Faster Cycle Detection in the Congested Clique

TL;DR

The paper addresses the faster detection of fixed-size cycles in the Congested Clique, delivering an approach whose runtime improves as the number of cycle occurrences $t$ grows. It introduces two complementary algorithms, Find-Cycle and Find-Vertex-In-Cycle, and a parallel small-matrix product framework that enables efficient, large-scale processing of random subgraphs; the analysis hinges on the number of participating cycle-vertices $x$ via $x^{h- abla}=2ht$. For undirected and directed graphs, the authors derive a runtime of ilde{O}ig(h^{O(h)} n^{0.1567}/(t^{0.4617/(h-1.82408)}+1)ig), and in the quantum setting they obtain a triangle-detection bound of ilde{O}ig((n/(t^2+1))^{3 ho/4}ig) rounds, leveraging Grover search and the FMM toolkit. Collectively, these results advance odd-cycle detection and directed-cycle detection in the congested regime, while also enriching the toolbox for parallel matrix computations in distributed settings. The work has potential implications for broader subgraph-detection tasks and quantum-distributed models, given its emphasis on balanced input routing, randomized sampling, and scalable matrix-multiplication primitives.

Abstract

We provide a fast distributed algorithm for detecting $h$-cycles in the \textsf{Congested Clique} model, whose running time decreases as the number of $h$-cycles in the graph increases. In undirected graphs, constant-round algorithms are known for cycles of even length. Our algorithm greatly improves upon the state of the art for odd values of $h$. Moreover, our running time applies also to directed graphs, in which case the improvement is for all values of $h$. Further, our techniques allow us to obtain a triangle detection algorithm in the quantum variant of this model, which is faster than prior work. A key technical contribution we develop to obtain our fast cycle detection algorithm is a new algorithm for computing the product of many pairs of small matrices in parallel, which may be of independent interest.

Figures (4)

• Figure 1: An illustrative comparison between our results and prior work, for the case of triangles. For each algorithm, we plot the base-$n$ logarithm of the number of rounds as a function of the base-$n$ logarithm of the number of triangles.
• Figure 2: An illustrative comparison between our results and prior work, for the case of triangles. For each algorithm, we plot the base-$n$ logarithm of the number of rounds as a function of the base-$n$ logarithm of the number of triangles. An additional axis represents the value of $\delta$ ranging from 0 to 2. For a fixed $t$, Find-Cycle performs faster as $\delta$ decreases, with its round complexity depicted by the area shaded in teal. Conversely, Find-Vertex-In-Cycle performs better as $\delta$ increases, and its round complexity is shown by the area shaded in violet.
• Figure 3: An illustration of the line $R(z)$ that bounds the step function we use in order to bound $\rho(z)$, and an image of an interpolation of what could be $\rho(z)$.
• Figure 4: Enlarged view of the plot around $z=1$ from the previous figure.

Theorems & Definitions (50)

• Theorem $\mathbf{}$: $h$-Cycle Detection
• Theorem $\mathbf{}$: Informal
• Definition 1: Balanced Input
• Theorem $\mathbf{}$
• Definition 2: $\delta$
• Claim 1
• Claim 2
• Lemma $\mathbf{}$: Lenzen's Routing Lemma lenzen2013optimal
• Theorem $\mathbf{}$: Chernoff Bound doerr2019theory
• Theorem $\mathbf{}$: Reverse Markov's inequality doerr2019theory