Logical Time and Logical Clocks

1
Logical Time and Logical Clocks

• Béat Hirsbrunner

References G. Coulouris, J. Dollimore and T.
Kindberg "Distributed Systems Concepts and
Design", Ed. 4, Addison-Wesley 2005, Chap.
11.4 Michel Raynal, Mukesh Singhal "Capturing
Causality in Distributed Systems", IEEE Computer,
Febr. 1996, pp. 49-56
Distributed SystemsBéat Hirsbrunner (UniFr),
Peter Kropf (UniNe) and Pierre Kuonen
(EiaFr) Summer Semester 2007, Lecture 2b, 30
March 2007
2
Causality

• In relativity, if L causes R, then the order of L
  and R must be the same for all observers
• Woudn't this be a nice property for a distributed
  computer system?

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Logical Time Basic Observation

1. If two events occurred at the same process pi (i
   1, 2, N) then they occurred in the order
   observed by pi, that is in the order ????
2. When a message m is sent between two processes,
   the event of sending the message occured before
   the event of receiving the message send(m) ??
   receive(m)

Fig. 11.5. Events occuring at three processes
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Lamport's Happened-Before Relation ?

• HB1 for any pair of events e and e, if there is
  a process pi such that e ?i e, then e ? e
• HB2 for any pair of events e and e and for any
  message m, if e send(m) and e receive(m),
  then e ? e
• HB3 if e, e and e are events and if e ? e
  and e ? e, then e ? e (transitivity)

• Remarks.
• HB defines only a partial order, i.e. not all
  events are related by the relation ?.
• Concurrency if not (e ? e) and not (e' ? e),
  events e and e' are concurrent, and are denoted
  as e e'.
• HB captures only potential causality, i.e. two
  events can be related by ? even though there is
  no real connection between them.

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Lamport's Logical Clock

• In a system of logical clock, every process has a
  logical clock that is advanced using a set of
  rules.
• Lamport's logical clocks capture the
  happen-before relation each process pi keeps
  its own clock Li.
• Lamport's timestamp of event e at pi is denoted
  by Li(e).

The rules to update Li are LC1 Li is
incremented before each event at pi Li Li
1. LC2a When a process pi sends a message m, it
piggybacks on m the value t Li. LC2b On
receiving (m,t), a process pj computes Lj
max(Lj, t) and then applies LC1 before
timestamping the event receive(m).
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Example
Fig. 11.6

• Remark 1
• Note that e ? e gt L(e) lt L(e').
• Unfortunately, the inverse is not true!

• Remark 2
• L generates only a partial order, i.e. some
  distincts events have numerically identical
  Lamport timestamps.
• A total order can be defined as follows
• we timestamps an event e occuring at pi with
  local timestamp Ti by (Ti, i)
• and define (Ti,i) lt (Tj,j) if and only if Ti lt Tj
  or Ti Tj and i lt j.

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Vector Clocks

• Vector clocks overcome the shortcoming of Lamport
  logical clocksL(e) lt L(e) does not imply e
  happened before e.

• As for Lamport's clock, each process keeps its
  own vector clock Vi
• Vii is the number of events that pi has
  timestamped,
• Vij, i?j, is the number of events that have
  occured at pj that pi has potentially been
  affected to.

The rules to update the vector Vi of N intergers
are VC1 Initially Vij 0 for i, j 1, 2,
N. VC2 Just before pi timestamps an event, it
sets Vii Vii 1. VC3 pi piggybacks t
Vi on every message it sends. VC4 when pi
receives (m,t) it sets Vij max(Vij , tj)
for all j, and then applies VC2 before
timestamping the event receive(m).
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Example
Fig. 11.7

• Vector comparison definition
• V V iff Vj Vj for all j
• V ? V iff Vj ? Vj for all j
• V lt V iff V ? V and V ? V

Example V(b) lt V(d), i.e. b ? d V(e) is
unorded to V(d), i.e. e d
A last remark matrix clocks have been defined,
whereby processes keep estimates of other
processes' times as well as their own, cf Raynal
96.
Theorem e ? e ltgt V(e) lt V(e')