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1. The Singular Optimality of Distributed Computation in LOCAL

   https://doi.org/10.4230/LIPIcs.OPODIS.2024.26  

   Dufoulon, Fabien; Pandurangan, Gopal; Robinson, Peter; Scquizzato, Michele (January 2025, Proceedings of OPODIS 2025. Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bonomi, Silvia; Galletta, Letterio; Rivière, Etienne; Schiavoni, Valerio (Ed.)

   It has been shown that one can design distributed algorithms that are (nearly) singularly optimal, meaning they simultaneously achieve optimal time and message complexity (within polylogarithmic factors), for several fundamental global problems such as broadcast, leader election, and spanning tree construction, under the KT₀ assumption. With this assumption, nodes have initial knowledge only of themselves, not their neighbors. In this case the time and message lower bounds are Ω(D) and Ω(m), respectively, where D is the diameter of the network and m is the number of edges, and there exist (even) deterministic algorithms that simultaneously match these bounds. On the other hand, under the KT₁ assumption, whereby each node has initial knowledge of itself and the identifiers of its neighbors, the situation is not clear. For the KT₁ CONGEST model (where messages are of small size), King, Kutten, and Thorup (KKT) showed that one can solve several fundamental global problems (with the notable exception of BFS tree construction) such as broadcast, leader election, and spanning tree construction with Õ(n) message complexity (n is the network size), which can be significantly smaller than m. Randomization is crucial in obtaining this result. While the message complexity of the KKT result is near-optimal, its time complexity is Õ(n) rounds, which is far from the standard lower bound of Ω(D). An important open question is whether one can achieve singular optimality for the above problems in the KT₁ CONGEST model, i.e., whether there exists an algorithm running in Õ(D) rounds and Õ(n) messages. Another important and related question is whether the fundamental BFS tree construction can be solved with Õ(n) messages (regardless of the number of rounds as long as it is polynomial in n) in KT₁. In this paper, we show that in the KT₁ LOCAL model (where message sizes are not restricted), singular optimality is achievable. Our main result is that all global problems, including BFS tree construction, can be solved in Õ(D) rounds and Õ(n) messages, where both bounds are optimal up to polylogarithmic factors. Moreover, we show that this can be achieved deterministically. 

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2. 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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3. On the fine-grained complexity of small-size geometric set cover and discrete k-center for small k

   https://doi.org/10.4230/LIPIcs.ICALP.2023.34  

   Chan, Timothy M.; He, Qizheng; Yu, Yuancheng (July 2023, Proc. 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time. - We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}). - We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2. - We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2. - We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound. 

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4. Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.55  

   Cook, Joshua; Moshkovitz, Dana (September 2024, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   We prove that for some constant a > 1, for all k ≤ a, MATIME[n^{k(1+o(1))}]/1 ⊄ SIZE[O(n^k)], for some specific o(1) function. This is a super linear polynomial circuit lower bound. Previously, Santhanam [Santhanam, 2007] showed that there exists a constant c>1 such that for all k>1: MATIME[n^{ck}]/1 ⊄ SIZE[O(n^k)]. Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c. Using ideas from Murray and Williams [Murray and Williams, 2018], one can show for all k>1: MATIME [n^{10 k²}]/1 ⊄ SIZE[O(n^k)]. To prove this result, we construct the first PCP for SPACE[n] with quasi-linear verifier time: our PCP has a Õ(n) time verifier, Õ(n) space prover, O(log(n)) queries, and polynomial alphabet size. Prior to this work, PCPs for SPACE[O(n)] had verifiers that run in Ω(n²) time. This PCP also proves that NE has MIP verifiers which run in time Õ(n). 

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5. Distributed MST: A Smoothed Analysis

   https://doi.org/10.1145/3369740.3369778  

   Chatterjee, Soumyottam; Pandurangan, Gopal; Pham, Nguyen Dinh (January 2020, ICDCN 2020: Proceedings of the 21st International Conference on Distributed Computing and Networking)

   We study smoothed analysis of distributed graph algorithms, focusing on the fundamental minimum spanning tree (MST) problem. With the goal of studying the time complexity of distributed MST as a function of the "perturbation" of the input graph, we posit a smoothing model that is parameterized by a smoothing parameter 0 ≤ ϵ(n) ≤ 1 which controls the amount of random edges that can be added to an input graph G per round. Informally, ϵ(n) is the probability (typically a small function of n, e.g., n--¼) that a random edge can be added to a node per round. The added random edges, once they are added, can be used (only) for communication. We show upper and lower bounds on the time complexity of distributed MST in the above smoothing model. We present a distributed algorithm that, with high probability, 1 computes an MST and runs in Õ(min{1/√ϵ(n)2O(√log n), D+ √n}) rounds2 where ϵ is the smoothing parameter, D is the network diameter and n is the network size. To complement our upper bound, we also show a lower bound of Ω(min{1/√ϵ(n), D + √n}). We note that the upper and lower bounds essentially match except for a multiplicative 2O(√log n) polylog(n) factor. Our work can be considered as a first step in understanding the smoothed complexity of distributed graph algorithms. 

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