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

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

The Lovász Local Lemma • «Bad» events 𝐹 1 , 𝐹 2 , … , 𝐹 𝑜 with Pr 𝐹 𝑗 < 1 mutually independent ⇒ Pr ¬𝐹 1 ∧ ¬𝐹 2 ∧ ⋯ ∧ ¬𝐹 𝑜 > 0 Lovász Local Lemma • Each event is independent of all but 𝑒 other events • Pr 𝐹 𝑗 < 𝑞 where 𝑓𝑞 𝑒 + 1 ≤ 1 ⇒ Pr ¬𝐹 1 ∧ ¬𝐹 2 ∧ ⋯ ∧ ¬𝐹 𝑜 > 0

The Constructive LLL • Mutually independent random variables 𝑌 1 , 𝑌 2 , … , 𝑌 𝑙 • Bad events 𝐹 1 , 𝐹 2 , … , 𝐹 𝑜 • Each event is independent of all but 𝑒 other events 𝑌 1 ∨ ¬𝑌 2 ∧ ¬𝑌 1 ∨ 𝑌 3 ∧ 𝑌 3 ∨ 𝑌 4 𝑌 1 , 𝑌 3 𝑌 3 , 𝑌 4 𝐹 1 𝐹 2 𝐹 3 • Dependency graph 𝐹 2 𝐹 3 𝑌 1 , 𝑌 2 𝐹 1

Distributed Computing

Distributed Computing

Distributed Computing

Distributed Computing

Distributed Computing

The Distributed LLL • Input: dependency graph • Additional input for each node 𝐹 𝑗 : the random variables that 𝐹 𝑗 depends on (and how 𝐹 𝑗 depends on them) • Output of each node 𝐹 𝑗 : an assignment of the variables it depends on such that: 1) it agrees with its neighbours 2) the bad event 𝐹 𝑗 is avoided

Our result • Moser and Tardos (2010): 𝑃 log 2 𝑜 • Chung et al. (2014): 𝑃(log 𝑜) for bounded-degree graphs Ω(log ∗ 𝑜) • Ω(log log 𝑜) (Monte-Carlo, w.h.p.)

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

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

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

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

Reduction from SO to LLL Instance for SO, Output for SO, 3 -regular 3 -regular Instance for SO, Output for SO, 4 -regular 4 -regular Instance for LLL Output for LLL

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!

𝑢 SO algorithm SC algorithm

𝑢 + 1 SO algorithm 𝑢 SO algorithm SC algorithm

𝑢 SO algorithm 𝑢 − 1 SC algorithm

𝑢 SO algorithm 𝑢 − 1 SC algorithm SO algorithm

𝑢 SO algorithm 𝑢 − 1 SC algorithm SO algorithm … … … … 0 SC algorithm

SO algorithm, 𝑢 Pr( failure ) ≤ 𝑞 SC algorithm, SO algorithm, 𝑢 − 1 Pr( failure ) ≤ 𝑞 ′ Pr( failure ) ≤ 𝑟 … … … … SC algorithm, 0 Pr( failure ) ≤ ?

𝑞 ′ = 𝐷 ⋅ 12 𝑞 SO algorithm, 𝑢 Pr( failure ) ≤ 𝑞 SC algorithm, SO algorithm, 𝑢 − 1 Pr( failure ) ≤ 𝑞 ′ Pr( failure ) ≤ 𝑟 … … … … SC algorithm, 0 Pr( failure ) ≤ ?

𝑞 ′ = 𝐷 ⋅ 12 𝑞 w.h.p. SO algorithm, 𝑢 ∈ Θ(log log 𝑜) Pr( failure ) ≤ 𝑞 SC algorithm, SO algorithm, 𝑢 − 1 Pr( failure ) ≤ 𝑞 ′ Pr( failure ) ≤ 𝑟 … … … … SC algorithm 0 1 Pr( failure ) ≤ 10

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

Any Monte-Carlo algorithm for the distributed LLL that gives a correct output 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 output w.h.p. needs Ω(log log 𝑜) rounds. Chang et al. (2016) The randomised time complexity of finding a node 𝑒 -colouring in trees with maximum degree 𝑒 is Θ(log d log 𝑜) , the deterministic complexity is Θ(log d 𝑜) .

Backup Slides

The Constructive LLL • Each 𝐹 𝑗 shares variables with at most 𝑒 other events • Pr 𝐹 𝑗 < 𝑞 where 𝑓𝑞 𝑒 + 1 ≤ 1 ⇒ An assignment of the random variables that avoids all bad events can be found efficiently Moser and Tardos, 2010 • Example: 𝑌 𝑗 binary 𝑌 1 ∨ ¬𝑌 2 ∧ ¬𝑌 1 ∨ 𝑌 3 ∨ ¬𝑌 4 ∧ 𝑌 2 ∨ 𝑌 4 ∧ (¬𝑌 3 ∨ 𝑌 4 ) 𝑒 = 3 , 𝑞 = 1 vbl 𝐹 1 = {𝑌 1 , 𝑌 2 } , vbl 𝐹 2 = {𝑌 1 , 𝑌 3 , 𝑌 4 } , vbl 𝐹 3 = {𝑌 2 , 𝑌 4 } , vbl 𝐹 4 = {𝑌 3 , 𝑌 4 } 4

The Dependency Graph • Nodes: events • Edges: the events share a variable • Example: vbl 𝐹 1 = {𝑌 1 } 𝐹 5 𝐹 4 vbl 𝐹 2 = {𝑌 1 , 𝑌 2 } vbl 𝐹 3 = {𝑌 1 , 𝑌 3 } vbl 𝐹 4 = {𝑌 2 , 𝑌 3 , 𝑌 4 } 𝐹 3 vbl 𝐹 5 = {𝑌 4 } 𝐹 2 • Maximum degree 𝑒 𝐹 1

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)

Reduction from SO to LLL

Reduction from SO to LLL

Reduction from SO to LLL 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7

Reduction from SO to LLL 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7 or

Reduction from SO to LLL 𝐹 6 𝐹 7 𝐹 8 𝐹 5 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7 𝐹 1 𝐹 2 𝐹 3 𝐹 4 or

Reduction from SO to LLL 𝐹 6 𝐹 7 𝐹 8 𝐹 5 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7 𝐹 1 𝐹 2 𝐹 3 𝐹 4 𝑔 4 ≤ 16 or

Reduction from SO to LLL 𝐹 6 𝐹 7 𝐹 8 𝐹 5 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7 𝐹 1 𝐹 2 𝐹 3 𝐹 4 𝑔 4 ≤ 16 𝑞 ⋅ 𝑔 𝑒 ≤ 1 or

Reduction from SO to LLL 𝐹 6 𝐹 7 𝐹 8 𝐹 5 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7 𝐹 1 𝐹 2 𝐹 3 𝐹 4 𝑔 4 ≤ 16 𝑞 ⋅ 𝑔 𝑒 ≤ 1 or 1 16 ⋅ 16 ≤ 1

Reduction from SO to LLL 𝐹 6 𝐹 7 𝐹 8 𝐹 5 𝑌 3 𝑌 5 𝑌 2 𝑌 6 𝑌 1 𝑌 4 𝑌 7 𝐹 1 𝐹 2 𝐹 3 𝐹 4 𝑔 4 ≤ 16 𝑞 ⋅ 𝑔 𝑒 ≤ 1 or 1 16 ⋅ 16 ≤ 1

Reduction from SO to LLL

Reduction from SO to LLL

Reduction from SO to LLL

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