Consensus Algorithms: Distributed Agreement Under Failure

Purpose: Algorithms and theoretical frameworks for achieving agreement among distributed processes despite failures, enabling fault-tolerant distributed systems.

Core Insight: Consensus is impossible in asynchronous systems with even one crash failure (FLP), but practical with timeouts, randomization, or synchrony assumptions.

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Table of Contents

1. Foundational Concepts
2. FLP Impossibility
3. Paxos
4. Raft
5. Byzantine Consensus
6. Practical Systems
7. Stack-Specific Implementations
8. Integration Points

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Foundational Concepts

The Consensus Problem

Goal: N processes must agree on a single value, despite failures.

Requirements (Traditional):

1. Termination: Every correct process eventually decides on a value
2. Agreement: All correct processes decide the same value
3. Validity: The decided value was proposed by some process

Failure Models

Crash Failures (Benign):

• Process stops executing permanently
• No incorrect behavior before crash
• Examples: power failure, OOM kill

Byzantine Failures (Malicious):

• Arbitrary behavior: send wrong messages, lie, collude
• Includes both malicious actors and software bugs
• Requires stronger protocols (BFT)

Omission Failures:

• Process fails to send or receive messages
• Includes network partitions

Timing Failures:

• Process responds too late
• Relevant in synchronous systems

System Models

Asynchronous:

• No bounds on message delay or processing speed
• No reliable failure detection
• Most realistic but hardest (FLP impossible)

Synchronous:

• Known bounds on delays and processing
• Reliable failure detection via timeouts
• Easiest but least realistic

Partially Synchronous (Practical):

• Eventually synchronous: bounds hold after some unknown time
• Models real systems (networks eventually recover)
• Algorithms like Paxos, Raft work here

---

FLP Impossibility

Theorem (Fischer, Lynch, Paterson, 1985):

“No deterministic consensus algorithm can guarantee termination in an asynchronous system with even one crash failure.”

FLP Proof Sketch

Setup:

• Asynchronous system, N processes
• At most 1 crash failure
• Binary consensus (decide 0 or 1)

Key Concept: Bivalent Configuration

• Configuration where both 0 and 1 are possible outcomes
• Univalent = only one outcome possible (0-valent or 1-valent)

Proof Steps:

1. Initial bivalent configuration exists: Some initial state allows either outcome
2. Bivalent configurations reachable forever: From any bivalent config, can always reach another bivalent config by delaying one message
3. No termination: If always bivalent configs reachable, never reach univalent (decided) state

Implication: In pure asynchronous model, can’t distinguish crashed process from slow process → can’t safely terminate.

Circumventing FLP

Real systems work because they relax assumptions:

Randomization:

• Use randomness to break symmetry
• Terminate with probability 1 (not deterministic guarantee)
• Example: Bitcoin’s proof-of-work

Timeouts (Partial Synchrony):

• Assume eventual synchrony: bounds hold eventually
• Timeout to detect failures (may be wrong temporarily)
• Examples: Paxos, Raft, PBFT

Failure Detectors:

• Abstract failure detection service
• Perfect FD: never wrong (requires synchrony)
• Eventually Perfect FD: eventually accurate (achievable)
• Examples: Chandra-Toueg consensus

Weakened Termination:

• Require only “live” processes terminate
• Non-faulty but slow processes may not decide
• Example: Paxos doesn’t guarantee termination for all

---

Paxos

Developed by: Leslie Lamport (1989, published 1998)

Key Idea: Use quorum-based voting with proposal numbering to ensure agreement even when processes crash or messages are delayed.

Single-Decree Paxos (Basic Paxos)

Goal: Agree on a single value.

Roles:

• Proposer: Proposes values
• Acceptor: Votes on proposals
• Learner: Learns the chosen value

Phases:

Phase 1: Prepare

1. Proposer selects unique proposal number N
2. Sends Prepare(N) to quorum of acceptors
3. Acceptor responds:
   • Promise(N) if N > any previous promise
   • Include highest-numbered accepted proposal (if any)

Phase 2: Accept

1. If proposer receives promises from quorum:
   • If any acceptor returned accepted value, use highest-numbered one
   • Otherwise, use proposer’s own value
2. Send Accept(N, value) to quorum
3. Acceptor accepts if N ≥ promised number
4. Respond Accepted(N, value)

Learning:

• Learner receives Accepted from quorum → value is chosen

Invariant: If quorums overlap, at most one value can be chosen.

Multi-Paxos

Problem: Basic Paxos requires 2 round trips per decision.

Solution: Elect a leader (stable proposer) who skips Phase 1 for subsequent proposals.

Leader Election:

• Use Phase 1 to establish leadership
• Leader has exclusive proposal rights while stable
• On leader failure, run Phase 1 again

Log Replication:

• Apply Paxos to each log slot independently
• Leader coordinates appending entries across replicas
• Achieves replicated state machine

Paxos Properties

Safety (Always):

• Agreement: Only one value chosen per instance
• Validity: Chosen value was proposed

Liveness (Eventually, with stable leader):

• Progress if majority available and leader stable

Quorum Requirement:

• Need majority: ⌈(N+1)/2⌉
• Tolerates ⌊N/2⌋ crash failures

Paxos Variants

Fast Paxos:

• Clients send directly to acceptors (skip proposer)
• Requires 3f+1 acceptors for f failures
• Faster when no conflicts

Cheap Paxos:

• Subset of acceptors (main quorum) + auxiliary acceptors
• Auxiliary only activated on main failure
• Reduces number of always-active acceptors

Flexible Paxos:

• Different quorum sizes for Phase 1 and Phase 2
• Requirement: Q1 ∩ Q2 ≠ ∅ (quorums must overlap)

---

Raft

Developed by: Diego Ongaro and John Ousterhout (2014)

Goal: Understandable alternative to Paxos for replicated state machines.

Raft Overview

Key Innovation: Decompose consensus into:

1. Leader Election
2. Log Replication
3. Safety

Roles:

• Leader: Handles all client requests, replicates log
• Follower: Passive, replicate leader’s log
• Candidate: Participates in leader election

Leader Election

Terms:

• Logical clock, monotonically increasing
• Each term has at most one leader

Election Process:

1. Follower timeout → becomes Candidate
2. Candidate increments term, votes for self
3. Requests votes from all peers
4. Peer grants vote if:
   • Candidate’s term ≥ peer’s term
   • Peer hasn’t voted this term
   • Candidate’s log at least as up-to-date
5. Candidate wins if receives majority votes
6. Becomes Leader, sends heartbeats

Split Vote:

• No candidate gets majority → timeout, retry with new term
• Randomized timeouts reduce split votes

Log Replication

Normal Operation:

1. Client sends command to Leader
2. Leader appends to local log
3. Leader sends AppendEntries RPCs to Followers
4. Followers append entry, respond success
5. Leader commits when majority replicated
6. Leader applies to state machine, replies to client
7. Leader piggybacks commitIndex in heartbeats

Log Matching Property:

• If two logs have entry with same index and term → all preceding entries identical
• Maintained by Leader’s consistency check

Handling Failures:

• Follower crash: Leader retries AppendEntries indefinitely
• Leader crash: New leader elected, log repaired
• Log inconsistencies: Leader forces followers to match its log

Safety

Election Safety:

• At most one leader per term

Leader Completeness:

• If log entry committed in term T, present in logs of leaders for all terms > T

Log Matching:

• If two logs contain entry with same index and term, they match up to that point

State Machine Safety:

• If server applies log entry at index, no other server applies different entry at same index

Commitment Rule:

• Leader can only commit entries from current term
• Once current term entry committed, all prior entries implicitly committed

Raft vs. Paxos

Aspect	Paxos	Raft
Structure	Single-decree + Multi-Paxos	Leader-based from start
Leader	Optional, emergent	Explicit, necessary
Log Gaps	Allowed	Not allowed
Commit Rule	Value-based	Log-index-based
Understandability	Complex	Simpler

---

Byzantine Consensus

Challenge: Achieve consensus when some nodes are Byzantine (malicious or faulty).

Requirement: ≥ 3f+1 nodes to tolerate f Byzantine failures (quorum = 2f+1).

PBFT (Practical Byzantine Fault Tolerance)

Developed by: Castro & Liskov (1999)

Key Idea: State machine replication with Byzantine fault tolerance.

Phases:

Pre-Prepare:

1. Client sends request to Primary
2. Primary assigns sequence number, broadcasts PrePrepare(v, n, m) where:
   • v = view number
   • n = sequence number
   • m = request message

Prepare:

1. Replica receives PrePrepare, validates
2. Broadcasts Prepare(v, n, d) where d = digest of m
3. Replica waits for 2f matching Prepare messages (quorum)
4. “Prepared” certificate established

Commit:

1. Replica broadcasts Commit(v, n, d)
2. Waits for 2f+1 Commit messages (including self)
3. “Committed” certificate established
4. Execute request, reply to client

Client:

• Waits for f+1 matching replies (one from correct replica guaranteed)

View Change (Leader Replacement):

• Replica timeout → suspect Primary faulty
• Initiate view change protocol
• New primary elected, state transferred

PBFT Properties:

• Safety: All correct replicas agree on order
• Liveness: Clients eventually receive replies
• Performance: ~3 round trips, similar to Paxos

Tendermint

Developed by: Jae Kwon (2014)

Key Idea: BFT consensus for blockchain with immediate finality.

Mechanism:

• Rounds with designated proposer (rotating)
• Three phases: Propose, Prevote, Precommit
• Two rounds of voting for each decision
• Block finalized when 2f+1 Precommits collected

Features:

• Immediate Finality: No forks, block final when committed
• Application Agnostic: ABCI interface for any application
• Light Clients: Efficient verification via Merkle proofs

Used In: Cosmos blockchain, many app-chains.

HotStuff

Developed by: Yin, Malkhi, et al. (2018)

Key Innovation: Linear communication complexity (O(n) per decision, not O(n²) like PBFT).

Mechanism:

• Threshold Signatures: Aggregate 2f+1 signatures into single signature
• Three-Phase Voting: Prepare, Pre-Commit, Commit
• Chained Structure: Each phase’s QC becomes next phase’s justify

Phases:

1. Prepare: Leader proposes, replicas vote
2. Pre-Commit: Leader aggregates votes into QC, replicas vote again
3. Commit: Leader aggregates, replicas commit
4. Decide: Block finalized

Advantages:

• O(n) communication (vs. O(n²) in PBFT)
• Responsive: Progresses at network speed (not fixed timeout)
• Used in LibraBFT (Diem blockchain)

---

Practical Systems

Chubby (Google)

Purpose: Distributed lock service

Algorithm: Paxos (Multi-Paxos)

Features:

• Coarse-grained locks (lease-based)
• Small file storage (config, leader election results)
• 5-replica typical deployment

Use Cases: GFS, Bigtable (master election, metadata storage)

ZooKeeper (Apache)

Purpose: Coordination service for distributed applications

Algorithm: Zab (ZooKeeper Atomic Broadcast) - similar to Paxos/Raft

Features:

• Hierarchical namespace (like filesystem)
• Watches for change notifications
• Ephemeral nodes (disappear when client session ends)

Use Cases: Kafka (broker coordination), HBase (master election)

etcd

Purpose: Distributed key-value store

Algorithm: Raft

Features:

• Strong consistency (linearizable reads/writes)
• Watch API for change notifications
• TTL for keys (lease-based)

Use Cases: Kubernetes (cluster state), service discovery

Consul (HashiCorp)

Purpose: Service mesh, service discovery, configuration

Algorithm: Raft

Features:

• Multi-datacenter support (federated Raft clusters)
• Health checking integrated
• DNS and HTTP interfaces

Use Cases: Microservice coordination, config management

---

Stack-Specific Implementations

Rust: Raft Implementation (Simplified)

use serde::{Deserialize, Serialize};
use std::collections::HashMap;
use std::sync::{Arc, Mutex};
use std::time::{Duration, Instant};

/// Raft node state
#[derive(Debug, Clone, PartialEq)]
enum Role {
    Follower,
    Candidate,
    Leader,
}

/// Log entry
#[derive(Debug, Clone, Serialize, Deserialize)]
struct LogEntry {
    term: u64,
    index: u64,
    command: String,
}

/// RequestVote RPC
#[derive(Debug, Serialize, Deserialize)]
struct RequestVoteArgs {
    term: u64,
    candidate_id: u64,
    last_log_index: u64,
    last_log_term: u64,
}

#[derive(Debug, Serialize, Deserialize)]
struct RequestVoteReply {
    term: u64,
    vote_granted: bool,
}

/// AppendEntries RPC
#[derive(Debug, Serialize, Deserialize)]
struct AppendEntriesArgs {
    term: u64,
    leader_id: u64,
    prev_log_index: u64,
    prev_log_term: u64,
    entries: Vec<LogEntry>,
    leader_commit: u64,
}

#[derive(Debug, Serialize, Deserialize)]
struct AppendEntriesReply {
    term: u64,
    success: bool,
    conflict_index: Option<u64>,
}

/// Raft node
struct RaftNode {
    id: u64,
    peers: Vec<u64>,

    // Persistent state
    current_term: u64,
    voted_for: Option<u64>,
    log: Vec<LogEntry>,

    // Volatile state
    commit_index: u64,
    last_applied: u64,
    role: Role,

    // Leader-specific
    next_index: HashMap<u64, u64>,
    match_index: HashMap<u64, u64>,

    // Election timing
    last_heartbeat: Instant,
    election_timeout: Duration,
}

impl RaftNode {
    fn new(id: u64, peers: Vec<u64>) -> Self {
        RaftNode {
            id,
            peers,
            current_term: 0,
            voted_for: None,
            log: vec![],
            commit_index: 0,
            last_applied: 0,
            role: Role::Follower,
            next_index: HashMap::new(),
            match_index: HashMap::new(),
            last_heartbeat: Instant::now(),
            election_timeout: Duration::from_millis(150),
        }
    }

    /// Handle RequestVote RPC
    fn handle_request_vote(
        &mut self,
        args: RequestVoteArgs,
    ) -> RequestVoteReply {
        let mut vote_granted = false;

        if args.term > self.current_term {
            self.current_term = args.term;
            self.role = Role::Follower;
            self.voted_for = None;
        }

        if args.term == self.current_term {
            let can_vote = self.voted_for.is_none()
                || self.voted_for == Some(args.candidate_id);

            let log_ok = self.is_log_up_to_date(
                args.last_log_term,
                args.last_log_index,
            );

            if can_vote && log_ok {
                self.voted_for = Some(args.candidate_id);
                vote_granted = true;
                self.last_heartbeat = Instant::now();
            }
        }

        RequestVoteReply {
            term: self.current_term,
            vote_granted,
        }
    }

    /// Check if candidate's log is at least as up-to-date
    fn is_log_up_to_date(
        &self,
        last_term: u64,
        last_index: u64,
    ) -> bool {
        let (my_last_term, my_last_index) = self.last_log_info();

        if last_term != my_last_term {
            last_term > my_last_term
        } else {
            last_index >= my_last_index
        }
    }

    fn last_log_info(&self) -> (u64, u64) {
        if let Some(last) = self.log.last() {
            (last.term, last.index)
        } else {
            (0, 0)
        }
    }

    /// Handle AppendEntries RPC
    fn handle_append_entries(
        &mut self,
        args: AppendEntriesArgs,
    ) -> AppendEntriesReply {
        if args.term > self.current_term {
            self.current_term = args.term;
            self.role = Role::Follower;
            self.voted_for = None;
        }

        let mut success = false;
        let mut conflict_index = None;

        if args.term == self.current_term {
            self.role = Role::Follower;
            self.last_heartbeat = Instant::now();

            // Check log consistency
            if self.log_matches(args.prev_log_index, args.prev_log_term) {
                success = true;

                // Append new entries
                let mut index = args.prev_log_index;
                for entry in args.entries {
                    index += 1;
                    if let Some(existing) =
                        self.log.get_mut(index as usize)
                    {
                        if existing.term != entry.term {
                            // Conflict: delete from here onward
                            self.log.truncate(index as usize);
                            self.log.push(entry.clone());
                        }
                    } else {
                        self.log.push(entry.clone());
                    }
                }

                // Update commit index
                if args.leader_commit > self.commit_index {
                    self.commit_index = std::cmp::min(
                        args.leader_commit,
                        self.log.len() as u64,
                    );
                }
            } else {
                // Find conflict index
                conflict_index = Some(args.prev_log_index);
            }
        }

        AppendEntriesReply {
            term: self.current_term,
            success,
            conflict_index,
        }
    }

    fn log_matches(&self, index: u64, term: u64) -> bool {
        if index == 0 {
            return true;
        }

        if let Some(entry) = self.log.get(index as usize - 1) {
            entry.term == term
        } else {
            false
        }
    }

    /// Start leader election
    fn start_election(&mut self) {
        self.current_term += 1;
        self.role = Role::Candidate;
        self.voted_for = Some(self.id);
        self.last_heartbeat = Instant::now();

        let (last_log_term, last_log_index) = self.last_log_info();

        let args = RequestVoteArgs {
            term: self.current_term,
            candidate_id: self.id,
            last_log_index,
            last_log_term,
        };

        // Send RequestVote RPCs to all peers
        // (In real implementation, send via network)
        println!(
            "Node {} starting election for term {}",
            self.id, self.current_term
        );
    }

    /// Become leader
    fn become_leader(&mut self) {
        self.role = Role::Leader;

        // Initialize leader state
        let next_index = self.log.len() as u64 + 1;
        for peer in &self.peers {
            self.next_index.insert(*peer, next_index);
            self.match_index.insert(*peer, 0);
        }

        println!("Node {} became leader for term {}", self.id, self.current_term);
    }

    /// Check if election timeout elapsed
    fn election_timeout_elapsed(&self) -> bool {
        self.last_heartbeat.elapsed() > self.election_timeout
    }
}

/// Simplified Raft example usage
async fn raft_example() {
    let mut node = RaftNode::new(1, vec![2, 3, 4, 5]);

    // Simulate election timeout
    node.start_election();

    // Simulate receiving votes (in real system, via network)
    let mut votes = 1; // Self-vote

    if votes > node.peers.len() / 2 {
        node.become_leader();
    }
}

TypeScript: Basic Paxos (Conceptual)

/**
 * Paxos proposer
 */
class Proposer {
  private proposalNumber = 0;
  private highestAccepted: {
    number: number;
    value: any;
  } | null = null;

  constructor(
    private id: number,
    private acceptors: Acceptor[]
  ) {}

  async propose(value: any): Promise<boolean> {
    // Phase 1: Prepare
    this.proposalNumber++;
    const promisesreceived = await this.sendPrepare(
      this.proposalNumber
    );

    const promises = promisesreceived.filter(p => p !== null);

    // Need majority
    if (promises.length < Math.floor(this.acceptors.length / 2) + 1) {
      return false; // Failed to get quorum
    }

    // Find highest accepted value
    let valueToPropose = value;
    for (const promise of promises) {
      if (
        promise.acceptedNumber !== null &&
        (this.highestAccepted === null ||
          promise.acceptedNumber > this.highestAccepted.number)
      ) {
        this.highestAccepted = {
          number: promise.acceptedNumber,
          value: promise.acceptedValue,
        };
        valueToPropose = promise.acceptedValue;
      }
    }

    // Phase 2: Accept
    const accepted = await this.sendAccept(
      this.proposalNumber,
      valueToPropose
    );

    const acceptCount = accepted.filter(a => a).length;

    // Need majority
    return acceptCount >= Math.floor(this.acceptors.length / 2) + 1;
  }

  private async sendPrepare(
    n: number
  ): Promise<PromiseResponse[]> {
    return Promise.all(
      this.acceptors.map(acceptor =>
        acceptor.receivePrepare(n).catch(() => null)
      )
    );
  }

  private async sendAccept(
    n: number,
    value: any
  ): Promise<boolean[]> {
    return Promise.all(
      this.acceptors.map(acceptor =>
        acceptor.receiveAccept(n, value).catch(() => false)
      )
    );
  }
}

/**
 * Paxos acceptor
 */
class Acceptor {
  private promisedNumber: number | null = null;
  private acceptedNumber: number | null = null;
  private acceptedValue: any = null;

  constructor(private id: number) {}

  async receivePrepare(n: number): Promise<PromiseResponse> {
    if (this.promisedNumber === null || n > this.promisedNumber) {
      this.promisedNumber = n;
      return {
        promised: true,
        acceptedNumber: this.acceptedNumber,
        acceptedValue: this.acceptedValue,
      };
    }

    return { promised: false, acceptedNumber: null, acceptedValue: null };
  }

  async receiveAccept(n: number, value: any): Promise<boolean> {
    if (this.promisedNumber === null || n >= this.promisedNumber) {
      this.promisedNumber = n;
      this.acceptedNumber = n;
      this.acceptedValue = value;
      return true; // Accepted
    }

    return false; // Rejected
  }

  getAcceptedValue(): any {
    return this.acceptedValue;
  }
}

interface PromiseResponse {
  promised: boolean;
  acceptedNumber: number | null;
  acceptedValue: any;
}

/**
 * Example: Run Paxos
 */
async function runPaxos() {
  // Create 5 acceptors
  const acceptors = [
    new Acceptor(1),
    new Acceptor(2),
    new Acceptor(3),
    new Acceptor(4),
    new Acceptor(5),
  ];

  // Create proposer
  const proposer = new Proposer(1, acceptors);

  // Propose value
  const success = await proposer.propose('my-value');

  if (success) {
    console.log('Consensus reached!');
  } else {
    console.log('Failed to reach consensus');
  }
}

PHP: Simple Consensus Coordinator

<?php

namespace Consensus;

/**
 * Simple leader election using database (similar to ZooKeeper)
 */
class LeaderElection
{
    private string $lockKey = 'leader_lock';
    private int $leaseSeconds = 10;

    public function tryBecomeLeader(string $nodeId): bool
    {
        // Try to acquire lock using Redis
        $acquired = Redis::set(
            $this->lockKey,
            $nodeId,
            'NX', // Only set if not exists
            'EX', // Expiration
            $this->leaseSeconds
        );

        return (bool) $acquired;
    }

    public function renewLease(string $nodeId): bool
    {
        $currentLeader = Redis::get($this->lockKey);

        if ($currentLeader === $nodeId) {
            Redis::expire($this->lockKey, $this->leaseSeconds);
            return true;
        }

        return false;
    }

    public function isLeader(string $nodeId): bool
    {
        return Redis::get($this->lockKey) === $nodeId;
    }

    public function getLeader(): ?string
    {
        return Redis::get($this->lockKey) ?: null;
    }
}

/**
 * Quorum-based configuration store
 */
class QuorumStore
{
    private array $nodes;
    private int $quorumSize;

    public function __construct(array $nodes)
    {
        $this->nodes = $nodes;
        $this->quorumSize = (int) floor(count($nodes) / 2) + 1;
    }

    public function write(string $key, mixed $value): bool
    {
        $successes = 0;
        $version = time(); // Simple versioning

        foreach ($this->nodes as $node) {
            if ($this->writeToNode($node, $key, $value, $version)) {
                $successes++;
            }

            if ($successes >= $this->quorumSize) {
                return true; // Quorum reached
            }
        }

        return false;
    }

    public function read(string $key): mixed
    {
        $responses = [];

        foreach ($this->nodes as $node) {
            $response = $this->readFromNode($node, $key);
            if ($response !== null) {
                $responses[] = $response;
            }

            if (count($responses) >= $this->quorumSize) {
                break;
            }
        }

        if (count($responses) < $this->quorumSize) {
            throw new \Exception('Failed to reach quorum for read');
        }

        // Return value with highest version
        usort($responses, fn($a, $b) => $b['version'] <=> $a['version']);

        return $responses[0]['value'];
    }

    private function writeToNode(
        string $node,
        string $key,
        mixed $value,
        int $version
    ): bool {
        // Simulate network write to node
        try {
            $response = Http::post("http://$node/write", [
                'key' => $key,
                'value' => $value,
                'version' => $version,
            ]);

            return $response->successful();
        } catch (\Exception $e) {
            return false;
        }
    }

    private function readFromNode(string $node, string $key): ?array
    {
        try {
            $response = Http::get("http://$node/read", [
                'key' => $key,
            ]);

            if ($response->successful()) {
                return $response->json();
            }
        } catch (\Exception $e) {
            // Node unavailable
        }

        return null;
    }
}

---

Integration Points

With Consistency Models

• Consensus provides linearizability: Agreed value visible to all
• Log replication: Consensus on operation order → consistent state
• Quorum intersection: Foundation for both consensus and consistency

With Actor Model

• Leader election: Select coordinator actor in actor system
• Distributed actors: Consensus for cross-node actor coordination
• Saga coordination: Consensus on distributed transaction outcome

With Streams

• Kafka: Log replication via quorum-based replication (Raft-like)
• Stream partitioning: Consensus on partition assignment
• Watermarks: Consensus on progress in distributed stream processing

---

Further Reading

Papers

• Fischer, Lynch, Paterson (1985) - “Impossibility of Distributed Consensus with One Faulty Process”
• Lamport (1998) - “The Part-Time Parliament” (Paxos)
• Ongaro & Ousterhout (2014) - “In Search of an Understandable Consensus Algorithm” (Raft)
• Castro & Liskov (1999) - “Practical Byzantine Fault Tolerance”

Books

• Van Steen & Tanenbaum - “Distributed Systems” (3rd ed.)
• Kleppmann - “Designing Data-Intensive Applications”

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End of Consensus Appendix