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Lecture 4 Introduction to Principles of Distributed Computing

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### Lecture 4Introduction to Principles of Distributed Computing

### Part II: Algorithms, and the Failure Detector Abstraction

### Part II: Algorithms, and the Failure Detector Abstraction

### A Simple Proof of the Uniform Consensus Synchronous Lower Bound[Keidar, Rajsbaum IPL 02]

Sergio Rajsbaum

Math Institute

UNAM, Mexico

Lecture 4

Consensus in partially synchronous systems, and failure detectors

- Part I: Realistic timing model and metric
- Part II: Failure detectors, algorithms
- Part III: this is the best possible
- Part IV: New directions and extensions

CONSENSUS A fundamental Abstraction

Each process has an input, should decide an output s.t.

Agreement: correct processes’ decisions are the same

Validity: decision is input of one process

Termination: eventually all correct processes decide

There are at least two possible input values 0 and 1.

all possible vectors over the input values V

L(X0)

X0

The lecture in a nutshell- Consensus solvability depends on how long connectivity preserved by a particular model
- In synchronous it is solvable, in asynchronous not. What about intermediate, more realistic models?

Connectivity

destroyed

Initial states

states after one round

Connectivity

preserved

states after 2 rounds

Basic Model

- Message passing (essentially equivalent to read/write shared memory model)
- Channels between every pair of processes
- Crash failures

t < n potential failures out of n >1 processes

- No message loss among correct processes

Is consensus solvable?If so, how long does it take to solve it?

- It depends on what exactly the model is
- But what is a realistic model?
- And what are the common scenarios within the model? The nature of a distributed system is to include complex combinations of failures and delays

How Fast Can We Solve Consensus?

Depends on the timing model:

- Message delays
- Processing times
- Clocks
- And on the metric used:
- Worst case
- Average
- etc

The Rest of This Lecture

- Part I: Realistic timing model and metric
- Part II: Upper bounds
- Part III: this is the best possible
- Part IV: New directions and extensions

Synchronous Model

- Algorithm runs in synchronous rounds:
- send messages to any set of processes,
- receive messages from previous round,
- do local processing (possibly decide, halt)

Round

- If process i crashes in a round, then any subset of the messages i sends in this round can be lost

Synchronous Consensus

- In a run with f failures (f<t)
- Processes can decide in f+1 rounds

[Lamport Fischer 82; Dolev, Reischuk, Strong 90](early-deciding)

- 1 round with no failures
- In this talk deciding
- halting takes min(f+2,t+1) [Dolev, Reischuk, Strong 90]

The Middle Ground

Many real networks are neither synchronous nor asynchronous

- During long stable periods, delays and processing times are bounded
- Like synchronous model
- Some unstable periods
- Like asynchronous model

Partial Synchrony Model [Dwork, Lynch, Stockmeyer 88]

- Processes have clocks (with bounded drift)
- D, upper bound on message delay
- r, upper bound on processing time
- GST, global stabilization time
- Until GST, unstable: bounds do not hold
- After GST, stable: bounds hold
- GST unknown

Partial Synchrony in Practice

- For D, r, choose bounds that hold with high probability
- Stability forever?
- We assume that once stable remains stable
- In practice, has to last “long enough” for given algorithm to terminate
- A commonly used model that alternates between stable and unstable times:

Timed Asynchronous Model [Cristian, Fetzer 98]

Consensus with Partial Synchrony

- Solvable
- requires t < n/2 [DLS88]

Unbounded running time

by [FLP85], because model can be asynchronous for unbounded time

Exercise

- Prove that consensus is not solvable in the partially synchronous model, if t ≥ n/2
- Prove that if t<n/2, it takes unbounded running time to be solved

Performance Metric

Number of rounds in well-behavedruns

- Well-behaved:
- No failures
- Stable from the beginning
- Motivation: common case

The Rest of This Lecture

- Part II: best known algorithms decide in 2 rounds in well-behaved runs
- 2 time (with delay bound , 0 processing time)
- Part III: this is the best possible
- Part IV: new directions and extensions

II.a Failure Detectors and Partial Synchrony

-=

II.b Algorithms

Time-Free Algorithms

- Goal: abstract away time, get simpler algorithms
- We describe the algorithms using failure detector abstraction [Chandra, Toueg 96]

Unreliable Failure Detectors [Chandra, Toueg 96]

- Each process has local failure detector oracle
- Typically outputs list of processes suspected to have crashed at any given time
- Unreliable: failure detector output can be arbitrary for unbounded (finite) prefix of run

Performance of Failure Detector Based Consensus Algorithms

- Implement a failure detector in the partial synchrony model
- Design an algorithm for the failure detector
- Analyze the performance in well-behaved runs of the combined algorithm

A Natural Failure Detector Implementation in Partial Synchrony Model

- Implement failure detector using timeouts:
- When expecting a message from a process i, wait D + r + clock skew before suspecting i
- In well-behaved runs, D, r always hold, hence no false suspicions

The resulting failure detector is <>P - Eventually Perfect

- Strong Completeness: From some point on, every faulty process is suspected by every correct process
- Eventual Strong Accuracy: From some point on, every correct process is not suspected

Weakest Failure Detectors for Consensus

- <>S - Eventually Strong
- Strong Completeness
- Eventual Weak Accuracy: From some point on, some correct process is not suspected
- W - Leader
- Outputs one trusted process
- From some point, all correct processes trust the same correct process

A Simple W Implementation

- Use <>P implementation
- Output lowest id non-suspected process

In well-behaved runs: process 1 always trusted

Exercise

- Write the algorithm code for this failure detector W, and prove it is correct

Relationships among Failure Detector Classes

- <>S is a subset of <>P
- <>S is strictly weaker than <>P
- <>S ~ W[Chandra, Hadzilacos, Toueg 96]

Food for thought:

What is the weakest timing model where <>S and/or W are implementable but <>P is not?

Relationships among Failure Detector Classes- Recent Results

Partial Answer: In PODC’03 Aguilera et al present a system with synchronous processes S :

- any number of them may crash, and
- only the output links of an unknown correct process are eventually timely (all other links can be asynchronous and/or lossy)

<>P is not implementable in S, W yes

New proof that: <>S is strictly weaker than <>P

Note on the Power of Consensus

- Consensus cannot implement <>P, interactive consistency, atomic commit, …
- So its “universality”, in the sense of
- wait-free objects in shared memory [Herlihy 93]
- state machine replication [Lamport 78; Schneider 90]

does not cover sensitivity to failures, timing, etc.

Other Failure Detector Implementations

Food for thought:

When is building <>P more costly than <>S or W?

Partial answer: Aguilera at al PODC’03 observe

- any implementation of <>P (even in a perfectly synchronous system) requires all alive processes to send messages forever, while W can be implemented such that eventually only the leader sends messages

Other Failure Detector Implementations

- Message efficient <>S implementation [Larrea, Fernández, Arévalo 00]
- QoS tradeoffs between accuracy and completeness [Chen, Toueg, Aguilera 00]
- Leader Election [Aguilera, Delporte, Fauconnier, Toueg 01]
- Adaptive <>P[Fetzer, Raynal, Tronel 01]

II.a Failure Detectors and Partial Synchrony

II.b Algorithms

Algorithms that Take 2 Rounds in Well-Behaved Runs

- <>S-based [Schiper 97; Hurfin, Raynal 99; Mostefaoui, Raynal 99]
- W-based for t < n/3[Mostefaoui, Raynal 00]
- W-based for t < n/2[Dutta, Guerraoui 01]
- Paxos (optimized version) [Lamport 89; 96]
- Leader-based (W)
- Also tolerates omissions, crash recoveries
- COReL - Atomic Broadcast [Keidar, Dolev 96]
- Group membership based (<>P)

Of This Laundry List, We Present Two Algorithms

- <>S-based [MR99]
- Paxos

<>S-based Consensus [MR99]

- val input v; est null

for r =1, 2, … do

coord(r mod n)+1

if I am coord,then send (r,val) to all

wait for ( (r, val)from coordOR suspect coord (by <>S))

if receive val from coord then estval elseest null

send (r, est)to all

wait for (r,est) from n-t processes

if any non-null est received thenvalest

if all ests have same vthen send (“decide”, v) to all; return(v)

od

- Upon receive (“decide”, v), forward to all, return(v)

1

2

In Case of Omissions

The algorithm can block in case of transient message omissions, waiting for a specific round message that will not arrive

Paxos [Lamport 88; 96; 01]

- Uses W failure detector
- Phase 1: prepare
- A process who trusts itself tries to become leader
- Chooses largest unique (using ids) ballot number
- Learns outcome of all smaller ballots
- Phase 2: accept
- Leader proposes a value with his ballot number.
- Leader gets majority to accept his proposal.
- A value accepted by a majority can be decided

Paxos - Variables

- Type Rank
- totally ordered set with minimum element r0
- Variables:

Rank BallotNum, initially r0

Rank AcceptNum, initially r0

Value {^} AcceptVal, initially ^

Paxos Phase I: Prepare

- Periodically, until decision is reached do:

if leader (by W) then

BallotNum (unique rank > BallotNum)

send (“prepare”, rank) to all

- Upon receive (“prepare”, rank) from i

if rank > BallotNum then

BallotNum rank

send (“ack”, rank, AcceptNum, AcceptVal) to i

Paxos Phase II: Accept

Upon receive (“ack”, BallotNum, b, val) from n-t

if all vals = ^ then myVal = initial value

else myVal = received val with highest b

send (“accept”, BallotNum, myVal) to all /* proposal */

Upon receive (“accept”, b, v) with b BallotNum

AcceptNum b; AcceptVal v /* accept proposal */

send (“accept”, b, v) to all (first time only)

Paxos – Deciding

Upon receive(“accept”, b, v) from n-t

decide v

periodically send (“decide”, v) to all

Upon receive (“decide”, v)

decide v

In Well-Behaved Runs

1

1

1

1

1

2

2

2

(“prepare”,1)

(“accept”,1 ,v1)

.

.

.

.

.

.

.

.

.

(“ack”,1,r0,^)

n

n

n

(“accept”,1 ,v1)

Our W implementation

always trusts process 1

decide v1

Optimization

- Allow process 1 (only!) to skip Phase 1
- use rank r0
- propose its own initial value
- Takes 2 rounds in well-behaved runs
- Takes 2 rounds for repeated invocations with the same leader

What About Message Loss?

- Does not block in case of a lost message
- Phase I can start with new rank even if previous attempts never ended
- But constant omissions can violate liveness
- Specify conditional liveness:

If n-t correct processes including the leader can communicate with each other

then they eventually decide

Synchronous Consensus

- In a run with f failures (f<t)
- Processes can decide in f+1 rounds
- And no less !

[Lamport Fischer 82; Dolev, Reischuk, Strong 90](early-deciding)

- 1 round with no failures
- In this talk deciding
- halting takes min(f+2,t+1) [Dolev, Reischuk, Strong 90]

Uniform Consensus

- Uniform agreement: decision of every two processes is the same

Recall: with consensus, only correct processes have to agree (disagreement with the dead is OK)

This version of consensus will be useful to extend the lower bound argument to asynchronous models

Synchronous Uniform Consensus

Every algorithm has a run with f failures (f<t-1), that takes at least f+2 rounds to decide

- [Charron-Bost, Schiper 00; KR 01]
- as opposed to f+1 for consensus

Theorem: f+2 Lower Bound

- Assume n>t, and f < t-1
- Lf(X0) - final states of runs with f failures
- connected
- in any state in Lf(X0) exist at least 3 non-failed processes and 2 can fail
- Take z, z’X0 s.t. val(z)val(z’),
- let x, x’ be failure-free extensions of z, z’: x=z.(i,[0])f Lf(X0)

Exercise

- Consider Modify the theorem and the proof of this talk for the consensus problem (instead of the uniform consensus problem)

Upper Bounds From Part II

We saw that there are algorithms that take 2 rounds todecide in well-behaved runs

- <>S-based, W-based, Paxos, COReL
- Presented two of them.

Why are there no 1-Round Algorithms?

There is a lower bound of 2 rounds in well-behaved executions

- Similar bounds shown in [Dwork, Skeen 83; Lamport 00]
- We will show that the bound follows from a similar bound on Uniform Consensus in the synchronous model

Uniform Consensus

- Uniform agreement: decision of every two processes is the same

Recall: with consensus, only correct processes have to agree

From Consensus to Uniform Consensus

In partial synchrony model, any algorithm A for consensus solves uniform consensus[Guerraoui 95]

Proof: Assume by contradiction that A does not solve uniform consensus

- in some run, p,q decide differently, p fails
- p may be non-faulty, and may wake up after q decides

Synchronous Uniform Consensus

Every algorithm has a well-behaved run that takes 2 rounds to decide

- More generally, it has a run with f failures (f<t-1), that takes at least f+2 rounds to decide[Charron-Bost, Schiper 00; KR 01]
- as opposed to f+1 for consensus

Bibliography

- Keidar and Rajsbaum, “A Simple Proof of the Uniform Consensus Synchronous Lower Bound,” in IPL, Vol. 85, pp. 47-52, 2003.
- Keidar and Rajsbaum, “Onthe Cost of Fault-Tolerant Consensus When There Are No Faults” in Keidar’s page, including slides and papers.
- Moses, Rajsbaum, “A Layered Analysis of Consensus,” SIAM J. Comput. 31(4): 989-1021, 2002.
- Mostéfaoui, Rajsbaum, Raynal: Conditions on input vectors for consensus solvability in asynchronous distributed systems. J. ACM, 2003

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