1 Causality

2 The “happens before” relation happens before (causes)

3 A message is sent from to

4 and transitivity

5

6 “happens before” is a partial order Parallel events:

7 “happens before” is a partial order Parallel events:

8 We can move parallel events The happens before relation doesn’t change

9 We want to find a mechanism that captures the “happens before” relationship so that we use causality in various computation problems

10 Logical Clocks In each process, the logical clock increases by 1 with each local event 0 0

Logical clocks are piggy-packed on messages 4 0 0

12 Logical clocks, seem that they capture the happens before relation

Example:

14 However, logical clocks cannot capture parallelism Parallel events

15 We need another mechanism that can capture the parallelism of events

16 Vector Clocks Process entry

17 Each process increases its entry at each event increment

18 Each process increases its entry at each event increment

19 vector clocks are piggy-packed on messages The maximum of each entry

20 vector clocks are piggy-packed on messages

21 vector clocks are piggy-packed on messages

22

23 Comparison of vector clocks if for all We write

24 Examples:

25 Incomparable vector clocks We write If neither nor

26 Examples:

27 Vector clocks capture causality If then

28 If then By examining the vector clocks we can determine the order of events

29

30

31

32

33 Cuts

35 consists from an event from each process Cut:

36 no messages cross the cut Consistent cut:

37 messages can cross from left to right of the cut Consistent cut:

38 messages cross from right to left of the cut Inconsistent cut:

messages cross from right to left of the cut Inconsistent cut:

40 A consistent cut such that Maximal Consistent Cut Maximal Consistent Cut of : Consider some (inconsistent) cut and contains most recent events

41 (inconsistent cut)

42 maximal consistent cut (inconsistent cut)

43 Theorem: For every cut, there is a unique maximal consistent cut Proof:Proof by contradiction

44 Assume for contradiction there are two (or more) maximal cuts of maximal consistent cuts

45 not maximal! It cannot be that and don’t cross

46 and cross

47 and cross new cut

48 and cross impossible is consistentsince

49 and cross impossible is consistentsince

50 and cross is consistent 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

OK

OK

OK

OK

59 Distributed Snapshot A set of processors initiate the computation for obtaining a global snapshot The cut contains the state of at least one processor in in the initiation (these processors receive special marker messages from the system)

60 A Distributed Snapshot Algorithm Upon receiving a marker message: Processor : Count local events in If then set send marker to all neighbors

62 m

63 m m m

64 m m m

65 m m m m m m m m

66 m m m m m m m m

67 m m m m m m m m m

68 m m m m m m m m m cut

69 Theorem:The cut obtained by the algorithm is consistent Proof: By contradiction

70 Suppose the cut is incosistent

71 It must be: m Impossible! End of proof (we assume FIFO)

Spring 2003Costas Busch72 FIFO-Broadcast

Spring 2003Costas Busch73 FIFO Order Messages from are received in the order they are sent from

Spring 2003Costas Busch74 Not FIFO-order

Spring 2003Costas Busch75 Messages from different processors may be received in different order FIFO order

Spring 2003Costas Busch76 Not FIFO-order

Spring 2003Costas Busch77 Why is FIFO important? Deposit $1000 ATM Bank Account Withdraw $500 $0 $1000$500 FIFO order

Spring 2003Costas Busch78 Why is FIFO important? Deposit $1000 ATM Bank Account Withdraw $500 $0 $500-$500 Not FIFO order Not an allowed transaction

Spring 2003Costas Busch79 2 A processor increments a counter when it broadcasts A simple FIFO algorithm 12

Spring 2003Costas Busch80 A processor keeps track of the highest values received from other processors

Spring 2003Costas Busch81 Wait until all messages with smaller values from the same sender are received waitok

Spring 2003Costas Busch82 receive

Spring 2003Costas Busch83 Totally Ordered - Broadcast

Spring 2003Costas Busch84 Total Order Messages are received in the same order in every processor

Spring 2003Costas Busch85 Not Total Order

Spring 2003Costas Busch86 Total Order

Spring 2003Costas Busch87 Not Total Order

Spring 2003Costas Busch88 An Asymmetric Algorithm 1 Request a counter value counter

Spring 2003Costas Busch counter Receive a counter value

Spring 2003Costas Busch counter Broadcast

Spring 2003Costas Busch counter

Spring 2003Costas Busch92 2 With the counter, broadcast messages are totally ordered 1 wait ok 2 1

Spring 2003Costas Busch93 2 With the counter, broadcast messages are totally ordered 1 wait ok received

Spring 2003Costas Busch94 A Symmetric Algorithm Builds on top of FIFO It is a form of Vector Clocks

Spring 2003Costas Busch95 Every process has a counter On broadcast: the process sends the counter value to every boby

Spring 2003Costas Busch96 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 2003Costas Busch97

Spring 2003Costas Busch

Spring 2003Costas Busch

Spring 2003Costas Busch Special broadcast message

Spring 2003Costas Busch

Spring 2003Costas Busch Actual reception of normal broadcast message

Spring 2003Costas Busch Another normal broadcast

Spring 2003Costas Busch104 Local vector of : receive a normal broadcast message if message’s counter is smaller or equal than each

Spring 2003Costas Busch105

Spring 2003Costas Busch

Spring 2003Costas Busch

Spring 2003Costas Busch

Spring 2003Costas Busch

Spring 2003Costas Busch Symmetry is broken with node ids

Spring 2003Costas Busch111 Casually Ordered - Broadcast

Spring 2003Costas Busch112 Casual Order Messages are received in the order they are caused

Spring 2003Costas Busch113 Not causal order

Spring 2003Costas Busch114 Casual order

Spring 2003Costas Busch115 Not casual order

Spring 2003Costas Busch116 The Total Order algorithms we described are also Casual Order algorithms Observation: There is a more efficient algorithm in terms of message complexity

Spring 2003Costas Busch117

Spring 2003Costas Busch118

Spring 2003Costas Busch119

Spring 2003Costas Busch120

Spring 2003Costas Busch121

Spring 2003Costas Busch122

Spring 2003Costas Busch123

Spring 2003Costas Busch124

Spring 2003Costas Busch125 Processor officially receives a message from with vector clock 1) All messages from smaller or equal to arrive 2) All messages from smaller than arrive When:

Spring 2003Costas Busch126 Relationship Between Broadcast Orders

Spring 2003Costas Busch127 Casual Order Casual Order does not imply Total Order a b ab ba

Spring 2003Costas Busch128 Casual Order Casual Order implies FIFO ab ab b a

Spring 2003Costas Busch129 Total Order Total Order does not imply Casual Order nor FIFO ab ab ba

Spring 2003Costas Busch130 FIFO FIFO does not imply Casual Order nor Total Order aab b a b