Synchronous counting and computational algorithm design Danny Dolev - - PowerPoint PPT Presentation

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Synchronous counting and computational algorithm design Danny Dolev - - PowerPoint PPT Presentation

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*010 *020 *011 *002 *221 *110 *121 Synchronous counting and computational algorithm design Danny Dolev Christoph Lenzen Hebrew University of Jerusalem MIT Janne H. Korhonen Joel Rybicki Jukka Suomela University of Helsinki &

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Synchronous counting and computational algorithm design

Janne H. Korhonen Joel Rybicki Jukka Suomela

University of Helsinki & HIIT

November 16, 2013 SSS 2013, Osaka, Japan

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Danny Dolev

Hebrew University of Jerusalem

Christoph Lenzen

MIT

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What is this talk about?

Developing compact fault-tolerant algorithms for a consensus-like problem using computational techniques.

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Algorithm design

Ask the computer scientist: “Is there an algorithm A for problem P?”

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Algorithm design

Ask the computer scientist: “Is there an algorithm A for problem P?”

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Computational algorithm design

Ask the computer: “Is there an algorithm A for problem P?”

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Verification vs synthesis

Verification: “Check that given A satisfies the specification S.” Synthesis: “Construct an A that satisfies a specification S.”

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Searching for algorithms

How to do a computer search? Intuitively, the task seems very difficult.

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An inductive approach

“Computers are good at boring calculations. People are good at generalizing.”

  • 1. Solve a difficult base case using computers
  • 2. Construct a general solution using the base case
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Synchronous counting

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The model

  • n processors
  • s states
  • arbitrary initial state

1 2 3

4

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The model

  • n processors
  • s states
  • arbitrary initial state

Synchronous step:

  • 1. send state to all neighbors
  • 2. update state

1 2 3

4

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The model

  • n processors
  • s states
  • arbitrary initial state

Synchronous step:

  • 1. send state to all neighbors
  • 2. update state

algorithm = transition function 1 2 3

4

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Self-stabilizing counting

Stabilization Counting

4 3 2 1

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Self-stabilizing counting

A simple algorithm solves the problem

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Self-stabilizing counting

Solution: Follow the leader.

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Self-stabilizing counting

Solution: Follow the leader.

4 3 2 1

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Self-stabilizing counting

Solution: Follow the leader.

4 3 2 1

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Self-stabilizing counting

Solution: Follow the leader.

4 3 2 1

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Tolerating Byzantine failures

Assume that at most f nodes may be Byzantine.

1

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Tolerating Byzantine failures

Assume that at most f nodes may be Byzantine. 1

2

3 4

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Tolerating Byzantine failures

Assume that at most f nodes may be Byzantine. 1 2

3

4

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Tolerating Byzantine failures

Assume that at most f nodes may be Byzantine. 1 2 3

4

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Tolerating Byzantine failures

can send different messages to non-faulty nodes! 1 2 3

4

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Tolerating Byzantine failures

can send different messages to non-faulty nodes! Note: Easy if self-stabilization is not required! 1 2 3

4

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Fault-tolerant counting

Stabilization Counting

4 3 2 1

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The model with failures

  • n processors
  • s states
  • arbitrary initial state
  • at most f Byzantine nodes

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4

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Some basic facts

  • How many states do we need?
  • s ≥ 2
  • How many faults can we tolerate?
  • f < n/3
  • How fast can we stabilize?
  • t > f

Pease et al., 1980 Fischer & Lynch, 1982

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Solving synchronous counting

Deterministic solutions with large s known for similar problems (e.g. D. Dolev & Hoch, 2007) Are there deterministic algorithms with small s and t? Focus on the first non-trivial case f = 1 Randomized solutions for counting with small s and large t in expectation (e.g. Shlomi Dolev’s book) Our work:

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Generalizing from a base case

For any fixed s, f and t:

⇒

There is an algorithm A for n nodes There is an algorithm B for n+1 nodes with same s, f and t

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Finding an algorithm

The size of the search space is sb where b = nsn. n = 4 s = 2 264 ≈ 1019 parameters search space

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Finding an algorithm

The size of the search space is sb where b = nsn. n = 4 s = 2 n = 4 s = 3 264 ≈ 1019 3324 ≈ 10154 parameters search space We need a clever way to do the search!

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The high-level idea

  • Express the existence of an algorithm as a finite

combinatorial problem

  • Solve a base case that implies a general solution
  • SAT solvers solve the decision problem
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SAT solving

Given a propositional formula Ψ, does there exist a satisfying variable assignment? Problem: Example 1: (x1 ∨ ¬x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

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SAT solving

Given a propositional formula Ψ, does there exist a satisfying variable assignment? Problem: Example 1:

Satisfiable! x1 = 0 x2 = 0 x3 = 1 (x1 ∨ ¬x2 ∨ x3) ∧ (¬x1 ∨ ¬x3)

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SAT solving

  • NP-hard
  • Surprisingly fast in practice
  • Complete: proves YES and NO instances
  • Several solvers available
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Verification is easy

  • Let F be a set of faulty nodes, |F| ≤ f
  • Construct a state graph GF from A:

Nodes = actual states Edges = possible state transitions

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execution = walk

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Even

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Odd

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Easy to reason about graphs!

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deadlock = loop

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livelock = cycle

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Verification is easy

Every GF is good no deadlocks GF is loopless

⇔

stabilization All nodes have a path to 0

⇔

counting {0,1} is the only cycle

⇔

A is correct

⇔

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From verification to synthesis

xi,u,s Ai(u) = s

The encoding uses the following variables:

eq,r

exists

(q, r) pq,r q r xi,u,s eq,r pq,r ⇔ ⇔

edge

⇔

path exists

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Main results, f = 1

  • lower bound: no 2-state algorithm
  • upper bound: 3 states suffice

If 4 ≤ n ≤ 5: If n ≥ 6:

  • 2 states always suffice
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Summary

  • We have algorithms that use the optimal number
  • f states for any n and f = 1
  • Computational techniques useful in design of

fault-tolerant algorithms

  • Solve a base case using computers; let people

generalize

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Summary

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Thanks!

  • We have algorithms that use the optimal number
  • f states for any n and f = 1
  • Computational techniques useful in design of

fault-tolerant algorithms

  • Solve a base case using computers; let people

generalize