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1. On the Distributed Complexity of Large-Scale Graph Computations

   https://doi.org/10.1145/3460900  

   Pandurangan, Gopal ; Robinson, Peter ; Scquizzato, Michele ( June 2021 , ACM Transactions on Parallel Computing)

   null (Ed.)

   Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower bounds for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds. 

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2. Adaptive Out-Orientations with Applications

   Chekuri, Chandra ; Christiansen, Aleksander Bjørn ; Holm, Jacob ; van der Hoog, Ivor ; Quanrud, Kent ; Rotenberg, Eva ; Schwiegelshohn, Chris. ( January 2024 , Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out-degree is proportional to the arboricity $\alpha$ of the graph, in, either, an amortised update time of $O(\log^2 n \log \alpha)$, or a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worst-case, for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $n$ and $\alpha$. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$, of the dynamic graph. Our algorithms have update times of $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case, and $O(\varepsilon^{-4}\log^2 n \log \rho)$ amortised, respectively. We may output a subgraph $H$ of the input graph where its density is a $(1+\varepsilon)$ approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the $O(\varepsilon^{-6}\log ^4 n)$ algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1~+~\varepsilon)\textnormal{OPT} + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$, improving the $O(\varepsilon^{-6}\alpha^2 \log^3 n)$ algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, $\Delta+1$ colouring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time $O(\alpha^2 + \log^2 n)$, and by Neiman and Solomon from STOC 2013 runs in time $O(\sqrt{m})$. We give improved running times whenever the arboricity $\alpha \in \omega( \log n\sqrt{\log\log n})$. 

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3. A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

   https://doi.org/10.1109/FOCS46700.2020.00111  

   Chuzhoy, Julia ; Gao, Yu ; Li, Jason ; Nanongkai, Danupon ; Peng, Richard ; Saranurak, Thatchaphol ( November 2020 , 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We consider the classical Minimum Balanced Cut problem: given a graph $G$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $n$-vertex $m$-edge graph $G$ and any parameter $1\leq r\leq O(\log n)$, computes a $(\log m)^{r^2}$-approximation for Minimum Balanced Cut on $G$, in time $O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$. In particular, we obtain a $(\log m)^{1/\epsilon}$-approximation in time $m^{1+O(1/\sqrt{\epsilon})}$ for any constant $\epsilon$, and a $(\log m)^{f(m)}$-approximation in time $m^{1+o(1)}$, for any slowly growing function $m$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $G$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $n$-vertex graph is $n^{o(1)}$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $n$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs. 

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4. A Distanced Matching Game, Decremental {APSP} in Expanders, and Faster Deterministic Algorithms for Graph Cut Problems

   Chuzhoy, Julia ( January 2023 , Proceedings of the 2023 {ACM-SIAM} Symposium on Discrete Algorithms,{SODA} 2023)

   Expander graphs play a central role in graph theory and algorithms. With a number of powerful algorithmic tools developed around them, such as the Cut-Matching game, expander pruning, expander decomposition, and algorithms for decremental All-Pairs Shortest Paths (APSP) in expanders, to name just a few, the use of expanders in the design of graph algorithms has become ubiquitous. Specific applications of interest to us are fast deterministic algorithms for cut problems in static graphs, and algorithms for dynamic distance-based graph problems, such as APSP. Unfortunately, the use of expanders in these settings incurs a number of drawbacks. For example, the best currently known algorithm for decremental APSP in constant-degree expanders can only achieve a (log n) O(1/ 2 ) -approximation with n 1+O( ) total update time for any . All currently known algorithms for the Cut Player in the Cut-Matching game are either randomized, or provide rather weak guarantees: expansion 1/(log n) 1/ with running time n 1+O( ) . This, in turn, leads to somewhat weak algorithmic guarantees for several central cut problems: the best current almost linear time deterministic algorithms for Sparsest Cut, Lowest Conductance Cut, and Balanced Cut can only achieve approximation factor (log n) ω(1). Lastly, when relying on expanders in distancebased problems, such as dynamic APSP, via current methods, it seems inevitable that one has to settle for approximation factors that are at least Ω(log n). In contrast, we do not have any negative results that rule out a factor-5 approximation with near-linear total update time. In this paper we propose the use of well-connected graphs, and introduce a new algorithmic toolkit for such graphs that, in a sense, mirrors the above mentioned algorithmic tools for expanders. One of these new tools is the Distanced Matching game, an analogue of the Cut-Matching game for well-connected graphs. We demonstrate the power of these new tools by obtaining better results for several of the problems mentioned above. First, we design an algorithm for decremental APSP in expanders with significantly better guarantees: in a constant-degree expander, the algorithm achieves (log n) 1+o(1)-approximation, with total update time n 1+o(1). We also obtain a deterministic algorithm for the Cut Player in the Cut-Matching game that achieves expansion 1 (log n) 5+o(1) in time n 1+o(1), deterministic almost linear-time algorithms for Sparsest Cut, Lowest-Conductance Cut, and Minimum Balanced Cut with approximation factors O(poly log n), as well as improved deterministic algorithm for Expander Decomposition. We believe that the use of well-connected graphs instead of expanders in various dynamic distance-based problems (such as APSP in general graphs) has the potential of providing much stronger guarantees, since we are no longer necessarily restricted to superlogarithmic approximation factors. 

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5. Time-Message Trade-Offs in Distributed Algorithms

   https://doi.org/10.4230/LIPIcs.DISC.2018.32  

   Gmyr, Robert ; Pandurangan, Gopal ( January 2018 , 32nd International Symposium on Distributed Computing (DISC 2018))

   This paper focuses on showing time-message trade-offs in distributed algorithms for fundamental problems such as leader election, broadcast, spanning tree (ST), minimum spanning tree (MST), minimum cut, and many graph verification problems. We consider the synchronous CONGEST distributed computing model and assume that each node has initial knowledge of itself and the identifiers of its neighbors - the so-called KT_1 model - a well-studied model that also naturally arises in many applications. Recently, it has been established that one can obtain (almost) singularly optimal algorithms, i.e., algorithms that have simultaneously optimal time and message complexity (up to polylogarithmic factors), for many fundamental problems in the standard KT_0 model (where nodes have only local knowledge of themselves and not their neighbors). The situation is less clear in the KT_1 model. In this paper, we present several new distributed algorithms in the KT_1 model that trade off between time and message complexity. Our distributed algorithms are based on a uniform and general approach which involves constructing a sparsified spanning subgraph of the original graph - called a danner - that trades off the number of edges with the diameter of the sparsifier. In particular, a key ingredient of our approach is a distributed randomized algorithm that, given a graph G and any delta in [0,1], with high probability constructs a danner that has diameter O~(D + n^{1-delta}) and O~(min{m,n^{1+delta}}) edges in O~(n^{1-delta}) rounds while using O~(min{m,n^{1+delta}}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. Using our danner construction, we present a family of distributed randomized algorithms for various fundamental problems that exhibit a trade-off between message and time complexity and that improve over previous results. Specifically, we show the following results (all hold with high probability) in the KT_1 model, which subsume and improve over prior bounds in the KT_1 model (King et al., PODC 2014 and Awerbuch et al., JACM 1990) and the KT_0 model (Kutten et al., JACM 2015, Pandurangan et al., STOC 2017 and Elkin, PODC 2017): 1) Leader Election, Broadcast, and ST. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,1]. 2) MST and Connectivity. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. In particular, for delta = 0.5 we obtain a distributed MST algorithm that runs in optimal O~(D+sqrt{n}) rounds and uses O~(min{m,n^{3/2}}) messages. We note that this improves over the singularly optimal algorithm in the KT_0 model that uses O~(D+sqrt{n}) rounds and O~(m) messages. 3) Minimum Cut. O(log n)-approximate minimum cut can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. 4) Graph Verification Problems such as Bipartiteness, Spanning Subgraph etc. These can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. 

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