Even-Cycle Detection in the Randomized and Quantum CONGEST Model

TL;DR

<3-5 sentence high-level summary> This work addresses the distributed problem of detecting even-length cycles (C_{2k}) in the CONGEST model, with extensions to the quantum CONGEST setting. It introduces a global-threshold colored BFS framework that partitions nodes into light, selected, and heavy categories, enabling efficient detection of 2k-cycles in O(n^{1-1/k}) rounds classically and a quadratic-speedup to ~n^{1/2-1/(2k)} rounds quantumly via distributed Monte-Carlo amplification. The approach also achieves tight or near-tight bounds for odd cycles and for detecting cycles up to length 2k under quantum resources, while building a robust framework for amplification and diameter-reduction to handle local versus global subgraph detection. Overall, the paper substantially advances subgraph-freeness in both classical and quantum distributed models, offering scalable algorithms and new amplification techniques with potential applicability to other distributed subgraph problems.

Abstract

We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the \CONGEST{} model by a randomized Monte-Carlo distributed algorithm with one-sided error probability $1/3$. This matches the best round-complexities of previously known algorithms for $k\in\{2,3,4,5\}$ by Drucker et al. [PODC'14] and Censor-Hillel et al. [DISC'20], but improves the complexities of the known algorithms for $k>5$ by Eden et al. [DISC'19], which were essentially of the form $\tilde O(n^{1-2/k^2})$. Our algorithm uses colored BFS-explorations with threshold, but with an original \emph{global} approach that enables to overcome a recent impossibility result by Fraigniaud et al. [SIROCCO'23] about using colored BFS-exploration with \emph{local} threshold for detecting cycles. We also show how to quantize our algorithm for achieving a round-complexity $\tilde O(n^{\frac{1}{2}-\frac{1}{2k}})$ in the quantum setting for deciding $C_{2k}$ freeness. Furthermore, this allows us to improve the known quantum complexities of the simpler problem of detecting cycles of length \emph{at most}~$2k$ by van Apeldoorn and de Vos [PODC'22]. Our quantization is in two steps. First, the congestion of our randomized algorithm is reduced, to the cost of reducing its success probability too. Second, the success probability is boosted using a new quantum framework derived from sequential algorithms, namely Monte-Carlo quantum amplification.

Figures (1)

• Figure 1: The case of a $10$-cycle (i.e., $k=5$). In the figure, ${\mathsf{IN}}(v,0)\neq\varnothing$ as $v\in V_2$. Here we have $q=1$, and thus the considered graphs for the proof are ${\mathsf{IN}}(v,0)\subseteq{\mathsf{IN}}(v,1)\subseteq{\mathsf{IN}}(v,2)\subseteq{\mathsf{IN}}(v)$. Regarding the proof of Claim \ref{['claim:path_S_W']}, we have $\deg_{{\mathsf{IN}}(v,1)}(s_1)>1$, and thus there exist vertices $w_2$ and $w'_2$ in ${\mathsf{IN}}(v,1)$. Since $\deg_{{\mathsf{IN}}(v,2)}(w_2)>2$, and since $\deg_{{\mathsf{IN}}(v,2)}(w'_2)>2$, there are two vertices $s_3$ and $s'_3$ in ${\mathsf{IN}}(v,2)$. And since $\deg_{{\mathsf{IN}}(v)}(s_3)>8$, there exists a vertex $w$ in ${\mathsf{IN}}(v)$. Regarding the proof of Claim \ref{['claim:vertical_paths']}, we have $\deg_{{\mathsf{IN}}(v)}(s)>8$ and $\deg_{{\mathsf{OUT}}(v'_1)}(s'_3)\leq 4$. Therefore, there exists a vertex $w"$ in ${\mathsf{IN}}(v)[s]\smallsetminus(\{w,w_2,w'_2\}\cup{\mathsf{OUT}}(v'_1))$.

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
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• Lemma 1
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• Lemma 2
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