smaller than that of each corresponding receive
  event.

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Adding Logical Time to Asynchronous Algorithms

• Logical times are generated for the events of a
  given asynchronous S/R network algorithm A.
• The given algorithm A transforms into a new
  asynchronous S/R algorithm L(A).
• The transformation works process by process.
• Whenever a process of L(A) simulates a step of A,
  it also generates a logical time.

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Advancing the clock (1.Lamport-Time
Transformation)

• In this algorithm local clocks are maintained and
  advanced when messages are received in order to
  keep them synchronized.
• Logical time domain T is a set of pairs (c,i)
  where c is a non-negative integer, i is a process
  index.
• The clock variable gets increased by at least 1
  at every event that occurs at process i.
• The logical time of any event is defined to be
  the value of the clock variable immediately after
  the event, paired with process index i as a tie
  breaker

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Lamports time transformation contd...

• Send-The clock value of send event is attached as
  time stamp to the mesg being sent.
• Receive-For every receive event, i increases its
  clock variable larger than its previous value and
  than the timestamp of the mesg.
• LamportTime(A) satisfies all the properties of
  definition of logical time.
• LamportTime transformation can be easily modified
  to work in asynchronous broadcast systems.

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Delaying Future Events(2.WelchTime
Transformation)

• Clocks are not advanced in response to mesg
  receipts, but mesgs that arrive early are
  delayed.
• Domain T is a set of triples (c,i,k), where c is
  a nonnegative real, i process index, k belongs to
  N.
• Logical time of any event is defined to be the
  value of the clock when the event occurs.
• Process i maintains a FIFO receive-buffer in
  order to hold mesgs whose timestamps are greater
  than or equal to the local clock value.

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WelchTime contd..

• For every mesg if timestamp is less than current
  clock value, mesg is processed, else it is placed
  in receive-buffer, they are processed in the
  order in which they appear in the receive-buffer.
• The unbounded ness of the local clock variables
  also implies that every mesg in a receive-buffer
  is eventually processed.
• WelchTime(A) satisfies all the properties of
  definition of logical time.
• WelchTime transformation can be modified to work
  in asynchronous broadcast systems.

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ApplicationsBanking Systems

• For counting the total amount of money that is
  transferred between process via mesg.
• Each process has a local variable that money that
  contains current amount at that location.
• CountMoney algorithm transformed from algorithm A
  finds the local balance

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CountMoney Algorithm

• It has a subroutine with predetermined logical
  time t belongs to T, assuming t is known, general
  strategy is
• 1)For each process of A, determine the value of
  the money variable after all events with logical
  times less than or equal to t and before all
  events with logical times greater than t.
• 2)For each channel, determine the amount of
  money in all the messages sent at logical times
  less than or equal to t but received at logical
  times strictly greater than t.

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Global Snapshots

• CountMoney generalised to get an instantaneous
  global snapshot of system state.
• useful for debugging,in case of failure,or
  detecting certain global properties.
• it is possible to get a global snapshot by
  delaying process and mesgs till it records all
  the needed information, but not practical.
• true global snapshot may not be needed, but a
  system state that look-like global snapshot to
  all processes is good enough.

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LogicalTimeSnapShot algorithm

• As in CountMoney,it has a subroutine with
  predetermined logical time t belongs to T,
  assuming t is known, general strategy is
• 1)determine the state of each process of A after
  all events with logical times less than or equal
  to t and before all events with logical times
  greater then t.
• 2)For each channel, determine the sequence of
  messages sent at logical times less than or equal
  to t but received at logical times strictly
  greater then t.
• --this information is determined by same
  distributed algo. (CountMoney)

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ReplicatedStateMachine Algorithm

• To simulate single state machine or single shared
  variable, say x.
• Consider there are n user processes submitting
  invocations to x and receiving responses from x.
• One user process Ui at each node i of the
  network.
• The network as a whole should be atomic object.
• ALGO- Each process Ai simply receives
  invocations from user Ui and broadcasts them.
• Each process has a local copy of x and a local
  variable invocation-buffer in which it stores all
  invocations.
• Process i places local invocation in its
  invocation-buffer when it performs the broadcast
  and places remote invocation when it performs the
  receive for that invocation.

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Continued

• Process i maintains a vector known-time, which
  keeps track of largest local time for each
  process.
• Known-time(i)i logical time of most recent
  event at process i.
• Known-time(j)i logical time of broadcast event
  received by i from j .
• Process i is permitted to apply an invocation pi
  in its invocation-buffer to its copy of x when
  following is true.
• Invocation pi has the smallest logical time of
  any invocation in invocation-buffer (i) .
• For every j, known-time(j)i gt logical time of pi.

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Transforming Real-Time Algorithms to Logical-Time
Algorithms

• S/R system A ,in which each process Ai has a
  real-time value
• Times occurring as arguments in its output event
  should be nondecreasing and unbounded in any fair
  execution.
• It is possible to transform each process Ai into
  a process Bi without real-time,but with logical
  time.