Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Round Elimination via Self-Reduction: Closing Gaps for Distributed Maximal Matching

Seri Khoury, Aaron Schild

TL;DR

This paper establishes a tight randomized lower bound for distributed maximal matching on Δ-ary trees, showing that any LOCAL algorithm requires Ω(min{ log Δ, sqrt(log n) }) rounds to achieve a nontrivial success probability. The authors introduce a novel self-reduction round-elimination framework based on a new metric, vertex survival probability, and analyze the algorithm’s behavior on edge-local inputs called r-flowers and r-neighborhoods. Central to the argument is a careful decomposition into near-deterministic local structures, a dominance-direction analysis, and a refined handling of a continuum of parameter choices to upgrade the bound from Ω(log log Δ) to Ω(min{ log Δ, sqrt(log n) }). The results imply near-optimal Δ-Dependent upper bounds for MM and MIS in wide Δ ranges, and reveal a fundamental separation between MIS and MM in trees, strengthening the understanding of distributed symmetry-breaking problems in sparse networks.

Abstract

In this work, we present an $Ω\left(\min\{\log Δ, \sqrt{\log n}\}\right)$ lower bound for Maximal Matching (MM) in $Δ$-ary trees against randomized algorithms. By a folklore reduction, the same lower bound applies to Maximal Independent Set (MIS), albeit not in trees. As a function of $n$, this is the first advancement in our understanding of the randomized complexity of the two problems in more than two decades. As a function of $Δ$, this shows that the current upper bounds are optimal for a wide range of $Δ\in 2^{O(\sqrt{\log n})}$, answering an open question by Balliu, Brandt, Hirvonen, Olivetti, Rabie, and Suomela [FOCS'19, JACM'21]. Moreover, our result implies a surprising and counterintuitive separation between MIS and MM in trees, as it was very recently shown that MIS in trees can be solved in $o(\sqrt{\log n})$ rounds. While MIS can be used to find an MM in general graphs, the reduction does not preserve the tree structure when applied to trees. Our separation shows that this is not an artifact of the reduction, but a fundamental difference between the two problems in trees. This also implies that MIS is strictly harder in general graphs compared to trees.

Round Elimination via Self-Reduction: Closing Gaps for Distributed Maximal Matching

TL;DR

This paper establishes a tight randomized lower bound for distributed maximal matching on Δ-ary trees, showing that any LOCAL algorithm requires Ω(min{ log Δ, sqrt(log n) }) rounds to achieve a nontrivial success probability. The authors introduce a novel self-reduction round-elimination framework based on a new metric, vertex survival probability, and analyze the algorithm’s behavior on edge-local inputs called r-flowers and r-neighborhoods. Central to the argument is a careful decomposition into near-deterministic local structures, a dominance-direction analysis, and a refined handling of a continuum of parameter choices to upgrade the bound from Ω(log log Δ) to Ω(min{ log Δ, sqrt(log n) }). The results imply near-optimal Δ-Dependent upper bounds for MM and MIS in wide Δ ranges, and reveal a fundamental separation between MIS and MM in trees, strengthening the understanding of distributed symmetry-breaking problems in sparse networks.

Abstract

In this work, we present an lower bound for Maximal Matching (MM) in -ary trees against randomized algorithms. By a folklore reduction, the same lower bound applies to Maximal Independent Set (MIS), albeit not in trees. As a function of , this is the first advancement in our understanding of the randomized complexity of the two problems in more than two decades. As a function of , this shows that the current upper bounds are optimal for a wide range of , answering an open question by Balliu, Brandt, Hirvonen, Olivetti, Rabie, and Suomela [FOCS'19, JACM'21]. Moreover, our result implies a surprising and counterintuitive separation between MIS and MM in trees, as it was very recently shown that MIS in trees can be solved in rounds. While MIS can be used to find an MM in general graphs, the reduction does not preserve the tree structure when applied to trees. Our separation shows that this is not an artifact of the reduction, but a fundamental difference between the two problems in trees. This also implies that MIS is strictly harder in general graphs compared to trees.

Paper Structure

This paper contains 56 sections, 53 theorems, 188 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our Contribution:
  3. Implications:
  4. Roadmap:
  5. Technical Overview
  6. Our Technique:
  7. Vertex Survival Probability:
  8. Randomness and edge neighborhoods:
  9. Round Elimination via Self-Reduction
  10. Strategy: Determinism on $r$-Flowers and $r$-Neighborhoods
  11. $\Delta$-ary tree structures: $r$-neighborhoods and $r$-flowers:
  12. Determinism of $f$ on $r$-flowers:
  13. Determinism of $f$ on $r$-neighborhoods:
  14. The behaviour of $f$ on $(r-1)$-flowers and $(r-1)$-neighborhoods:
  15. Warmup: An $\Omega(\log\log\Delta)$-Round Lower Bound
  16. ...and 41 more sections

Key Result

Theorem 1

Any $C_0\min\{\log\Delta, \sqrt{\log n}\}$-round randomized LOCAL algorithm cannot produce a maximal matching with probability greater than $\Delta^{-1/1000}$ in $n$-vertex $\Delta$-regular graphs, where $C_0 :=10^{-10}$. The same lower bound applies to $\Delta$-ary trees.

Figures (5)

  • Figure 1: 2-flowers and 2-neighborhoods when $\Delta=3$ without edge labels. For edge-labeled versions, see Figure \ref{['fig:rN-rF']}.
  • Figure 2: Depicting the scenario in which there are two different $\text{dir}(f,x)$-directions for the 1-neighborhood $x$ (for the sake of contradiction). $x$ consists of three labels $x_{11}$, $x_{12}$, and $x_{13}$ in directions 1, 2, and 3 respectively. Suppose that $f$ is a 1-round matching-certified algorithm, and suppose that 1 and 2 are valid possible values for $\text{dir}(f,x)$. This would imply that labels for the green edges exist for which $f$ applied to the 1-flower centered at $\{A,B_1\}$ returns 1. Separately, it would imply that labels for the red edges exist for which $f$ applied to the 1-flower centered at $\{A,B_2\}$ returns 1. In this case, the red and green edges do not overlap, so they can both be set. However, this would imply that both $\{A,B_1\}$ and $\{A,B_2\}$ are in the matching, contradicting the fact that $f$ is matching-certified.
  • Figure 3: The assignment of scalars within $z$ and $w$ to edges in a $\Delta$-ary tree. The nested indices are never actually used in our proofs -- we manipulate flowers and neighborhoods entirely using the functions $\text{end}_v$ and $\text{res}_i$ described later in this section. We state the properties of $\text{end}_v$ and $\text{res}_i$ that we require in Section \ref{['sec:VertexSurvivalProbability']} as propositions and lemmas in Section \ref{['sec:PrelimDistributions']}.
  • Figure 4: The 2-neighborhood $z = \text{end}_A(w)$ with $\Delta = 3$ and the 2-flower $w$ given in Figure \ref{['fig:rN-rF']}. The $\{A,B\}$ edge became the edge pointing in the 1 direction.
  • Figure 5: The 1-flower $y = \text{res}_2(z)$ with $\Delta = 3$ and the 2-neighborhood $z$ given in Figure \ref{['fig:rN-rF']}.

Theorems & Definitions (104)

  • Theorem 1
  • Definition 1: $r$-Flowers $\mathcal{F}_r$
  • Definition 2: $r$-Neighborhoods $\mathcal{R}_r$
  • Definition 3: $\sigma$-Shuffles
  • Definition 4: The special permutations $\sigma_i$
  • Definition 5: Endpoint $end_v$ of an $r$-Flower
  • Definition 6: Restrictions $\text{res}_i$ of $r$-Neighborhoods to $(r-1)$-Flowers
  • Definition 7: Reversals $\overline{w}$ of $r$-Flowers
  • Definition 8: Projection
  • Definition 9: Incidence of $r$-Flowers
  • ...and 94 more