Distributed Lower Bounds for Ruling Sets

Given a graph G=(V, E), an ( α, β) -ruling set is a subset S ⊆ V such that the distance between any two vertices in S is at least α, and the distance between any vertex in V and the closest vertex in S is at most β. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a ( 2, β) - ruling set (and hence also any ( α, β) -ruling set with ) in the LOCAL model of distributed computing, we show the following, where n denotes the number of vertices, Δ the maximum degree, and c is some universal constant independent of n and Δ. · Any deterministic algorithm requires Ω(min{[(logΔ)/(β log log Δ)], log Δ n}) rounds, for all β ≤ c·min{√{[(log Δ)/(log log Δ)]}, log Δ n}. By optimizing Δ, this implies a deterministic lower bound of Ω(√{[log n/(β log log n)]}) for all β ≤ c 3 √{[log n/log log n]}. ·Any randomized algorithm requires Ω(min{[(log Δ)/(β log log Δ)], log log n}) rounds, for all β ≤ c·min{√{[(log Δ)/(log log Δ)]}, log log n}. By optimizing Δ, this implies a randomized lower bound of Ω(√{[log log n/(βlog log log n)]}) for all β ≤ c 3 √{[log log n/log log log n]}. For, this improves on the previously best lower bound of Ω(log*n) rounds that follows from the 30-year-old bounds of Linial [FOCS'87] and Naor [J. Disc. Math. '91] (resp. Ω(1) rounds if β ∈ ω(log*n)). For β = 1, i. e. , for the problem of computing a maximal independent set (which is nothing else than a (2, 1)-ruling set), our results improve on the previously best lower bound of Ω(log*n) on trees, as our bounds already hold on trees.

Authors

• Alkida Balliu
• Sebastian Brandt 0002
• Dennis Olivetti

Keywords

• Computational modeling
• Complexity theory
• Upper bound
• Distributed computing
• Transforms
• Task analysis
• Indexes
• Lower Bound
• Deterministic
• Maximum Degree
• Maximum Independent Set
• State Of The Art
• Color Space
• Problem Description
• Problem Of Finding
• Line Graph
• Disjunction
• Family Problems
• Number Of Labels
• Sequence Of Problems
• Automatic Sequencer
• Maximum Matching
• Output Label
• Input Graph
• Regular Graphs
• Message Size
• Polylogarithmic
• Incident Edges
• Random Bits
• Interesting Problem
• Time Complexity
• Distributed Algorithm
• Ruling set
• maximal independent set
• distributed graph algorithms
• lower bounds