CPSC 668 Distributed Algorithms and Systems

CPSC 668 Set 15: Broadcast 1

CPSC 668Distributed 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-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


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)


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


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


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


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.


Example of Happens Before for Broadcast Messagesm1

m2

m3

m4

m1 happens before m3 and m4

m2 happens before m4

m3 happens before m4


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


Example

a

b

single-source FIFO?

totally ordered?

causally ordered?


Example

a b

single-source FIFO?

totally ordered?

causally ordered?


Example

a

b

single-source FIFO?

totally ordered?

causally ordered?


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


…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


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


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


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


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


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


Asymmetric Algorithm Discussion

• Simple

• Only requires basic broadcast

• But pc is a bottleneck

• Alternative approach next…


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


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.


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


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


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).


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.


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.


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…


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


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.


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)


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.


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…


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.


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.


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…


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.


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)


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.

CPSC 668 Distributed Algorithms and Systems

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