cpsc 668 distributed algorithms and systems
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CPSC 668 Distributed Algorithms and Systems. Fall 2009 Prof. Jennifer Welch. Broadcast Specifications. Recall the specification of a broadcast service given in the last set of slides: Inputs : bc-send i ( m ) an input to the broadcast service - PowerPoint PPT PresentationTRANSCRIPT
CPSC 668 Set 15: Broadcast 1
CPSC 668Distributed Algorithms and Systems
Fall 2009
Prof. Jennifer Welch
CPSC 668 Set 15: Broadcast 2
Broadcast Specifications
• Recall the specification of a broadcast service given in the last set of slides:
• Inputs: bc-sendi(m)– an input to the broadcast service– pi wants to use the broadcast service to send m to
all the procs
• Outputs: bc-recvi(m,j)– an output of the broadcast service– broadcast service is delivering msg m, sent by pj,
to pi
CPSC 668 Set 15: Broadcast 3
Broadcast Specifications
• A sequence of inputs and outputs (bc-sends and bc-recvs) is allowable iff there exists a mapping from each bc-recvi(m,j) event to an earlier bc-sendj(m) event s.t. is well-defined: every msg bc-recv'ed was
previously bc-sent (Integrity) restricted to bc-recvi events, for each i, is one-to-
one: no msg is bc-recv'ed more than once at any single proc. (No Duplicates)
restricted to bc-recvi events, for each i, is onto: every msg bc-sent is received at every proc. (Liveness)
CPSC 668 Set 15: Broadcast 4
Ordering Properties
• Sometimes we might want a broadcast service that also provides some kind of guarantee on the order in which messages are delivered.
• We can add additional constraints on the mapping :– single-source FIFO or– totally ordered or– causally ordered
CPSC 668 Set 15: Broadcast 5
Single-Source FIFO Ordering
• For all messages m1 and m2 and all pi and pj, if pi sends m1 before it sends m2, and if pj receives m1 and m2, then pj receives m1 before it receives m2.
• Phrased carefully to avoid requiring that both messages are received.– that is the responsibility of a liveness
property
CPSC 668 Set 15: Broadcast 6
Totally Ordered
• For all messages m1 and m2 and all pi and pj, if both pi and pj receive both messages, then they receive them in the same order.
• Phrased carefully to avoid requiring that both messages are received by both procs.– that is the responsibility of a liveness property
CPSC 668 Set 15: Broadcast 7
Happens Before for Broadcast Messages• Earlier we defined "happens before" relation for
events.• Now extend this definition to broadcast
messages.• Assume all communication is through
broadcast sends and receives.• Msg m1 happens before msg m2 if
– some bc-recv event for m1 happens before the bc-send event for m2, or
– m1 and m2 are bc-sent by the same proc. and m1 is bc-sent before m2 is bc-sent.
CPSC 668 Set 15: Broadcast 8
Example of Happens Before for Broadcast Messagesm1
m2
m3
m4
m1 happens before m3 and m4
m2 happens before m4
m3 happens before m4
CPSC 668 Set 15: Broadcast 9
Causally Ordered
• For all messages m1 and m2 and all pi, if m1 happens before m2, and if pi receives both m1 and m2, then pi receives m1 before it receives m2.
• Phrased carefully to avoid requiring that both messages are received.– that is the responsibility of a liveness
property
CPSC 668 Set 15: Broadcast 10
Example
a
b
single-source FIFO?
totally ordered?
causally ordered?
CPSC 668 Set 15: Broadcast 11
Example
a b
single-source FIFO?
totally ordered?
causally ordered?
CPSC 668 Set 15: Broadcast 12
Example
a
b
single-source FIFO?
totally ordered?
causally ordered?
CPSC 668 Set 15: Broadcast 13
Algorithm BB to Simulate Basic Broadcast on Top of Point-to-Point
• When bc-sendi(m) occurs:
– pi sends a separate copy of m to every processor (including itself) using the underlying point-to-point message passing communication system
• When can pi perform bc-recvi(m)?
– when it receives m from the underlying point-to-point message passing communication system
CPSC 668 Set 15: Broadcast 14
…Alg BB
Basic Broadcast Simulation
BB0
bc-sendi bc-recvi
sendi recvi
asynch pt-to-pt message passing
BBn-1
bc-sendj bc-recvj
sendj recvjbasic broadcast
CPSC 668 Set 15: Broadcast 15
Correctness of Basic Broadcast Algorithm• Assume the underlying point-to-point
message passing system is correct (i.e., conforms to the spec given in previous set of slides).
• Check that the simulated broadcast service satisfies:– Integrity– No Duplicates– Liveness
CPSC 668 Set 15: Broadcast 16
Single-Source FIFO Algorithm• Assume the underlying communication system
is basic broadcast.• when ssf-bc-sendi(m) occurs:
– pi uses the underlying basic broadcast service to bcast m together with a sequence number
– pi increments sequence number by 1 each time it initiates a bcast
• when can pi perform ssf-bc-recvi(m)?– when pi has bc-recv'ed m with sequence number T
and has ssf-bc-recv'ed messages from pj (the ssf-bc-sender of m) with all smaller sequence numbers
CPSC 668 Set 15: Broadcast 17
Single-Source FIFO Algorithm
SSF alg(timestamps)
basic bcastalg (n copies)
point-to-point message passing
user of SSF bcast
ssf-bc-send ssf-bc-recv
bc-send
send
bc-recv
recv
basicbcast
ssfbcast
CPSC 668 Set 15: Broadcast 18
Asymmetric Algorithm for Totally Ordered Broadcast• Assume underlying communication service is
basic broadcast.• There is a distinguished proc. pc
• when to-bcasti(m) occurs:– pi sends m to pc (either assume the basic broadcast
service also has a point-to-point mechanism, or have recipients other than pc ignore the msg)
• when pc receives m from pi from the basic broadcast service:– append a sequence number to m and bc-send it
CPSC 668 Set 15: Broadcast 19
Asymmetric Algorithm for Totally Ordered Broadcast
• when can pi perform to-bc-recv(m)?
– when pi has bc-recv'ed m with sequence number T and has to-bc-recv'ed messages with all smaller sequence numbers
CPSC 668 Set 15: Broadcast 20
Asymmetric Algorithm Discussion
• Simple
• Only requires basic broadcast
• But pc is a bottleneck
• Alternative approach next…
CPSC 668 Set 15: Broadcast 21
Symmetric Algorithm for Totally Ordered Broadcast
• Assume the underlying communication service is single-source FIFO broadcast.
• Each proc. tags each msg it sends with a timestamp (increasing).– Break ties using proc. ids.
• Each proc. keeps a vector of estimates of the other proc's timestamps:– If pi 's estimate for pj is k, then pi will not receive any
later msg from pj with timestamp k.– Estimates are updated based on msgs received and
"timestamp update" msgs
CPSC 668 Set 15: Broadcast 22
Symmetric Algorithm for Totally Ordered Broadcast
• Each proc. keeps its timestamp to be ≥ all its estimates:– when pi has to increase its timestamp because of the
receipt of a message, it sends a timestamp update msg
• A proc. can deliver a msg with timestamp T once every entry in the proc's vector of estimates is at least T.
CPSC 668 Set 15: Broadcast 23
Symmetric Algorithm
when to-bc-sendi(m) occurs:
ts[i]++
add (m,ts[i],i) to pending
invoke ssf-bc-sendi((m,ts[i]))
when ssf-bc-recvi((m,T)) from pj
occurs:
ts[j] := T
add (m,T,j) to pending
if T > ts[i] then
ts[i] := T
invoke ssf-bc-sendi("ts-up",T)
when ssf-bc-recvi("ts-up",T)
from pj occurs:
ts[j] := T
invoke to-bc-recvi(m,j) when:
(m,T,j) is entry in pending with
smallest (T,j)
T ≤ ts[k] for all k
result: remove (m,T,j) from
pending
CPSC 668 Set 15: Broadcast 24
SSF alg(timestamps)
basic bcastalg (n copies)
point-to-point message passing
symmetric TO alg
ssf-bc-send ssf-bc-recv
bc-send
send
bc-recv
recv
basicbcast
user of TO bcast
to-bc-send to-bc-recv
ssfbcast
TObcast
CPSC 668 Set 15: Broadcast 25
Correctness of Symmetric AlgorithmLemma (8.2): Timestamps assigned to msgs
form a total order (break ties with id of sender).
Theorem (8.3): Symmetric algorithm simulates totally ordered broadcast service.
Proof: Must show top-level outputs of symmetric algorithm satisfy 4 properties, in every admissible execution (relies on underlying ssf-bcast service being correct).
CPSC 668 Set 15: Broadcast 26
Correctness of Symmetric Alg.Integrity: follows from same property for ssf-bcast.No Duplicates: follows from same property for ssf-bcast.Liveness: • Suppose in contradiction some pi has some entry (m,T,j)
stuck in its pending set forever, where (T,j) is the smallest timestamp of all stuck entries.
• Eventually (m,T,j) has the smallest timestamp of all entries in pi's pending set.
• Why is (m,T,j) stuck at pi? Because pi's estimate of some pk's timestamp is stuck at some value T' < T.
• But that would mean either pk never receives (m,T,j) or pk's timestamp-update msg resulting from pk receiving (m,T,j) is never received at pi, contradicting correctness of the SSF broadcast.
CPSC 668 Set 15: Broadcast 27
Correctness of Symmetric Alg.Total Ordering: Suppose pi does to-bc-recv for msg m
with timestamp (T,j), and later it does to-bc-recv for msg m' with timestamp (T',j'). Show (T,j) < (T',j').
• By the code, if (m',T',j') is in pi's pending set when pi does the to-bc-recv for m, then (T,j) < (T',j').
• Suppose (m',T',j') is not yet in pi's pending set at that time.
• When pi does the to-bc-recv for m, precondition ensures that T ≤ ts[j']. So pi has received a msg from pj' with timestamp ≥ T.
• By the SSF property, every subsequent msg pi receives from pj' will have timestamp > T, so T' must be > T.
CPSC 668 Set 15: Broadcast 28
Causal Ordering Algorithms
• The symmetric total ordering algorithm ensures causal ordering:– timestamp order extends the happens-
before order on messages.
• Causal ordering can also be attained without the overhead of total ordering, by using an algorithm based on vector clocks…
CPSC 668 Set 15: Broadcast 29
Causal Order Algorithm
when co-bc-sendi(m) occurs:
vt[i]++
invoke co-bc-recvi(m)
invoke bc-sendi((m,vt))
when bc-recvi((m,w)) from pj occurs:
add (m,w,j) to pending
invoke co-bc-recvi(m,j) when:
(m,w,j) is in pending
w[j] = vt[j] + 1
w[k] ≤ vt[k] for all k ≠ j
result:
remove (m,w,j) from pending
vt[j]++
Note: vt[j] records how many msgs from pj have been co-recv'ed
CPSC 668 Set 15: Broadcast 30
Causal Order Algorithm Discussion• Vector clocks are implemented slightly
differently than in the point-to-point case.
• In point-to-point case, we exploited indirect (transitive) information about messages received by other procs.
• In the broadcast case, we don't need to do that, since very proc will eventually receive every message directly.
CPSC 668 Set 15: Broadcast 31
Causal Order Algorithm Example• Algorithm delays the delivery of the C.O.
msgs until causal order property won't be violated.
(0,1,0) (0,2,0) (0,3,0)
(1,3,0)
CPSC 668 Set 15: Broadcast 32
Correctness of Causal Order Algorithm (Sketch)Lemma (8.6): The local array variables vt
serve as vector clocks.Theorem (8.7): The algorithm simulates
causally ordered broadcast, if the underlying communication system satisfies (basic) broadcast.
Proof: Integrity and No Duplicates follow from the same properties of the basic broadcast. Liveness requires some arguing. Causal Ordering follows from the lemma.
CPSC 668 Set 15: Broadcast 33
Reliable Broadcast
• What do we require of a broadcast service when some of the procs can be faulty?
• Specifications differ from those of the corresponding non-fault-tolerant specs in two ways:
1. proc indices are partitioned into "faulty" and "nonfaulty"
2. Liveness property is modified…
CPSC 668 Set 15: Broadcast 34
Reliable Broadcast Specification
• Nonfaulty Liveness: Every msg bc-sent by a nonfaulty proc is eventually bc-recv'ed by all nonfaulty procs.
• Faulty Liveness: Every msg bc-sent by a faulty proc is bc-recv'ed by either all the nonfaulty procs or none of them.
CPSC 668 Set 15: Broadcast 35
Discussion of Reliable Bcast Spec
• Specification is independent of any particular fault model.
• We will only consider implementations for crash faults.
• No guarantee is given concerning which messages are received by faulty procs.
• Can extend this spec to the various ordering variants:– msgs that are received by nonfaulty procs must
conform to the relevant ordering property.
CPSC 668 Set 15: Broadcast 36
Spec of Failure-Prone Point-to-Point Message Passing System• Before we can design an algorithm to
implement reliable (i.e., fault-tolerant) broadcast, we need to know what we can rely on from the lower layer communication system.
• Modify the previous point-to-point spec from the no-fault case in two ways:
1. partition proc indices into "faulty" and "nonfaulty"
2. Liveness property is modified…
CPSC 668 Set 15: Broadcast 37
Spec of Failure-Prone Point-to-Point Message Passing System• Nonfaulty Liveness: every msg sent by a
nonfaulty proc to any nonfaulty proc is eventually received.
Note that this places no constraints on the eventual delivery of messages to faulty procs.
CPSC 668 Set 15: Broadcast 38
Reliable Broadcast Algorithm
when rel-bc-sendi(m) occurs:
invoke sendi(m) to all procs
when recvi(m) from pj occurs:
if m has not already been recv'ed then
invoke sendi(m) to all procs
invoke rel-bc-recvi(m)
CPSC 668 Set 15: Broadcast 39
Correctness of Reliable Bcast Alg
• Integrity: follows from Integrity property of underlying point-to-point msg system.
• No Duplicates: follows from No Duplicates property of underlying point-to-point msg system and the check that this msg was not already received.
• Nonfaulty Liveness: follows from Nonfaulty Liveness property of underlying point-to-point msg system.
• Faulty Liveness: follows from relaying and underlying Nonfaulty Liveness.