Solving Sequential Greedy Problems Distributedly with Sub-Logarithmic Energy Cost

TL;DR

It is shown that any problem belonging to the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity [EQUATION], which leads to a polynomial improvement over the state of the art when [EQUATION], e.g., Δ = n∊ for some arbitrarily small ∊ > 0.

Abstract

We study the awake complexity of graph problems that belong to the class O-LOCAL, which includes a subset of problems solvable by sequential greedy algorithms, such as $(Δ+1)$-coloring and maximal independent set. It is known from previous work that, in $n$-node graphs of maximum degree $Δ$, any problem in the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\logΔ+\log^\star n)$. In this paper, we show that any problem belonging to the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\sqrt{\log n}\cdot\log^\star n)$. This leads to a polynomial improvement over the state of the art when $Δ\gg 2^{\sqrt{\log n}}$, e.g., $Δ=n^ε$ for some arbitrarily small $ε>0$. The key ingredient for achieving our results is the computation of a network decomposition, that uses a small-enough number of colors, in sub-logarithmic time in the Sleeping model, which can be of independent interest.

Figures (4)

• Figure 1: The tree used in the proof of Lemma \ref{['lem:mapping-and-function']}. We have $\phi(2)=3$ as the second smallest label of a leaf is 3, and $r(2)=\{2,3,4,8\}$. Similarly, $\phi(4)=7$, and $r(4)=\{4,6,7,8\}$. Note that the lowest common ancestor of the nodes labeled 3 and 7 is the node labeled 4, and indeed $3<4<7$.
• Figure 2: An example of application of \ref{['lem:virtvirt-to-virt']}.
• Figure 3: Sum up of the Clustering Algorithm of \ref{['thm:nd']}. We compute iteratively a clustering of the remaining clusters. Some parts get their final cluster at each iteration, while the other goes back into the loop.
• Figure 4: In (a), we have a distance-2 coloring $c_1$ of the nodes, with $b=3$ and $k=100$. Notice that nodes of degree $\le3$ have added 100 to their colors. In red, each node $u$ is connected to $p_1(u)$ (with a self loop if $p_1(u)=\bot$). In (b), each node $u$ has computed its new color $c_2(u)$ (that is not a proper coloring, it is a decreasing coloring from node to parent) and its new parent $p_2(u)$ (in blue if it differs from $p_1(u)$). The trees formed with a root of degree at least 4 form the new clusters. (Grey) Nodes in a tree of root of degree at most 3 become singleton clusters (dotted tree edges to illustrate that the parent relation is forgotten). These nodes will compute some $ab^2$ coloring.

Theorems & Definitions (24)

• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 6: Barenboim and Maimon barenboimM21
• proof : Sketch of proof
• Lemma 7
• proof
• Lemma 8