Chapter 10 Global Properties

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Chapter 10Global Properties
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Unstable Predicate Detection

• A predicate is stable if, once it becomes true it
  remains true
• Snapshot algorithm is not useful for detection of
  global properties
• Not applicable for unstable predicates
• Can not compute the least global state that
  satisfies a given predicate
• Excessive overhead if frequency of snapshots is
  high

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Predicates

• Any predicate B constructed from local predicates
  using boolean connectives can be written in a
  disjunctive normal form i.e.
• where q1, q2, ,qn are conjunctive predicates
• E.g. x y (where x and y are boolean) can be
  written as

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Weak Conjunctive Predicate (WCP)

• A Weak Conjunctive Predicate (WCP) is true iff
  there exists a consistent global cut in which all
  the conjuncts are true
• Disjunctive predicates are easy to detect
• Given an algorithm for detecting WCP we can
  detect any predicate B constructed from local
  predicates using boolean connectives

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WCP Algorithm outline

• Non-Checker process
• Maintains a vector clock
• Sends vector clock to checker process when
  predicate becomes true

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WCP Algorithm outline

• Checker process
• Maintains a separate queue for each non-checker
  process
• Maintain a cut1..N (array of states of the
  processes)
• If state cuti ! cutj, then cuti
  queuei.getNext()
• Repeat above statement till all states in cut
  are concurrent
• cut is the least CGS for which the predicate
  holds

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WCP Detection - Checker Process
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Overhead Analysis

• n number of processes involved
• m max number of messages sent or received by any
  process
• Space
• Each local snapshot O(n)
• At most O(mn) local snapshots
• O(n2m) total space
• Time n2m comparisons
• O(n2m)

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Is the time complexity optimal ?

• Lemma
• Let there be n elements in a set S . Any
  algorithm that determines whether all elements
  are incomparable must make at least n(n-1)/2
  comparisons.

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Is the time complexity optimal ?

• Theorem
• Let S be any partially ordered finite set of size
  mn. We are given a decomposition of S into n
  sets P0 Pn-1 such that Pi is a chain of size m
  . Any algorithm that determines whether there
  exists an anti-chain of size n must make at least
  mn(n-1)/2 comparisons

Adversary algorithm
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A Token based algorithm for WCP

• Monitor process runs on each node along with the
  application
• Monitor processes pass the token to each other
• Token stores candidate cut and information to
  determine if it is consistent

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A Token based algorithm for WCP

• A token is sent to a process Pi when current
  state from Pi happened before some other state in
  the candidate cut
• Once the monitor process for Pi has eliminated
  the current state
• receive a new state from the application process
• check for consistency conditions again.
• This process is repeated until
• all states are eliminated from some process Pi or
  
• the WCP is detected.

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A Token based algorithm for WCP

• Token consists of two vectors G and color
• G represents the candidate global cut
• Gi k indicates that state (i,k) is part of
  the current cut
• Invariant Gi k implies that any global cut C
  with state (i,s) 2 C and s lt k cannot satisfy the
  WCP
• color indicates which states have been eliminated
• If colorired, then state (i,Gi) has been
  eliminated and can never satisfy the global
  predicate

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A Token based algorithm for WCP
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Applications

• Distributed debugging
• Detect a bad condition
• E.g. There is no leader ,i.e.,
• P1 does not have a token and
• P2 does not have a token and
• . . .
• Pn does not have a token