Set 9: Fault Tolerant Consensus
DISTRIBUTED ALGORITHMS
Spring 2014
Prof. Jennifer Welch
1
Processor Failures in Message Passing
2

Crash: at some point the processor stops taking
steps
 at
the processor's final step, it might succeed in sending
only a subset of the messages it is supposed to send

Byzantine: processor changes state arbitrarily and
sends messages with arbitrary content
Set 9: Fault Tolerant Consensus
Consensus Problem
3


Every processor has an input.
Termination: Eventually every nonfaulty processor
must decide on a value.
 decision


is irrevocable!
Agreement: All decisions by nonfaulty processors
must be the same.
Validity: If all inputs are the same, then the decision
of a nonfaulty processor must equal the common
input.
Set 9: Fault Tolerant Consensus
Examples of Consensus
4

Binary inputs:

input vector 1,1,1,1,1


input vector 0,0,0,0,0


decision must be 0
input vector 1,0,0,1,0


decision must be 1
decision can be either 0 or 1
Multi-valued inputs:

input vector 1,2,3,2,1

decision can be 1 or 2 or 3
Set 9: Fault Tolerant Consensus
Overview of Consensus Results
5



Synchronous system
At most f faulty processors
Tight bounds for message passing:
crash failures
Byzantine failures
number of
rounds
f+1
f+1
total number of
processors
f+1
3f + 1
polynomial
polynomial
message size
Set 9: Fault Tolerant Consensus
Overview of Consensus Results
6



Impossible in asynchronous case.
Even if we only want to tolerate a single crash
failure.
True both for message passing and shared readwrite memory.
Set 9: Fault Tolerant Consensus
Modeling Crash Failures
7

Modify failure-free definitions of admissible
execution to accommodate crash failures:
 All but a set of at most f processors (the
faulty ones) taken an infinite number of
steps.
 In
synchronous case: once a faulty processor fails
to take a step in a round, it takes no more steps.

In a faulty processor's last step, an arbitrary
subset of the processor's outgoing messages
make it into the channels.
Set 9: Fault Tolerant Consensus
Modeling Byzantine Failures
8


Modify failure-free definitions of admissible
execution to accommodate Byzantine failures:
A set of at most f processors (the faulty ones) can
send messages with arbitrary content and change
state arbitrarily (i.e., not according to their
transition functions).
Set 9: Fault Tolerant Consensus
Consensus Algorithm for Crash Failures
9
Code for each processor:
v := my input
at each round 1 through f+1:
if I have not yet sent v then send v to all
wait to receive messages for this round
v := minimum among all received values and
current value of v
if this is round f+1 then decide on v
Set 9: Fault Tolerant Consensus
Execution of Algorithm
10

round 1:







send my input
receive round 1 msgs
compute value for v
in channels initially
deliver events
compute events
round 2:


Relation to Formal Model
send v (if this is a new value)
receive round 2 msgs
compute value for v
due to previous compute events
deliver events
compute events
…
round f + 1:




send v (if this is a new value)
receive round f + 1 msgs
compute value for v
decide v
due to previous compute events
deliver events
compute events
part of compute events
Set 9: Fault Tolerant Consensus
Correctness of Crash Consensus Algorithm
11
Termination: By the code, finish in round f+1.
Validity: Holds since processors do not introduce
spurious messages: if all inputs are the same, then
that is the only value ever in circulation.
Set 9: Fault Tolerant Consensus
Correctness of Crash Consensus Algorithm
12
Agreement:

Suppose in contradiction pj decides on a smaller value, x, than does pi.

Then x was hidden from pi by a chain of faulty processors:
q1
round
1
q2
round
2
…
qf
round
f
qf+1
round
f+1
pj
pi

There are f + 1 faulty processors in this chain, a contradiction.
Set 9: Fault Tolerant Consensus
Performance of Crash Consensus Algorithm
13



Number of processors n > f
f + 1 rounds
at most n2 •|V| messages, each of size log|V| bits,
where V is the input set.
Set 9: Fault Tolerant Consensus
Lower Bound on Rounds
14
Assumptions:
 n > f + 1
 every processor is supposed to send a message to
every other processor in every round
 Input set is {0,1}
Set 9: Fault Tolerant Consensus
Failure-Sparse Executions
15




Bad behavior for the crash algorithm was when
there was one crash per round.
This is bad in general.
A failure-sparse execution has at most one crash
per round.
We will deal exclusively with failure-sparse
executions in this proof.
Set 9: Fault Tolerant Consensus
Valence of a Configuration
16



The valence of a configuration C is the set of all
values decided by a nonfaulty processor in some
configuration reachable from C by an admissible
(failure-sparse) execution.
Bivalent: set contains 0 and 1.
Univalent: set contains only one value
 0-valent
or 1-valent
Set 9: Fault Tolerant Consensus
Valence of a Configuration
17
C
0
D
0/1
E
0/1
1
F
G
0/1
0 0 0 0 0 1 0 11 1 1 1 1 0 0 1
0/1 : bivalent
1 : 1-valent
0 : 0-valent
Set 9: Fault Tolerant Consensus
 decisions
Statement of Round Lower Bound
18
Theorem (5.3): Any crash-resilient consensus
algorithm requires at least f + 1 rounds in the
worst case.
Proof Strategy:
round
1
show 
bivalent
initial
config.
round
2
…
round
f-2
round
f-1
show we can keep things bivalent
through round f - 1
Set 9: Fault Tolerant Consensus
round
f
show we can
keep a n.f.
proc. from
deciding in
round f
Existence of Bivalent Initial Config.
19

Suppose in contradiction all initial configurations
are univalent.
inputs
valency
000…00
0
000…01
?
000…11
?
by validity condition
…
001…11
? 0
011…11
? 1
111…11
1
Set 9: Fault Tolerant Consensus
Existence of Bivalent Initial Config.
20

Let
 I0
be a 0-valent initial config
 I1 be a 1-valent initial config
 s.t. they differ only in pi 's input

I0
pi fails initially, no other failures.
By termination, eventually rest decide.
all but pi
decide 0

I1
This execution looks the same as the one
above to all the processors except pi.
Set 9: Fault Tolerant Consensus
all but pi
decide 0
Keeping Things Bivalent
21

Let ' be a (failure-sparse) k-1 round execution
ending in a bivalent config.
 for

Show there is a one-round (f-s) extension  of '
ending in a bivalent config.
 so

k-1<f-1
 has k < f rounds
Suppose in contradiction every one-round (f-s)
extension of ' is univalent.
Set 9: Fault Tolerant Consensus
Keeping Things Bivalent
22
failure-free round k
pi fails to send to 
1-val
…
bival
'
pi fails to send to q1,…,qj
rounds 1 to k-1
pi fails to send to q1,…,qj+1
0-val
…
now focus
in on these
two extensions
1-val
pi crashes
pi fails to send to q1,…,qm
Set 9: Fault Tolerant Consensus
0-val
Keeping Things Bivalent
23
pi fails to send
to q1,…,qj
'
rounds 1 to k-1
Since k-1 < f-1 and α’ is
failure-sparse, less than
f-1 procs fail in α’. Even
with pi failing in round k,
less than f procs have failed.
So we can have qj+1 fail
in round k+1 without exceeding
our budget of f failures.

1-val
round k
qj+1 fails in
rd. k+1;
no other
failures
n.f.
decide
1
only qj+1 can
tell difference
0-val
pi fails to send
to q1,…,qj+1
Set 9: Fault Tolerant Consensus

n.f.
decide
1
Cannot Decide in Round f
24



We've shown there is an f - 1 round (failuresparse) execution, call it , ending in a bivalent
configuration.
Extending this execution to f rounds might not
preserve bivalence.
However, we can keep a processor from explicitly
deciding in round f, thus requiring at least one more
round (f+1).
Set 9: Fault Tolerant Consensus
Cannot Decide in Round f
25
Case 1: There is a 1-round (f-s) extension of 
ending in a bivalent config. Then we are done.
Case 2: All 1-round (f-s) extensions of  end in
univalent configs.
Set 9: Fault Tolerant Consensus
Cannot Decide in Round f
26
round f
1-val
failure free
pi sends to pj
and pk

rounds 1 to f-1
pk either
undecided
or decided 1
look same to pk
bival.
pi fails to send to nf pj ,
sends to another nf pk
0-val
pi fails to send to nf pj
pi might send to pk
Set 9: Fault Tolerant Consensus
look same
to pj
pj either
undecided
or decided 0