SNAPSHOT

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SNAPSHOT
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Motivation Termination detection

• a ring topology, token based algorithm
• a process is inactive
• a process can be activated
• after accepting a message
• First traversal paints nodes white
• If Pi sends a message to Pj , i gt j,
• gt change a color
• Second traversal checks the color
• The token changes a color after passing
  a colored node

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Snapshot

• observing a system from within the same system
• applications of a snapshot
• 1. analysis of stable properties
• (P is a stable property, ? is a reachable
  configuration from ?)
• P( ? ) ? ? gt ? gt P( ? )
• termination
• loss of tokens
• deadlock
• 2. restart of a system
• 3. debugging distributed programs

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Preliminaries

• A distributed algorithm may have many possible
  executions.
• Equivalence class of executions is called
• C ... computation .
• Computation is defined by
• Ev... set of events
• and their partial causal order.
• Assumptions
• every message is delivered in finite time (weak
  fairness)
• a network is strongly connected

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States of processes

• cp(0), cp(1), ... states of local process p
• 
  P ... set of processes in a
  distributed system
• cp(0) ... initial state
• Ev ?p? P ep(1) , ep(2) , ...
• denotation for local causal order ?p for p
  ep(i) ?p ep(j) ltgt i lt j
• global causal order ?
• events internal, send, receive
• The communication history of a process is
  recorded in its state

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Global snapshot

• for the channel pq
• cp(i) contains the list of messages
  sent(i)pq
• sent in the events
• cq(i) contains the list of messages
  rcvd(i)pq
• received in the events
• 
  
• local snapshot of state of p cp cp(i)
• preshot j ? i
  postshot events j gt i
• global snapshot S (cp1 , ... cpN )

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Channels

• Snapshot S includes communication histories
• so it defines a configuration ?
• if the state of channel pq can be obtained
  from
• sentpq \
  rcvdpq
• Then the system configuration ? can be obtained
  from such a snapshot
• ? local states
  channel states

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Feasible snapshot

• cp1, cp2 not feasible
• m1 ? rcvdp1p2 ? m1? sentp1p2
  rcvdp1p2 ? sentp1p2
• 
  (cp2 ) (cp1 )
• cp1, cp3 feasible sentp1p3 \ rcvdp1p3
  - message in transit m2

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Feasibility, a meaningful snapshot

• Snapshot S is feasible if for each two
  (neighbor) processes p and q,
• rcvdpq ? sentpq
• Preshot resp. postshot message is sent in a
  preshot resp. postshot event
• Snapshot S is meaningful in computation C if
  there
• exists an execution E ? C such that
• ? is a configuration of E.
• Algorithm has to take a snapshot in such a way
  that it is meaningful.
• Feasibility concerns neighboring processes, while
  meaningfulness is a global property

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Cuts

• A cut of Ev is a set L ? Ev such that
  
• e ? L ? e ?p e gt
  e ? L
• One-to-one correspondence between cuts and
  snapshots
• For every snapshot, the preshot events constitute
  a cut
• Every cut either doesnot include any event from a
  process or it does
• if it does, it also includes all preceding
  events therefore it contains all preshot
  events and the preshot events constitute a
  snapshot
• Cut L2 is said to be later than cut L1 if
  
• L1 ? L2
• A consistent cut of Ev is a set L ? Ev such
  that
• e ? L ? e ? e gt e ? L

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Equivalent properties

• Let S be a snapshot and L the cut implied by
  S. The following three statements are
  equivalent
• (1) S is feasible
• (2) L is a consistent cut
• (3) S is meaningful

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Proof

• (1) implies (2)
• S is feasible gt L is a consistent cut
• rcvdpq ? sentpq e ? L ?
  e ? e gt e ? L
• consider e ? L and e ? e
• a. e ?p e gt e ? L , as L is a cut
  (only proving feasible)
• b. e ... send, e ... receive p ? q
• e ? L gt e is preshot gt m ?
  rcvdpq
• gt m ? sentpq as S is
  feasible
• gt e ? L

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Proof contd

• (2) implies (3)
• L is a consistent cut gt S is meaningful
• The cut corresponds to a snaphot (we are proving
  only that it is meaningful)
• The snaphot divides events to pre and postshot
• Meaningful snaphot corresponds to a possible
  execution
• Lets create the possible execution F of the
  observed algorithm,
• 1. create permutation f of events in Ev
  (computation for this algorithm), in which the
  cut (preshot events) creates an introductory part
  of the execution
• 2. prove that the permutation f of events
  corresponds to a possible execution F, i.e. prove
  that it is consistent with ?

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Proof contd

• f (f0, f1, ...) enumeration of events from
  Ev, preshot events first
• Both preshot and postshot events are listed in
  any order that is possible according to ?
• preshot events
  postshot events

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Proof contd

• Show that the whole sequence is consistent with
  ?
• Lets have fi ? fj
• fi , fj are both preshot events i lt j
  follows because preshot resp. postshot events
  are enumerated in an order consistent with ?
• fi , fj are both postshot events i lt j
  follows for the same reason
• fi . . preshot event, fj . . postshot event ,
  i lt j
• because as all preshot events are before
  postshot events (snapshot)
• fi . . postshot event , fj . . preshot event
  cannot happen because L is consistent (contains
  only preshot events)
  if fj ? L and fi ?
  fj , then fi ? L , i.e. fi is a preshot
  event

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Proof contd

• We conclude that f is consistent with ?
• If events in f are in order consistent with ?
  , then there exists a computation F, that has
  events in Ev in the same order as they are in f.
  
• F reaches a configuration ? after performing
  all preshot events.
• We built an a possible execution F based on the
  fact that the cut is consistent (defines all
  preshot events)

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Proof contd

• (3) implies (1)
• S is meaningful gt S is feasible
• if S is meaningful then ? occures in an
  execution of C
• in each execution rcvdpq ? sentpq for each p
  and q
• hence S is feasible

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Chandy-Lamports algorithm

• the taking of a local snaphot must be triggered
  in each process
• no postshot message is received in a preshot
  event
• channels are FIFO

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CHANDY-LAMPORT
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Lemma

• If at least one process initiates the algorithm,
  all processes take a local snapshot within finite
  time.
• Proof Because each process takes a snapshot and
  sends ltmkrgt at most once, the algorithm ceases
  within finite time.
• The connectivity of the network implies that each
  process that has not taken the snapshot yet will
  take it after receiving the ltmkrgt message.
• -------------------
• Time O(D) Messages 2E

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Theorem

• The algorithm of Chandy and Lamport computes a
  meaningful snapshot within finite time after its
  initialization by at least one process.
• Proof Snaphot is feasible
• Let m be a postshot message sent from p to q.
• Before sending m, p took a local snapshot and
  sent a ltmkrgt message to q.
• Because the channel is FIFO, q received ltmkrgt
  before m.
• q took its snapshot upon receipt of this message
  or earlier.
• conseq. gt the receipt of m is a postshot event
  

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Lai Yang algorithm

• the algorithm does not ensure that each process
  eventually records its state
• the algorithm can be augmented with control
  messages (no FIFO)
• every basic message of an algorithm is tagged
  with an information on preshot resp. postshot
  status

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Lai Yang algorithm
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Lai Yang algorithm

• The algorithm of Lai and Yang only computes
  meaningful snapshots.
• Proof Let m ltmess, cgt be a postshot message
  from p to q
• gt c true gt process q records its state before
  receipt of m at the latest
• gt acceptance of m is a postshot event

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Computation of the channel state

• In a feasible snapshot, sentpq \ rcvdpq
• equal the set of messages sent by p in a
  preshot event and received by q in a postshot
  event.
• Proof
• if m is sent in a preshot event m ?
  sentpq
• if m is received in a postshot event
  
• m ? rcvdpq
• gt m ? sentpq \ rcvdpq

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Computation of the channel state

• Process q constructs the state of channel pq by
  recording all preshot-sent messages that are
  received postshot.
• Chandy-Lamport postshot-received messages that
  are received before ltmkrgt message
• Record a local state
• Record all messages received through a channel
  before a marker
• These messages define a state of the channel
• Lai-Yang postshot-received messages tagged with
  false process q has to know the number of
  preshot messages sent to q, so that it can finish
  recording of messages.

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Channel states in Chandy-Lamport algorithm
S11
recording
record empty
S2 0
stop recording
record 1
Two-dollar bank algorithm
the snaphot