A Lower Bound for the Distributed Lovsz Local Lemma Sebastian - - PowerPoint PPT Presentation

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A Lower Bound for the Distributed Lovsz Local Lemma Sebastian - - PowerPoint PPT Presentation

Mar 14, 2024 153 likes •624 views

A Lower Bound for the Distributed Lovsz Local Lemma Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiinen, Joel Rybicki, Jukka Suomela, Jara Uitto Aalto University, Comerge AG, ETH Zurich, Tel Aviv University The

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Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, Jara Uitto

A Lower Bound for the Distributed Lovász Local Lemma

Aalto University, Comerge AG, ETH Zurich, Tel Aviv University

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The Lovász Local Lemma

  • «Bad» events 𝐹1, 𝐹2, … , 𝐹𝑜 with Pr 𝐹𝑗 < 1

⇒ Pr ¬𝐹1 ∧ ¬𝐹2 ∧ ⋯ ∧ ¬𝐹𝑜 > 0

  • Each event is independent of all but 𝑒 other events
  • Pr 𝐹𝑗 < 𝑞 where 𝑓𝑞 𝑒 + 1 ≤ 1

⇒ Pr ¬𝐹1 ∧ ¬𝐹2 ∧ ⋯ ∧ ¬𝐹𝑜 > 0 mutually independent Lovász Local Lemma

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The Constructive LLL

  • Mutually independent random variables 𝑌1, 𝑌2, … , 𝑌𝑙
  • Bad events 𝐹1, 𝐹2, … , 𝐹𝑜
  • Each event is independent of all but 𝑒 other events

𝑌1 ∨ ¬𝑌2 ∧ ¬𝑌1 ∨ 𝑌3 ∧ 𝑌3 ∨ 𝑌4

  • Dependency graph

𝐹1 𝐹3 𝐹2 𝐹1 𝐹2 𝐹3 𝑌1, 𝑌2 𝑌1, 𝑌3 𝑌3, 𝑌4

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Distributed Computing

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Distributed Computing

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Distributed Computing

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Distributed Computing

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Distributed Computing

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The Distributed LLL

  • Input: dependency graph
  • Additional input for each node 𝐹𝑗: the random variables that

𝐹𝑗 depends on (and how 𝐹𝑗 depends on them)

  • Output of each node 𝐹𝑗: an assignment of the variables it

depends on such that: 1) it agrees with its neighbours 2) the bad event 𝐹𝑗 is avoided

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Our result

  • Moser and Tardos (2010): 𝑃 log2 𝑜
  • Chung et al. (2014): 𝑃(log 𝑜) for bounded-degree graphs

Ω(log∗ 𝑜)

  • Ω(log log 𝑜) (Monte-Carlo, w.h.p.)
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Sinkless Orientation

  • Input: edge 𝑒-coloured, 𝑒-regular graph
  • Output of each node: non-conflicting orientations of the

incident edges such that the node itself is not a sink

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Sinkless Orientation

  • Input: edge 𝑒-coloured, 𝑒-regular graph
  • Output of each node: non-conflicting orientations of the

incident edges such that the node itself is not a sink

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Sinkless Orientation

  • Input: edge 𝑒-coloured, 𝑒-regular graph
  • Output of each node: non-conflicting orientations of the

incident edges such that the node itself is not a sink

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Sinkless Orientation

  • Input: edge 𝑒-coloured, 𝑒-regular graph
  • Output of each node: non-conflicting orientations of the

incident edges such that the node itself is not a sink

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Reduction from SO to LLL

Instance for SO, 3-regular Output for SO, 4-regular Instance for LLL Output for SO, 3-regular Instance for SO, 4-regular Output for LLL

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Sinkless Colouring

  • Input: edge 𝑒-coloured, 𝑒-regular graph
  • Output of each node: one of the 𝑒 colours such that no

forbidden configuration occurs Forbidden! Fine!

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SO algorithm 𝑢 SC algorithm

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SO algorithm 𝑢 SO algorithm 𝑢 + 1 SC algorithm

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SO algorithm 𝑢 𝑢 − 1 SC algorithm

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SO algorithm 𝑢 𝑢 − 1 SO algorithm SC algorithm

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SO algorithm 𝑢 𝑢 − 1 … SO algorithm SC algorithm SC algorithm … … …

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SO algorithm, Pr(failure) ≤ 𝑞 𝑢 𝑢 − 1 … SO algorithm, Pr(failure) ≤ 𝑞′ SC algorithm, Pr(failure) ≤ ? SC algorithm, Pr(failure) ≤ 𝑟 … … …

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SO algorithm, Pr(failure) ≤ 𝑞 𝑢 𝑢 − 1 … SO algorithm, Pr(failure) ≤ 𝑞′ SC algorithm, Pr(failure) ≤ ? SC algorithm, Pr(failure) ≤ 𝑟 … … … 𝑞′ = 𝐷 ⋅ 12 𝑞

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SO algorithm, Pr(failure) ≤ 𝑞 𝑢 ∈ Θ(log log 𝑜) 𝑢 − 1 … SO algorithm, Pr(failure) ≤ 𝑞′ SC algorithm SC algorithm, Pr(failure) ≤ 𝑟 … … … 𝑞′ = 𝐷 ⋅ 12 𝑞 w.h.p. Pr(failure) ≤

1 10

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Any Monte-Carlo algorithm for the distributed LLL that gives a correct

  • utput w.h.p. needs Ω(log log 𝑜) rounds.
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Any Monte-Carlo algorithm for the distributed LLL that gives a correct

  • utput w.h.p. needs Ω(log log 𝑜) rounds.

Any Monte-Carlo algorithm for finding a node 𝑒-colouring in 𝑒-regular, bipartite, Ω(log 𝑜)-girth graphs that gives a correct

  • utput w.h.p. needs Ω(log log 𝑜) rounds.

The randomised time complexity of finding a node 𝑒-colouring in trees with maximum degree 𝑒 is Θ(logd log 𝑜), the deterministic complexity is Θ(logd 𝑜). Chang et al. (2016)

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Backup Slides

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The Constructive LLL

  • Each 𝐹𝑗 shares variables with at most 𝑒 other events
  • Pr 𝐹𝑗 < 𝑞 where 𝑓𝑞 𝑒 + 1 ≤ 1

⇒ An assignment of the random variables that avoids all bad events can be found efficiently

  • Example: 𝑌𝑗 binary

𝑌1 ∨ ¬𝑌2 ∧ ¬𝑌1 ∨ 𝑌3 ∨ ¬𝑌4 ∧ 𝑌2 ∨ 𝑌4 ∧ (¬𝑌3 ∨ 𝑌4) vbl 𝐹1 = {𝑌1, 𝑌2}, vbl 𝐹2 = {𝑌1, 𝑌3, 𝑌4}, vbl 𝐹3 = {𝑌2, 𝑌4}, vbl 𝐹4 = {𝑌3, 𝑌4} Moser and Tardos, 2010 𝑒 = 3, 𝑞 = 1 4

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The Dependency Graph

  • Nodes: events
  • Edges: the events share a variable
  • Example:

vbl 𝐹1 = {𝑌1} vbl 𝐹2 = {𝑌1, 𝑌2} vbl 𝐹3 = {𝑌1, 𝑌3} vbl 𝐹4 = {𝑌2, 𝑌3, 𝑌4} vbl 𝐹5 = {𝑌4}

  • Maximum degree 𝑒

𝐹2 𝐹4 𝐹3 𝐹5 𝐹1

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Distributed Computing

  • Input: simple undirected graph (+ some task-specific input)
  • Nodes: computational entities
  • Edges: communication channels
  • Synchronous rounds
  • In each round, each node ...

1) sends an arbitrarily large message to each neighbour 2) receives sent messages 3) performs local computations

  • Each node has to output a correct answer
  • Time complexity: number of rounds (worst-case input)
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Reduction from SO to LLL

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Reduction from SO to LLL

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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

  • r
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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

  • r

𝐹3 𝐹2 𝐹4 𝐹5 𝐹1 𝐹7 𝐹6 𝐹8

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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

  • r

𝐹3 𝐹2 𝐹4 𝐹5 𝐹1 𝐹7 𝐹6 𝐹8 𝑔 4 ≤ 16

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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

  • r

𝐹3 𝐹2 𝐹4 𝐹5 𝐹1 𝐹7 𝐹6 𝐹8 𝑔 4 ≤ 16 𝑞 ⋅ 𝑔 𝑒 ≤ 1

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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

  • r

𝐹3 𝐹2 𝐹4 𝐹5 𝐹1 𝐹7 𝐹6 𝐹8 𝑔 4 ≤ 16 𝑞 ⋅ 𝑔 𝑒 ≤ 1 1 16 ⋅ 16 ≤ 1

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Reduction from SO to LLL

𝑌2 𝑌1 𝑌4 𝑌3 𝑌6 𝑌5 𝑌7

  • r

𝐹3 𝐹2 𝐹4 𝐹5 𝐹1 𝐹7 𝐹6 𝐹8 𝑔 4 ≤ 16 𝑞 ⋅ 𝑔 𝑒 ≤ 1 1 16 ⋅ 16 ≤ 1

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Reduction from SO to LLL

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Reduction from SO to LLL

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Reduction from SO to LLL

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Technicalities

  • Monte-Carlo algorithms, w.h.p.
  • Girth ≥ 2𝑢 + 1
  • 𝑒 = 3
  • Failure probability 𝑞𝑔 𝑤 resp. 𝑞𝑔 𝑓