Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deterministic Even-Cycle Detection in Broadcast CONGEST

Pierre Fraigniaud, Maël Luce, Frédéric Magniez, Ioan Todinca

TL;DR

This work proves that for every $k\ge 2$, $C_{2k}$-freeness can be decided deterministically in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, improving the state of knowledge for even cycles and matching known lower bounds in key cases. The algorithm combines parallel BFS explorations with deterministic path-forwarding choices and introduces a Density Theorem that bounds local graph density to force a $2k$-cycle when forwarding becomes too congested. A two-phase approach handles light cycles via flooding on the light subgraph and heavy cycles via filtered flooding guided by the Representative Lemma, enabling a sublinear-round, deterministic detection for all $k$. The results strengthen the understanding of distributed subgraph detection, align with the performance of randomized algorithms for several $k$, and open avenues for tighter constants and potential quantum extensions.

Abstract

We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without $2k$-cycles, which we believe is interesting on its own.

Deterministic Even-Cycle Detection in Broadcast CONGEST

TL;DR

This work proves that for every , -freeness can be decided deterministically in rounds in the Broadcast CONGEST model, improving the state of knowledge for even cycles and matching known lower bounds in key cases. The algorithm combines parallel BFS explorations with deterministic path-forwarding choices and introduces a Density Theorem that bounds local graph density to force a -cycle when forwarding becomes too congested. A two-phase approach handles light cycles via flooding on the light subgraph and heavy cycles via filtered flooding guided by the Representative Lemma, enabling a sublinear-round, deterministic detection for all . The results strengthen the understanding of distributed subgraph detection, align with the performance of randomized algorithms for several , and open avenues for tighter constants and potential quantum extensions.

Abstract

We show that, for every , -freeness can be decided in rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for up to logarithmic factors thanks to the lower bound for -freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for , and by Fraigniaud et al. [PODC 2024] for . Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without -cycles, which we believe is interesting on its own.

Paper Structure

This paper contains 16 sections, 9 theorems, 50 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and Preliminary Results
  3. The Broadcast CONGEST Model
  4. Deciding $C_4$-Freeness
  5. Detection of Light $2k$-Cycles
  6. Filtered Flooding
  7. Representative Lemma
  8. Application to Cycle Detection
  9. Deterministic Even-Cycle Detection
  10. Algorithm Description
  11. Proof of Theorem \ref{['theo:c2k']}
  12. Proof of Density Theorem
  13. General Construction
  14. The Sets In and Out.
  15. Core Graphs.
  16. ...and 1 more sections

Key Result

Theorem 1

For every $k\geq 2$, there is a deterministic algorithm solving $C_{2k}$-freeness in $O(n^{1-1/k})$ rounds in the Broadcast $\textsf{CONGEST}$ model.

Figures (3)

  • Figure 1: Construction of the set ${\mathsf{in}}_i(u)$ from the sets ${\mathsf{out}}_{i-1}(w)$, $w\in N(u)$. In the figure, ${\mathsf{in}}_i(u)={\mathsf{out}}_{i-1}(w) \cup {\mathsf{out}}_{i-1}(w')$ because there is a simple path $P$ (resp., $P'$) of length $i-1$ from $w$ (resp., $w'$) to $X$ that can be extended to a simple path of length $i$ from $u$ to $X$. For every $y\in Y$, ${\mathsf{in}}_i(y,u)=E_H(y)\cap{\mathsf{in}}_i(u)$.
  • Figure 2: Construction of the set ${\mathsf{out}}_i(u)=\cup_{y\in Y}{\mathsf{out}}_i(y,u)$ from the sets ${\mathsf{in}}_i(y,u)$, $y\in Y$.
  • Figure 3: Construction of the paths $P,P'$, and $P"$ in the proof of Lemma \ref{['lem:cycle']}

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Lemma 3: Monien monien85
  • Lemma 4: S. Burr burr1982
  • proof
  • ...and 12 more