Distributed Algorithms

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Distributed Algorithms

• A Scattered Survey
• Seth Gilbert

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Overview

• Introduction
• Models
• Algorithms
• Formal Methods

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Introduction

• What is a distributed algorithm?
• Why study distributed algorithms?
• Is it practical?
• Examples
• TCP/IP, ethernet, Napster/Gnutella, NFS, Amazon

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Model Basics
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Model Basics - Anonymous
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Model Basics Unique IDs
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Model Basics - Network
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Model Basics - Unidirectional
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Model Basics - Ring
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Model Basics Shared Memory
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read/write
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A
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B
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Synchronous Model
Network
i
j
time td
time t
Clock
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Asynchronous Model
Network
i
j
time t???
time t
Clock
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Partially Synchronous Models

• Many different models
• Good clocks / unbounded communication
• Eventually synchronous
• Some communication links good
• Failure Detectors
• Eventual notification of failures

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Algorithms Overview

• Leader Election
• Shortest Path
• Consensus
• Other building-block algorithms
• Atomic Memory, Mutual Exclusion, Logical time /
  Synchronizers
• Note no randomization!!

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1. Leader Election

• Assumptions
• Ring topology
• Synchronous network
• No failures
• Goal
• Exactly one process says, I am the leader

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Leader Election Basics

• Impossible without unique IDs
• Symmetry
• Randomization helps
• Impossible with any failures
• ring topology

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Basic Algorithm

• Send ID to the right
• Remember largest ID and resend
• Discard smaller IDs
• Time O(n) rounds
• Message complexity O(n2)

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Better Algorithm

• Goal O(n log n) messages
• Add time-to-live to ID
• Proceed by levels
• TTL 2, TTL 4, TTL 8,
• log(n) rounds
• O(n) messages each round
• Optimal for comparison-based algorithms

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Algorithms

• Basic
• Time O(n) rounds
• Message complexity O(n2)
• More Complicated
• Time O(n) rounds
• Message complexity O(n log n)

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Lower Bounds Messages

• Comparison based algorithms
• O(n log n) messages
• In general
• O(n) messages
• Asynchronous Networks
• O(n log n)

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2. Shortest Path Routes

• Input
• Connected, undirected graph
• Destination
• Output
• Shortest route to destination
• Model Self-Stabilizing
• Asynchronous
• Arbitrary failures! (e.g. corrupt state)
• Eventual stabilization

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Bellman-Ford

• Every so often
• Ask all neighbors distance
• Set distance to minimum
• Destination always distance0
• Stabilizes after n rounds

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3. Consensus

• Input
• 1 or 0 to each node
• Output
• Agreement all nodes decide either 0 or 1
• Termination all nodes eventually decide
• Validity if all inputs 1, then decide 1

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Motivation

• Many problems equivalent to consensus
• Read-Modify-Write Memory
• Database commit
• Transactional filesystem
• Totally ordered broadcast
• Distributed firing squad

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Universality Result

• Assume consensus service
• Assume non-fault tolerant sequential algorithm
  (data object) A
• Then
• There exists a faul-tolerant, wait-free
  distributed implementation A of algorithm A

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Asynchronous Consensus

• Assume 1 stopping failure
• Impossible! FLP 1985
• If consensus algorithm C guarantees agreement and
  validity
• Then
• There exists an execution of C that does not
  terminate

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More Consensus Models

• Synchronous
• f1 rounds if f failures
• Asynchronous plus eventual sychrony
• eventual synchronized clocks
• eventual message delivery bound d
• Consensus terminates
• O((f4)d) after stabilization

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Failure Detectors

• Assume total asynchrony
• Assume failure detector service
• Notifies node i when node j fails
• Eventually
• Allow solving consensus
• Weakest failure-detector?
• Leader-election failure-detector

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Types of Failure Detectors

• All failed nodes suspected?
• All correct nodes not suspected?
• One correct node not suspected?
• Eventually Perfect 1 2
• Eventually Correct 1 3
• Eventual Leader 3

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Byzantine Consensus

• What if some nodes lie?
• Synchronous model
• f stopping failures
• n nodes
• 2f1 n

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Other Consensus Problems

• Weak Consensus
• not feasible
• Weak Byzantine Consensus
• not feasible
• k-set Consensus
• fltk gt feasible
• Approximate Consensus

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Current Research

• Dynamic Networks
• Joins / Leaves / Failures
• Peer-to-Peer networks
• Mobile networks
• Unknown participants
• Large scale networks
• Data warehouses
• Internet sized applications
• Non-blocking, wait-free implementations

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Formal Methods

• IOA Language and compiler
• Simulator
• Proof tools
• Larch Prover
• Daikon invariant detection
• Java code generator (in progress)

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Summary of Results
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Summary of Results
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The End
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Backup Slides
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Models

• Nodes
• bounded / unbounded
• unique ids
• known / unknown participants
• Communication
• Network links
• known / unknown topology
• Shared memory

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Atomic Memory

• Definition
• Motivation
• Algorithms?

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Mutual Exclusion

• Dijkstra
• Dining Philosophers
• Resource Allocation