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Near-optimal distributed computation of small vertex cuts

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Abstract

We present near-optimal algorithms for detecting small vertex cuts in the \({\textsf{CONGEST}}\) model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, \(\Delta \). Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a single cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing \(\Delta \) barrier. As a warm-up to our approach, we show a simple \(\widetilde{O}(D)\)-round randomized algorithm for computing all cut vertices in a D-diameter n-vertex graph. This improves upon the \(O(D+\Delta /\log n)\)-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an \(\widetilde{O}(D)\)-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art \(O(\Delta \cdot D)^4\)-round algorithm by [Parter, DISC ’19]. Note that even for the considerably simpler setting of edge cuts, currently \(\widetilde{O}(D)\)-round algorithms are known only for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of \(G {\setminus } \{x,y\}\) for every pair \(x,y \in V\), using \(\widetilde{O}(D)\)-rounds. We believe that the tools provided in this paper are useful for omitting the \(\Delta \)-dependency even for larger cut values.

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Notes

  1. Throughout the paper, we use the notation \(\widetilde{O}\) to hide poly-logarithmic in n terms.

  2. As usual, all presented randomized algorithms in this paper have success guarantee of \(1-1/n^c\), for any given constant \(c>1\).

  3. The edge congestion of a given algorithm is the worst-case bound on the total number of messages exchanged through a given edge e in the graph.

  4. We exploit this bounded congestion for detecting cut pairs.

  5. Here we mean the strong diameter of \(G\setminus \{x,y\}\), i.e. the diameter of the graph induced by G on \(V \setminus \{x,y\}\). Its weak diameter, defined as the maximal distance in G between any \(u,v \in V {\setminus } \{x,y\}\), remains at most D, which we crucially exploit.

  6. A maximal spanning forest is defined as the union of spanning trees for all connected components.

  7. This is a critical point where only x learns if \(y \in \pi (s,x)\) is its cut-mate (by running Alg. \({{\mathcal {A}}}_y\)), but y might not learn its descendant cut-mates, such as x.

  8. Ties are broken arbitrarily and consistently.

  9. Over the choice of the random seeds \(\mathcal {S}_{ID}\) and \(\mathcal {S}_{h}\).

  10. Notice that these notations are not symmetric in x, y, e.g. \(\mathcal {S}(x,y)\) is different than \(\mathcal {S}(y,x)\).

  11. A part can be both x-heavy and y-heavy.

  12. Recall that each sketch contains \(L=c\log n\) basic sketch units. Hence, by taking c to be a sufficiently large constant, we can guarantee that \(O(\log n)\) fresh basic sketch units exist.

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Acknowledgements

We thank the anonymous reviewers of Distributed Computing for their insightful comments and suggestions that considerably improved the presentation of this work.

Funding

Research funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 949083), and by the Israeli Science Foundation (ISF), grant No. 2084/18.

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  1. Weizmann Institute, Rehovot, Israel

    Merav Parter & Asaf Petruschka

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  1. Merav Parter
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Parter, M., Petruschka, A. Near-optimal distributed computation of small vertex cuts. Distrib. Comput. (2023). https://doi.org/10.1007/s00446-023-00455-z

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