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Given a graph $G = (V,E)$, an $(alpha, beta)$-ruling set is a subset $S subseteq V$ such that the distance between any two vertices in $S$ is at least $alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is at most $beta$. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a $(2, beta)$-ruling set in the LOCAL model, we show the following, where $n$ denotes the number of vertices, $Delta$ the maximum degree, and $c$ is some universal constant independent of $n$ and $Delta$. $bullet$ Any deterministic algorithm requires $Omegaleft(min left{ frac{log Delta}{beta log log Delta} , log_Delta n right} right)$ rounds, for all $beta le c cdot minleft{ sqrt{frac{log Delta}{log log Delta}} , log_Delta n right}$. By optimizing $Delta$, this implies a deterministic lower bound of $Omegaleft(sqrt{frac{log n}{beta log log n}}right)$ for all $beta le c sqrt[3]{frac{log n}{log log n}}$. $bullet$ Any randomized algorithm requires $Omegaleft(min left{ frac{log Delta}{beta log log Delta} , log_Delta log n right} right)$ rounds, for all $beta le c cdot minleft{ sqrt{frac{log Delta}{log log Delta}} , log_Delta log n right}$. By optimizing $Delta$, this implies a randomized lower bound of $Omegaleft(sqrt{frac{log log n}{beta log log log n}}right)$ for all $beta le c sqrt[3]{frac{log log n}{log log log n}}$. For $beta > 1$, this improves on the previously best lower bound of $Omega(log^* n)$ rounds that follows from the 30-year-old bounds of Linial [FOCS87] and Naor [J.Disc.Math.91]. For $beta = 1$, i.e., for the problem of computing a maximal independent set, our results improve on the previously best lower bound of $Omega(log^* n)$ on trees, as our bounds already hold on trees.
Recently, Balliu, Brandt, and Olivetti [FOCS 20] showed the first $omega(log^* n)$ lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking
There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(Delta + log^* n)$ communication rounds; here $n$ is the number of nodes and $Delta$ is the maximum degree. The lower bound by Linial (1987, 1992)
In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth constraints,
In this paper, we study lower bounds for randomized solutions to the maximal independent set (MIS) and connected dominating set (CDS) problems in the dual graph model of radio networks---a generalization of the standard graph-based model that now inc
We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassens algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds