Near-Optimal Distributed Computation of Small Vertex Cuts

We present near-optimal algorithms for detecting small vertex cuts in the {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, ?. 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 ? barrier. - As a warm-up to our approach, we show a simple O?(D)-round randomized algorithm for computing all cut vertices in a D-diameter n-vertex graph. This improves upon the O(D+?/log n)-round algorithm of [Pritchard and Thurimella, ICALP 2008]. - Our key technical contribution is an O?(D)-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art O(? ? D)?-round algorithm by [Parter, DISC \u2719]. Note that even for the considerably simpler setting of edge cuts, currently O?(D)-round algorithms are currently 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 ? {x,y} for every pair x,y ? V, using O?(D)-rounds. We believe that the tools provided in this paper are useful for omitting the ?-dependency even for larger cut values

Connectivity Labeling for Multiple Vertex Failures

We present an efficient labeling scheme for answering connectivity queries in graphs subject to a specified number of vertex failures. Our first result is a randomized construction of a labeling function that assigns vertices $O(f^3\log^5 n)$-bit labels, such that given the labels of $F\cup \{s,t\}$ where $|F|\leq f$, we can correctly report, with probability $1-1/\mathrm{poly}(n)$, whether $s$ and $t$ are connected in $G-F$. However, it is possible that over all $n^{O(f)}$ distinct queries, some are answered incorrectly. Our second result is a deterministic labeling function that produces $O(f^7 \log^{13} n)$-bit labels such that all connectivity queries are answered correctly. Both upper bounds are polynomially off from an $\Omega(f)$-bit lower bound. Our labeling schemes are based on a new low degree decomposition that improves the Duan-Pettie decomposition, and facilitates its distributed representation. We make heavy use of randomization to construct hitting sets, fault-tolerant graph sparsifiers, and in constructing linear sketches. Our derandomized labeling scheme combines a variety of techniques: the method of conditional expectations, hit-miss hash families, and $\epsilon$-nets for axis-aligned rectangles. The prior labeling scheme of Parter and Petruschka shows that $f=1$ and $f=2$ vertex faults can be handled with $O(\log n)$- and $O(\log^3 n)$-bit labels, respectively, and for $f>2$ vertex faults, $\tilde{O}(n^{1-1/2^{f-2}})$-bit labels suffice

Color Fault-Tolerant Spanners

We initiate the study of spanners in arbitrarily vertex- or edge-colored graphs (with no "legality" restrictions), that are resilient to failures of entire color classes. When a color fails, all vertices/edges of that color crash. An $f$-color fault-tolerant ($f$-CFT) $t$-spanner of an $n$-vertex colored graph $G$ is a subgraph $H$ that preserves distances up to factor $t$, even in the presence of at most $f$ color faults. This notion generalizes the well-studied $f$-vertex/edge fault-tolerant ($f$-V/EFT) spanners. The size of an $f$-V/EFT spanner crucially depends on the number $f$ of vertex/edge faults to be tolerated. In the colored variants, even a single color fault can correspond to an unbounded number of vertex/edge faults. The key conceptual contribution of this work is in showing that the size (number of edges) required by an $f$-CFT spanner is in fact comparable to its uncolored counterpart, with no dependency on the size of color classes. We provide optimal bounds on the size required by $f$-CFT spanners, revealing an interesting phenomenon: while (individual) edge faults are "easier" than vertex faults in terms of spanner size, edge-color faults are "harder" than vertex-color faults. Our upper bounds are based on a generalization of the blocking set technique of [Bodwin and Patel, PODC 2019] for analyzing the (exponential-time) greedy algorithm for FT spanners. We complement them by providing efficient constructions of CFT spanners with similar size guarantees, based on the algorithm of [Dinitz and Robelle, PODC 2020].Comment: ITCS 2024, shortened abstract for arxi