130
1 Causality
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1
Causality
2
The “happens before” relation1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
21 happens before (causes)
1 2
3
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
61
A message is sent from to 1 6
4
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
61 and
76 71
transitivity
5
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
81
810
161
92
6
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
“happens before” is a partial order
101
110
Parallel events: 101
101
7
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
“happens before” is a partial order
813
138
Parallel events: 138
138
8
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
We can move parallel events
The happens before relationdoesn’t change
9
We want to find a mechanism thatcaptures the “happens before” relationship
so that we use causality in various computation problems
10
Logical Clocks
1 2 3 4 50p
1p
1 2 3
1 2
1)( 1 LT 21)()( 12 LTLT
3
4 5
4
6 7 8 9
In each process, the logical clockincreases by 1 with each local event
0
0
11
1 2 30p
1p
1 2
1 2
514
1)4,2max(
1))(),(max()( 723
LTLTLT
3
5
4
4 5 6 7
Logical clocks are piggy-packed on messages
40
0
12
Logical clocks, seem that they capture the happens before relation
ji )()( ji LTLT
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
1 2 3 7 8
2 4 5 6
1 2 3 4 5 6 7
13
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
1 2 3 7 8
2 4 5 6
1 2 3 4 5 6 7
71 )(41)( 71 LTLT
Example:
14
However, logical clocks cannot capture parallelism
)(54)( 813 LTLT
Parallel events
138
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
2p
1 2 3 7 8
2 4 5 6
1 2 3 4 5 6 7
???)( 813
15
We need another mechanism that can capture the parallelism of events
16
Vector Clocks
1 2 3 4 50p
1p6 7 8 9
0
0
0
0
0
1
0
2
0
3
0
4
0
5
1
0
2
0
3
0
4
0
Process entry
Process entry
0p
1p
0
1)( 1VC
17
1 2 3 4 50p
1p6 7 8 9
0
0
0
0
0
1
0
2
0
3
0
4
0
5
1
0
2
0
3
0
4
0
Each process increases its entry at each event
0
3
0
1
0
2
0
1)()( 23 VCVC
increment
18
1 2 3 4 50p
1p6 7 8 9
0
0
0
0
0
1
0
2
0
3
0
4
0
5
1
0
2
0
3
0
4
0
Each process increases its entry at each event
4
0
1
0
3
0
1
0)()( 89 VCVC
increment
19
1 2 30p
1p4 5 6 7
vector clocks are piggy-packed on messages
0
0
0
0
0
1
0
2
1
0
2
0
3
0
4
0
4
0
4
3
4
2
4
0,
0
2max
The maximum of each entry
20
1 2 30p
1p4 5 6 7
vector clocks are piggy-packed on messages
0
0
0
0
0
1
0
2
1
0
2
0
3
0
4
0
4
0
4
3
4
3
0
1
4
2
0
1
4
0,
0
2max
0
1))(),(max()( 723 VCVCVC
21
30p
1p7
vector clocks are piggy-packed on messages
4
0
4
0
4
3
6
4
1
0
5
4
1
0
4
4,
5
0max
1
0))(),(max()( 1089 VCVCVC
4
4
5
0
6
4
4
4
8 9
10
22
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
3p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
23
Comparison of vector clocks
3
2
1
3
2
1
b
b
b
a
a
a
if for all
We write
ii ba i
24
5
7
9
4
7
3
Examples:
1
1
6
0
0
1
3
5
1
3
5
1
25
Incomparable vector clocks
3
2
1
3
2
1
b
b
b
a
a
aWe write
3
2
1
3
2
1
b
b
b
a
a
a
3
2
1
3
2
1
b
b
b
a
a
aIf neither nor
26
5
8
2
4
7
3
Examples:
1
1
0
0
0
1
27
Vector clocks capture causality
If then 21 )()( 21 VCVC
If then 21 )()( 21 VCVC
28
If then 21 )()( 21 VCVC
If then 21 )()( 21 VCVC
By examining the vector clockswe can determine the order of events
29
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
3p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 )()( 21 VCVC
30
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
3p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
132 )()( 132 VCVC
31
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
3p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
101 )()( 101 VCVC
32
1 2 3
6 7 8 9
10 11 12 13 14 15 16
4 50p
1p
3p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
148 )()( 148 VCVC
33
Cuts
34
0p
1p
2p
21 3 4 5
1 2 3 4
21 3 4 5 6 7
35
0p
1p
2p
consists from an event from each process
Cut:
3
2
1
21 3 4 5
1 2 3 4
21 3 4 5 6 7
36
0p
1p
2p
no messages cross the cutConsistent cut:
21 3 4 5
1 2 3 4
21 3 4 5 6 7
3
2
1
37
0p
1p
2p
messages can crossfrom left to right of the cut
Consistent cut:
21 3 4 5
1 2 3 4
21 3 4 5 6 7
5
3
2
38
0p
1p
messages cross from right to left of the cut
Inconsistent cut:
21 3 4 5
1 2 3 4
21 3 4 5 6 7
4
2
1
2p
39
0p
1p
21 3 4 5
1 2 3 4
21 3 4 5 6 7
5
3
4
messages cross from right to left of the cut
Inconsistent cut:
2p
40
A consistent cut such that
Maximal Consistent Cut
Maximal Consistent Cut of :
Consider some (inconsistent) cut r
r
s rs
and contains most recent events s
41
0p
1p
(inconsistent cut)
21 3 4 5
1 2 3 4
21 3 4 5 6 7
4
2
1
r
2p
42
0p
1p
maximal consistent cuts
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
(inconsistent cut)
4
2
1
3
2
1
2p
43
Theorem: For every cut ,there is a unique maximal consistent cut
Proof: Proof by contradiction
r
44
Assume for contradiction there are two(or more) maximal cuts of r
1s 2s
maximal consistent cuts
r
45
1s 2snot maximal!
It cannot be that and don’t cross 1s 2s
r
46
r1s 2s
and cross 1s 2s
47
r1s 2s
and cross 1s 2s
snew cut
48
r1s 2s
and cross 1s 2s
s
impossible
is consistent1ssince
49
r1s 2s
and cross 1s 2s
s
impossible
is consistent2ssince
50
r1s 2s
and cross 1s 2s
s is consistentnot maximal! not maximal!
Contradiction!
End of proof
51
A distributed algorithm for computing the maximum consistent cut of :
•Use vector clocks
•For each processor: Find most recent event with vector clock
r
rv
52
0p
1p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 72p
53
0p
1p
2p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
6
2
2
54
0p
1p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
6
2
2
6
2
2
0
0
2
OK
:0p
2p
55
0p
1p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
6
2
2
6
2
2
3
2
1
OK
:1p
OK
2p
56
0p
1p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
6
2
2
6
2
2
6
3
2
OK
:2p
OK
2p
57
0p
1p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
6
2
2
6
2
2
5
0
2
OK
:2p
OK
2pOK
58
0p
1p
0
0
1
0
0
2
0
0
3
5
4
4
5
4
5
0
1
1
3
2
1
3
3
1
5
4
2
1
0
0
2
0
0
3
0
0
4
0
2
5
0
2
6
3
2
7
3
2
21 3 4 5
1 2 3 4
21 3 4 5 6 7
r
6
2
2
2p
s
5
2
2
59
Distributed Snapshot
•A set of processors initiate the computation for obtaining a global snapshot
S
•The cut contains the state of at least one processor in in the initiation S
(these processors receive special marker messages from the system)
60
A Distributed Snapshot Algorithm
•Upon receiving a marker message:
Processor : 0inum nilansi ip
•Count local events in inum
If thennilansi set nilansi
send marker to all neighbors
61
0p
1p
2p
21 3 4 5
1 2 3 4
21 3 4 5 6 7
inum
62
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
63
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
m
m
64
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
m
m
65
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
m
m mm
m
mm
66
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
m
m mm
m
mm
67
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
m
m mm
m
mm
m
68
0p
2p
1p
3p
}{ 0pS
0p
1p
2p
3p
m
m
m mm
m
mm
m
cut
69
Theorem:The cut obtained by the algorithmis consistent
Proof: By contradiction
70
Suppose the cut is incosistent
71
It must be:
m
Impossible!
End of proof
(we assume FIFO)
Spring 2003 Costas Busch 72
FIFO-Broadcast
Spring 2003 Costas Busch 73
0p
1p
2p
1m 2mFIFO Order
Messages from are received in the order they are sent from
ip
ip
3p
Spring 2003 Costas Busch 74
Not FIFO-order
0p
1p
2p
1m 2m
3p
Spring 2003 Costas Busch 75
0p
1p
2p
1m 2m
3p
3m 4m
Messages from differentprocessors may be received indifferent order
FIFO order
Spring 2003 Costas Busch 76
Not FIFO-order
0p
1p
2p
1m 2m
3p
3m 4m
Spring 2003 Costas Busch 77
Why is FIFO important?
Deposit$1000
ATM
BankAccount
Withdraw$500
$0 $1000 $500
FIFO order
Spring 2003 Costas Busch 78
Why is FIFO important?
Deposit$1000
ATM
BankAccount
Withdraw$500
$0 $500-$500
Not FIFO order Not an allowedtransaction
Spring 2003 Costas Busch 79
0p
1p
2p
3p
2
A processor increments a counterwhen it broadcasts
1
11 22
1 2
A simple FIFO algorithm
1m 2m1 2
Spring 2003 Costas Busch 80
A processor keeps track of the highest valuesreceived from other processors
00
pV 10
pV 20
pV
0p
1p
2p
3p
21
11 22
1 21m 2m1 2
Spring 2003 Costas Busch 81
00
pV
Wait until all messages with smaller values from the same sender are received
wait ok
0p
1p
2p
3p
21
1 1 22
1 21m 2m1 2
Spring 2003 Costas Busch 82
00
pV
receive
10
pV 20
pV
0p
1p
2p
3p
21
1 1 22
1 21m 2m1 2
2
Spring 2003 Costas Busch 83
Totally Ordered - Broadcast
Spring 2003 Costas Busch 84
0p
1p
2p
1m
2m
Total Order
Messages are received in the same order in every processor
3p
Spring 2003 Costas Busch 85
0p
1p
2p
1m
2m
3p
Not Total Order
Spring 2003 Costas Busch 86
0p
1p
2p
1m
2m
3p
Total Order
Spring 2003 Costas Busch 87
0p
1p
2p
1m
2m
3p
Not Total Order
Spring 2003 Costas Busch 88
0p
1p
2p
3p
An Asymmetric Algorithm
1
Request a counter value
counter
Spring 2003 Costas Busch 89
0p
1p
2p
3p
1
1
counter
Receive a counter value
Spring 2003 Costas Busch 90
0p
1p
2p
3p
1
1
counter
Broadcast
1
1
1
1
Spring 2003 Costas Busch 91
0p
1p
2p
3p
1
1
counter
1
1
1
1
22 2
2
2
2
Spring 2003 Costas Busch 92
0p
1p
2p
3p
2
With the counter,broadcast messages are totally ordered
1
wait ok
2
1
Spring 2003 Costas Busch 93
0p
1p
2p
3p
2
With the counter,broadcast messages are totally ordered
1
wait ok
received
2
2
1
2
Spring 2003 Costas Busch 94
A Symmetric Algorithm
Builds on top of FIFO
It is a form of Vector Clocks
Spring 2003 Costas Busch 95
Every process has a counter
On broadcast: the process sends the counter value to every boby
Spring 2003 Costas Busch 96
On reception of a message:
If received counter value is higher than local value, then:
•Update local counter to new value
•Broadcast new counter value to everybody else
Spring 2003 Costas Busch 97
2p
1p
0p
0
0
0
0
0
0
0
0
0
Spring 2003 Costas Busch 98
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
Spring 2003 Costas Busch 99
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
1
0
0
1
Spring 2003 Costas Busch 100
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
1
0
1
1
0
0
1
Special broadcast message
Spring 2003 Costas Busch 101
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
Spring 2003 Costas Busch 102
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
Actual reception of normal broadcast message
Spring 2003 Costas Busch 103
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1 2
2
2
Another normal broadcast
Spring 2003 Costas Busch 104
2
1
0
a
a
a
Local vector of :
receive a normal broadcast message if message’s counter is smaller or equal than each ja
ip
ip
Spring 2003 Costas Busch 105
2p
1p
0p
0
0
0
0
0
0
0
0
0
Spring 2003 Costas Busch 106
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1 1
Spring 2003 Costas Busch 107
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1 1
1
1
1
1
0
1
1
0
1
Spring 2003 Costas Busch 108
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1 1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
Spring 2003 Costas Busch 109
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1 1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
Spring 2003 Costas Busch 110
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
111
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
Symmetry is broken with node ids
Spring 2003 Costas Busch 111
Casually Ordered - Broadcast
Spring 2003 Costas Busch 112
0p
1p
2p
1m
2m
Casual Order
Messages are received in the order they are caused
3p
Spring 2003 Costas Busch 113
0p
1p
2p
1m
2m
3p
Not causal order
Spring 2003 Costas Busch 114
0p
1p
2p1m
2m
3p
Casual order
Spring 2003 Costas Busch 115
0p
1p
2p1m
2m
3p
Not casual order
Spring 2003 Costas Busch 116
The Total Order algorithmswe described are also Casual Order algorithms
Observation:
There is a more efficient algorithmin terms of message complexity
Spring 2003 Costas Busch 117
2p
1p
0p
0
0
0
0
0
0
0
0
0
Spring 2003 Costas Busch 118
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
Spring 2003 Costas Busch 119
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
2
0
0
2
0
0
2
0
Spring 2003 Costas Busch 120
2p
1p
0p
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
2
0
0
2
0
0
2
0
0
2
0
Spring 2003 Costas Busch 121
2p
1p
0p
0
2
0
0
2
0
0
2
0
Spring 2003 Costas Busch 122
2p
1p
0p
0
2
0
0
2
0
0
2
0
0
3
0
0
3
0
0
3
0
0
3
0
Spring 2003 Costas Busch 123
2p
1p
0p
0
2
0
0
2
0
0
2
0
0
3
0
0
3
0
0
3
0
0
3
0
0
3
1
0
3
1
0
3
1
Spring 2003 Costas Busch 124
2p
1p
0p
0
2
0
0
2
0
0
2
0
0
3
0
0
3
0
0
3
0
0
3
0
0
3
1
0
3
1
0
3
1
0
3
1
Spring 2003 Costas Busch 125
Processor officially receives a message from with vector clock
ip
2
1
0
v
v
v
1) All messages from smaller or equal to arrive
2) All messages from smaller than arrive
kv
jp
kp )( jk
jp
jv
When:
Spring 2003 Costas Busch 126
Relationship Between Broadcast Orders
Spring 2003 Costas Busch 127
Casual Order
Casual Order does not imply Total Order
a
b
a b
b a
Spring 2003 Costas Busch 128
Casual Order
Casual Order implies FIFO
a b a b
ba
Spring 2003 Costas Busch 129
Total Order
Total Order does not imply Casual Order nor FIFO
a b ab
b a
Spring 2003 Costas Busch 130
FIFO
FIFO does not imply Casual Order nor Total Order
a ab
ba b
