Post on 03-Jan-2016
transcript
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Logical Clocks
Paul Krzyzanowskipxk@cs.rutgers.edu
ds@pk.org
Distributed Systems
Except as otherwise noted, the content of this presentation is licensed under the Creative Commons Attribution 2.5 License.
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Logical clocks
Assign sequence numbers to messages– All cooperating processes can agree on order
of events– vs. physical clocks: time of day
Assume no central time source– Each system maintains its own local clock– No total ordering of events
• No concept of happened-when
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Happened-before
Lamport’s “happened-before” notation
a b event a happened before event be.g.: a: message being sent, b: message
receipt
Transitive:if a b and b c then a c
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Logical clocks & concurrency
Assign “clock” value to each event– if ab then clock(a) < clock(b)– since time cannot run backwards
If a and b occur on different processes that do not exchange messages, then neither a b nor b a are true
– These events are concurrent
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Event counting example
• Three systems: P0, P1, P2
• Events a, b, c, …
• Local event counter on each system
• Systems occasionally communicate
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Event counting example
a b
i
kj
P1
P2
P3
1 2
1 3
21
d f
g3
c
2
4 6
Bad ordering:
e h
f k
h
e5
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Lamport’s algorithm
• Each message carries a timestamp of the sender’s clock
• When a message arrives:– if receiver’s clock < message timestamp
set system clock to (message timestamp + 1)
– else do nothing
• Clock must be advanced between any two events in the same process
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Lamport’s algorithm
Algorithm allows us to maintain time ordering among related events
– Partial ordering
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Summary
• Algorithm needs monotonically increasing software counter
• Incremented at least when events that need to be timestamped occur
• Each event has a Lamport timestamp attached to it
• For any two events, where a b:L(a) < L(b)
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Problem: Identical timestamps
ab, bc, …: local events sequenced
ic, fd , dg, … : Lamport imposes asendreceive relationship
Concurrent events (e.g., a & i) may have the same timestamp … or not
a b
h i
kj
P1
P2
P3
1 2
1 7
71
d f
g3
c
6
4 6
e5
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Unique timestamps (total ordering)
We can force each timestamp to be unique– Define global logical timestamp (Ti, i)
• Ti represents local Lamport timestamp
• i represents process number (globally unique)– E.g. (host address, process ID)
– Compare timestamps:(Ti, i) < (Tj, j)
if and only ifTi < Tj or
Ti = Tj and i < j
Does not relate to event ordering
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Unique (totally ordered) timestamps
a b
i
kj
P1
P2
P3
1.1 2.1
1.2 7.2
7.31.3
d f
g3.1
c
6.2
4.1 6.1h
e5.1
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Problem: Detecting causal relations
If L(e) < L(e’)– Cannot conclude that ee’
Looking at Lamport timestamps– Cannot conclude which events are causally
related
Solution: use a vector clock
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Vector clocksRules:
1. Vector initialized to 0 at each processVi [j] = 0 for i, j =1, …, N
2. Process increments its element of the vector in local vector before timestamping event: Vi [i] = Vi [i] +1
3. Message is sent from process Pi with Vi attached to it
4. When Pj receives message, compares vectors element by element and sets local vector to higher of two values
Vj [i] = max(Vi [i], Vj [i]) for i=1, …, N
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Comparing vector timestamps
DefineV = V’ iff V [i ] = V’[i ] for i = 1 … N V V’ iff V [i ] V’[i ] for i = 1 … N
For any two events e, e’if e e’ then V(e) < V(e’)
• Just like Lamport’s algorithm
if V(e) < V(e’) then e e’
Two events are concurrent if neitherV(e) V(e’) nor V(e’) V(e)
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0)
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0)
(2,0,0)
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0)
(2,0,0)
(2,1,0)
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0)
(2,0,0)
(2,1,0) (2,2,0)
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1)
(2,0,0)
(2,1,0) (2,2,0)
(0,0,1)
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) f (2,2,2)
(2,0,0)
(2,1,0) (2,2,0)
(0,0,1) (2,2,2)
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(0,0,1)
(1,0,0)
Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) f (2,2,2)
(2,0,0)
(2,1,0) (2,2,0)
(2,2,2)
concurrentevents
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) f (2,2,2)
(2,0,0)
(2,1,0) (2,2,0)
(0,0,1) (2,2,2)
concurrentevents
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) f (2,2,2)
(2,0,0)
(2,1,0) (2,2,0)
(0,0,1) (2,2,2)
concurrentevents
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Vector timestamps
a b
c d
fe
(0,0,0)P1
P2
P3
(0,0,0)
(0,0,0)
(1,0,0)
Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) f (2,2,2)
(2,0,0)
(2,1,0) (2,2,0)
(0,0,1) (2,2,2)
concurrentevents
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Summary: Logical Clocks & Partial Ordering
• Causality– If a->b then event a can affect event b
• Concurrency– If neither a->b nor b->a then one event cannot
affect the other
• Partial Ordering– Causal events are sequenced
• Total Ordering– All events are sequenced