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1. 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 amore »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].« less
2. New hardness results for planar graph problems in p and an algorithm for sparsest cut

   https://doi.org/10.1145/3357713.3384310  

   Abboud, Amir ; Cohen-Addad, Vincent ; Klein, Philip N. ( June 2020 , Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing)

   The Sparsest Cut is a fundamental optimization problem that have been extensively studied. For planar inputs the problem is in P and can be solved in Õ(n 3 ) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90’s and can only achieve O(log n)-approximation in Õ(n) time or 3.5-approximation in Õ(n 2 ) time [Rao, STOC92]. Our main result is an Ω(n 2−ε ) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approxima- tions are inevitable in the near-linear time regime. To complement the lower bound, we provide a 3.3-approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. We also show that our lower bound is not far from optimal by observing an exact algorithm with running time Õ(n 5/2 ) improving upon the Õ(n 3 ) algorithm of Park and Phillips [STOC93]. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Building on our construction we prove near-quadratic lower bounds under SETHmore »for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGET model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.« less
3. Can We Break Symmetry with o(m) Communication?

   https://doi.org/10.1145/3465084  

   Pai, Shreyas ; Pandurangan, Gopal ; Pemmaraju, Sriram ; Robinson, Peter ( July 2021 , PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing)

   We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT-1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparison-based (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT-1 Congest model if non-comparison-based algorithms are permitted. Anmore »important implication of this result is that one can use the synchronous nature of the KT-1 Congest model, using silence to convey information,and solve any graph problem using non-comparison-based algorithms with Õ(n) messages, but this takes an exponential number of rounds. In the asynchronous model, even this is not possible. In contrast, much less is known about the message complexity of local symmetry breaking problems such as coloring and MIS. Our paper fills this gap by presenting the following results. Lower bounds: In the KT-1 CONGEST model, we show that any comparison-based algorithm, even a randomized Monte Carlo algorithm with constant success probability, requires Ω(n 2) messages in the worst case to solve either (△ + 1)-coloring or MIS, regardless of the number of rounds. We also show that Ω(n) is a lower bound on the number ofmessages for any (△ + 1)-coloring or MIS algorithm, even non-comparison-based, and even with nodes having initial knowledge of up to a constant radius. Upper bounds: In the KT-1 CONGEST model, we present the following randomized non-comparison-based algorithms for coloring that, with high probability, use o(m) messages and run in polynomially many rounds.(a) A (△ + 1)-coloring algorithm that uses Õ(n1.5) messages, while running in Õ(D + √ n) rounds, where D is the graph diameter. Our result also implies an asynchronous algorithm for (△ + 1)-coloring with the same message bound but running in Õ(n) rounds. (b) For any constantε > 0, a (1+ε)△-coloring algorithm that uses Õ(n/ε 2 ) messages, while running in Õ(n) rounds. If we increase our input knowledge slightly to radius 2, i.e.,in the KT-2 CONGEST model, we obtain:(c) A randomized comparison-based MIS algorithm that uses Õ(n 1.5) messages. while running in Õ( √n) rounds. While our lower bound results can be viewed as counterparts to the classical result of Awerbuch, Goldreich, Peleg, and Vainish [JACM 90], but for local problems, our algorithms are the first-known algorithms for coloring and MIS that take o(m) messages and run in polynomially many rounds.« less
4. The Communication Complexity of Optimization

   https://doi.org/10.1137/1.9781611975994.106  

   Vempala, Santosh S. ; Wang, Ruosong ; Woodruff, David P. ( January 2020 , ACM-SIAM Symposium on Discrete Algorithms)

   We consider the communication complexity of a number of distributed optimization problems. We start with the problem of solving a linear system. Suppose there is a coordinator together with s servers P1, …, Ps, the i-th of which holds a subset A(i) x = b(i) of ni constraints of a linear system in d variables, and the coordinator would like to output an x ϵ ℝd for which A(i) x = b(i) for i = 1, …, s. We assume each coefficient of each constraint is specified using L bits. We first resolve the randomized and deterministic communication complexity in the point-to-point model of communication, showing it is (d2 L + sd) and (sd2L), respectively. We obtain similar results for the blackboard communication model. As a result of independent interest, we show the probability a random matrix with integer entries in {–2L, …, 2L} is invertible is 1–2−Θ(dL), whereas previously only 1 – 2−Θ(d) was known. When there is no solution to the linear system, a natural alternative is to find the solution minimizing the ℓp loss, which is the ℓp regression problem. While this problem has been studied, we give improved upper or lower bounds for every value ofmore »p ≥ 1. One takeaway message is that sampling and sketching techniques, which are commonly used in earlier work on distributed optimization, are neither optimal in the dependence on d nor on the dependence on the approximation ε, thus motivating new techniques from optimization to solve these problems. Towards this end, we consider the communication complexity of optimization tasks which generalize linear systems, such as linear, semi-definite, and convex programming. For linear programming, we first resolve the communication complexity when d is constant, showing it is (sL) in the point-to-point model. For general d and in the point-to-point model, we show an Õ(sd3L) upper bound and an (d2 L + sd) lower bound. In fact, we show if one perturbs the coefficients randomly by numbers as small as 2−Θ(L), then the upper bound is Õ(sd2L) + poly(dL), and so this bound holds for almost all linear programs. Our study motivates understanding the bit complexity of linear programming, which is related to the running time in the unit cost RAM model with words of O(log(nd)) bits, and we give the fastest known algorithms for linear programming in this model. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975994.106« less
5. Sampling Multiple Edges Efficiently

   Eden, T. ; Mossel, S. ; Rubinfeld, R. ( January 2021 , Random)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ϵ-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ∗(n/ √ m) for sampling a single edge in general graphs (where O ∗(·) suppresses poly(1/ϵ) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O∗(√ q · (n/ √ m)), which is strictly preferable to O∗(q · (n/ √ m)) cost resulting from q invocations ofmore »the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.« less