Simple, Fast and Deterministic Gossip and Rumor Spreading Main - - PowerPoint PPT Presentation

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Simple, Fast and Deterministic Gossip and Rumor Spreading Main - - PowerPoint PPT Presentation

Jun 20, 2023 677 likes •1.58k views

Simple, Fast and Deterministic Gossip and Rumor Spreading Main paper by: B. Haeupler, MIT Talk by: Alessandro Dovis, ETH Presentation Outline What is gossip? Applications Basic Algorithms Advanced Algorithms Other

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Simple, Fast and Deterministic Gossip and Rumor Spreading

Main paper by: B. Haeupler, MIT Talk by: Alessandro Dovis, ETH

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Presentation Outline

  • What is gossip?
  • Applications
  • Basic Algorithms
  • Advanced Algorithms
  • Other Results & Current Research
  • Q & A
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What is gossip?

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What is gossip?

Broadcast strategies

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What is gossip?

STRUCTURED

Broadcast strategies

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What is gossip?

Broadcast strategies

STRUCTURED

ISSUES

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What is gossip?

Broadcast strategies

STRUCTURED

ISSUES

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What is gossip?

Broadcast strategies

STRUCTURED

ISSUES

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What is gossip?

Broadcast strategies

STRUCTURED

ISSUES

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What is gossip?

Broadcast strategies

STRUCTURED

ISSUES

?

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What is gossip?

Broadcast strategies

STRUCTURED FLOODING

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What is gossip?

Broadcast strategies

FLOODING

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What is gossip?

Broadcast strategies

FLOODING ISSUES

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What is gossip?

Broadcast strategies

FLOODING ISSUES

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What is gossip?

Broadcast strategies

FLOODING ISSUES

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What is gossip?

Broadcast strategies

FLOODING ISSUES COMPLEXITY

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What is gossip?

Broadcast strategies

FLOODING ISSUES COMPLEXITY

TIME: O(D) MESSAGE: O(m) (or O(D*m))

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What is gossip?

STRUCTURED FLOODING

Broadcast strategies

GOSSIP

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What is gossip?

GOSSIP

Broadcast strategies

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What is gossip?

GOSSIP

Broadcast strategies

  • choice of active edge
  • randomized vs. deterministic
  • time complexity
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What is gossip?

GOSSIP

Broadcast strategies

  • choice of active edge
  • randomized vs. deterministic
  • time complexity

?

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Applications

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Applications

DATABASE REPLICATION

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Applications

DATABASE REPLICATION

1. Direct mail 2. Anti-entropy 3. Rumor mongering

  • PUSH
  • PULL
  • PUSH-PULL
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Applications

RESOURCE DISCOVERY

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Applications

RESOURCE DISCOVERY DISTRIBUTED COMPUTATION

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Applications

RESOURCE DISCOVERY DISTRIBUTED COMPUTATION NODE FAILURE DETECTION

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Basic algorithms

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Naive solution: simulated flooding

R[v] = rumor of v REPEAT D times R’ = ∅ FOR t = 1 to ∆ exchange rumors in R[v] with n[v][t] add all received rumors to R’ R[v] = R[v] ∪ R’ 1 3 2 4

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Naive solution: simulated flooding

R[v] = rumor of v REPEAT D times R’ = ∅ FOR t = 1 to ∆ exchange rumors in R[v] with n[v][t] add all received rumors to R’ R[v] = R[v] ∪ R’ 1 3 2 4

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Naive solution: simulated flooding

R[v] = rumor of v REPEAT D times R’ = ∅ FOR t = 1 to ∆ exchange rumors in R[v] with n[v][t] add all received rumors to R’ R[v] = R[v] ∪ R’ 1 3 2 4

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Naive solution: simulated flooding

R[v] = rumor of v REPEAT D times R’ = ∅ FOR t = 1 to ∆ exchange rumors in R[v] with n[v][t] add all received rumors to R’ R[v] = R[v] ∪ R’ 1 3 2 4

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Naive solution: simulated flooding

COMPLEXITY

TIME: O(∆*D) MESSAGE: O(m*D)

1 3 2 4

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Classic solution: uniform gossip

REPEAT ? times choose a uniformly random neighbor PUSH-PULL rumors in R[v] with n[v][t] add received rumors to R[v]

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Classic solution: uniform gossip

REPEAT ? times choose a uniformly random neighbor PUSH-PULL rumors in R[v] with n[v][t] add received rumors to R[v]

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Classic solution: uniform gossip

REPEAT ? times choose a uniformly random neighbor PUSH-PULL rumors in R[v] with n[v][t] add received rumors to R[v]

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Classic solution: uniform gossip

REPEAT ? times choose a uniformly random neighbor PUSH-PULL rumors in R[v] with n[v][t] add received rumors to R[v]

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Classic solution: uniform gossip

GIAKKOUPIS ‘12

TIME: O(log n / φ)

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Classic solution: uniform gossip

GIAKKOUPIS ‘12

TIME: O(log n / φ)

φ ?!

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Graph conductance

VOLUME CUT CONDUCTANCE GRAPH CONDUCTANCE

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Graph conductance

It measures how much the network is bottlenecked

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Graph conductance

It measures how much the network is bottlenecked

θ(1/n)

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Graph conductance

It measures how much the network is bottlenecked

θ(1/n) θ(1)

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Graph conductance

It measures how much the network is bottlenecked

θ(1/n) θ(1) θ(1/n^2)

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Advanced algorithms

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Conductance Independent results

NEIGHBOR EXCHANGE PROBLEM

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Conductance Independent results

NEIGHBOR EXCHANGE PROBLEM COMMON IDEA Solve NEP + compose it D times

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Conductance Independent results

NEIGHBOR EXCHANGE PROBLEM COMMON IDEA Solve NEP + compose it D times RESULTS (global) RANDOMIZED O(D*log^3 n) DETERMINISTIC O(D*log n + log^2 n)

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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM

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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM QUALITATIVE IDEA Run uniform gossip for a while… + … remove some edges … + do it again

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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM

Superstep(G, τ):

  • 1. UniformGossip algorithm with respect to F

[i] for τ rounds. K[i]: order of the random activated edges

  • 2. UniformGossip Krev[i], the reverse

process of the one realized in Step 1

  • 3. (Pruning) Set of pruned directed edges P

[i] = (u, w) : u received from v

  • 4. Set F[i+1] := F[i] − P[i] and i := i + 1
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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM

Superstep(G, τ):

  • 1. UniformGossip algorithm with respect to F

[i] for τ rounds. K[i]: order of the random activated edges

  • 2. UniformGossip Krev[i], the reverse

process of the one realized in Step 1

  • 3. (Pruning) Set of pruned directed edges P

[i] = (u, w) : u received from v

  • 4. Set F[i+1] := F[i] − P[i] and i := i + 1
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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM

Superstep(G, τ):

  • 1. UniformGossip algorithm with respect to F

[i] for τ rounds. K[i]: order of the random activated edges

  • 2. UniformGossip Krev[i], the reverse

process of the one realized in Step 1

  • 3. (Pruning) Set of pruned directed edges P

[i] = (u, w) : u received from v

  • 4. Set F[i+1] := F[i] − P[i] and i := i + 1
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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM

Superstep(G, τ):

  • 1. UniformGossip algorithm with respect to F

[i] for τ rounds. K[i]: order of the random activated edges

  • 2. UniformGossip Krev[i], the reverse

process of the one realized in Step 1

  • 3. (Pruning) Set of pruned directed edges P

[i] = (u, w) : u received from v

  • 4. Set F[i+1] := F[i] − P[i] and i := i + 1
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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM

Superstep(G, τ):

  • 1. UniformGossip algorithm with respect to F

[i] for τ rounds. K[i]: order of the random activated edges

  • 2. UniformGossip Krev[i], the reverse

process of the one realized in Step 1

  • 3. (Pruning) Set of pruned directed edges P

[i] = (u, w) : u received from v

  • 4. Set F[i+1] := F[i] − P[i] and i := i + 1
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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM RESULT If τ = θ(log^2 n), we need only θ(log n) cycles

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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM RESULT If τ = θ(log^2 n), we need only θ(log n) cycles

NEP: θ(log^3 n)

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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM RESULT If τ = θ(log^2 n), we need only θ(log n) cycles PROOF

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NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM RESULT If τ = θ(log^2 n), we need only θ(log n) cycles PROOF

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φ=Ω(1/log n)

NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM RESULT If τ = θ(log^2 n), we need only θ(log n) cycles PROOF

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φ=Ω(1/log n)

NEP: randomized (1)

NEIGHBOR EXCHANGE PROBLEM RESULT If τ = θ(log^2 n), we need only θ(log n) cycles PROOF

F ← F/2

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NEP: randomized (2)

NEIGHBOR EXCHANGE PROBLEM

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NEP: randomized (2)

NEIGHBOR EXCHANGE PROBLEM QUALITATIVE IDEA Run simulated flooding for a while… + … add some edges … + do it again

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NEP: randomized (2)

NEIGHBOR EXCHANGE PROBLEM

R[v] = v WHILE Γ[v]\R[v] = ∅ pick Θ(log^2 n) random edges in Γ[v]\R[v] d = Θ(log^2 n); E’ = all newly picked edges Flood in R[v] along E’-edges for d-hops add all received rumors to R[v]

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NEP: randomized (2)

NEIGHBOR EXCHANGE PROBLEM

R[v] = v WHILE Γ[v]\R[v] = ∅ pick Θ(log^2 n) random edges in Γ[v]\R[v] d = Θ(log^2 n); E’ = all newly picked edges Flood in R[v] along E’-edges for d-hops add all received rumors to R[v]

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NEP: randomized (2)

NEIGHBOR EXCHANGE PROBLEM

R[v] = v WHILE Γ[v]\R[v] = ∅ pick Θ(log^2 n) random edges in Γ[v]\R[v] d = Θ(log^2 n); E’ = all newly picked edges Flood in R[v] along E’-edges for d-hops add all received rumors to R[v]

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NEP: randomized (2)

NEIGHBOR EXCHANGE PROBLEM

R[v] = v WHILE Γ[v]\R[v] = ∅ pick Θ(log^2 n) random edges in Γ[v]\R[v] d = Θ(log^2 n); E’ = all newly picked edges Flood in R[v] along E’-edges for d-hops add all received rumors to R[v]

O(log^6 n)

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NEP: deterministic

NEIGHBOR EXCHANGE PROBLEM

R[v] = v WHILE Γ[v]\R[v] = ∅ pick Θ(log^2 n) random edges in Γ[v]\R[v] d = Θ(log^2 n); E’ = all newly picked edges Flood in R[v] along E’-edges for d-hops add all received rumors to R[v]

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NEP: deterministic

NEIGHBOR EXCHANGE PROBLEM

R[v] = v WHILE Γ[v]\R[v] = ∅ arbitrarily pick one edge in Γ[v]\R[v] d = 2log n; E’ = all newly picked edges Flood in R[v] along E’-edges for d-hops add all received rumors to R[v]

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NEP: deterministic

NEIGHBOR EXCHANGE PROBLEM RESULT We need only log n cycles

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NEP: deterministic

NEIGHBOR EXCHANGE PROBLEM RESULT We need only log n cycles

NEP: 2log^3 n

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NEP: deterministic

RESULT We need only log n cycles PROOF

  • in cycle i, vertex v creates a binomial i-tree

and floods for 2i hops

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NEP: deterministic

RESULT We need only log n cycles PROOF

  • in cycle i, vertex v creates a binomial i-tree

and floods for 2i hops

  • in each cycle (at least) all the nodes in the

binomial tree get the information from the root

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NEP: deterministic

RESULT We need only log n cycles PROOF

  • in cycle i, vertex v creates a binomial i-tree

and floods for 2i hops

  • in each cycle (at least) all the nodes in the

binomial tree get the information from the root

  • if 2 neighbours are strangers, their current

trees must be disjoint

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NEP: deterministic

RESULT We need only log n cycles PROOF

  • in cycle i, vertex v creates a binomial i-tree

and floods for 2i hops

  • in each cycle (at least) all the nodes in the

binomial tree get the information from the root

  • if 2 neighbours are strangers, their current

trees must be disjoint

  • a tree at step i contains 2^i nodes
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NEP: deterministic

RESULT We need only log n cycles PROOF

  • in cycle i, vertex v creates a binomial i-tree

and floods for 2i hops

  • in each cycle (at least) all the nodes in the

binomial tree get the information from the root

  • if 2 neighbours are strangers, their current

trees must be disjoint

  • a tree at step i contains 2^i nodes

at most log n cycles

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NEP: faster deterministic

Can we exploit the structure of the binomial tree?

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NEP: faster deterministic

Can we exploit the structure of the binomial tree?

PUSH in inverse order + PULL in order + symmetric PULL & PUSH

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NEP: faster deterministic

Can we exploit the structure of the binomial tree?

PUSH in inverse order + PULL in order + symmetric PULL & PUSH

NEP: 2log n(log n + 1)

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NEP composition

NEIGHBOR EXCHANGE PROBLEM NAIVE COMPOSITION

O(D*log^2 n)

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NEP composition

NEIGHBOR EXCHANGE PROBLEM NAIVE COMPOSITION

O(D*log^2 n)

REUSING TREE

O(D*log n + log^2 n)

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Other results & Current research

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Other results & Current research

SPANNERS & HEREDITARY DENSITY

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Other results & Current research

SPANNERS & HEREDITARY DENSITY ROBUSTNESS & ASYMMETRY

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Other results & Current research

MAXIMUM MESSAGE SIZE

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Other results & Current research

MAXIMUM MESSAGE SIZE NEP LOWER BOUND

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References

  • Simple, Fast, and Deterministic Gossip and Rumor Spreading,
  • B. Haeupler
  • Global Computation in a Poorly Connected World, K. Censor-Hillel

et al.

  • Tight bounds for rumor spreading in graphs of a given conductance, G. Giakkoupis
  • Epidemic Algorithms for Replicated Database Maintainance, A. Demers et al.
  • Resource Discovery in Distributed Networks, M. Harchol-Balter et al.
  • Gossip Algorithms: Design, Analysis and Applications, S. Boyd et al.
  • A Gossip-Style Failure Detection Service, R. van Renesse et al.
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Q & A