Identifying the connected components of a graph, apart from being a fundamental problem with countless applications, is a key primitive for many other algorithms. In this paper, we consider this problem in parallel settings. Particularly, we focus on the Massively Parallel Computations (MPC) model, which is the standard theoretical model for modern parallel frameworks such as MapReduce, Hadoop, or Spark. We consider the truly sublinear regime of MPC for graph problems where the space per machine is n δ for some desirably small constant δ ∈ (0, 1).We present an algorithm that for graphs with diameter D in the wide range [log n, n], takes O(log D) rounds to identify the connected components and takes O(log log n) rounds for all other graphs. The algorithm is randomized, succeeds with high probability 1 , does not require prior knowledge of D, and uses an optimal total space of O(m). We complement this by showing a conditional lower-bound based on the widely believed 2-Cycle conjecture that Ω(log D) rounds are indeed necessary in this setting.Studying parallel connectivity algorithms received a resurgence of interest after the pioneering work of Andoni et al. [FOCS 2018] who presented an algorithm with O(log D · log log n) round-complexity. Our algorithm improves this result for the whole range of values of D and almost settles the problem due to the conditional lower-bound.Additionally, we show that with minimal adjustments, our algorithm can also be implemented in a variant of the (CRCW) PRAM in asymptotically the same number of rounds. * A preliminary version of this paper is O(1) round algorithm if e.g. D = O( √ n). We refute this possibility and show that indeed for any choice of D ∈ [log 1+Ω(1) , n], there are family of graphs with diameter D on which Ω(log D) rounds are necessary in this regime of MPC, if the 2-Cycle conjecture holds.Theorem 2. Fix some D ≥ log 1+ρ n for a desirably small constant ρ ∈ (0, 1). Any MPC algorithm with n 1−Ω(1) space per machine that w.h.p. identifies each connected component of any given n-vertex graph with diameter D requires Ω(log D ) rounds, unless the 2-Cycle conjecture is wrong.We note that proving any unconditional super constant lower bound for any problem in P, in this regime of MPC, would imply NC 1 P which seems out of the reach of current techniques [59].Extention to PRAM. As a side result, we provide an implementation of our connectivity algorithm in O(log D + log log m/n n) depth in the multiprefix CRCW PRAM model, a parallel computation model that permits concurrent reads and concurrent writes. This implementation of our algorithm performs O((m+n)(log D +log log m/n n)) work and is therefore nearly work-efficient. The following theorem states our result. We defer further elaborations on this result to Appendix B.3.

The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, however, we still have a far more limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of n 1+Ω(1) per machine.In this work, we close this gap by providing a novel analysis of an extremely simple algorithm a variant of which was conjectured to work by Czumaj et al. [STOC'18]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among some other results, leads to an O(log log ∆) round algorithm for maximal matching with O(n) space (or even mildly sublinear in n using standard techniques).As an immediate corollary, we get a 2 approximate minimum vertex cover in essentially the same rounds and space. This is the best possible approximation factor under standard assumptions, culminating a long line of research. It also leads to an improved O(log log ∆) round algorithm for 1 + ε approximate matching. All these results can also be implemented in the congested clique model within the same number of rounds. * A preliminary version of this paper is

We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms [28,9] using MapReduce and a distributed hash table service [17].This extension allows us to give new graph algorithms with much lower round complexities compared to the best known solutions in the MPC model. In particular, in the AMPC model we show how to solve maximal independent set in O(1) rounds and connectivity/minimum spanning tree in O(log log m/n n) rounds both using O(n δ ) space per machine for constant δ < 1. In the same memory regime for MPC, the best known algorithms for these problems require poly log n rounds. Our results imply that the 2-Cycle conjecture, which is widely believed to hold in the MPC model, does not hold in the AMPC model.

We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph-which undergoes both edge insertions and deletions-in polylogarithmic time. Our algorithm is randomized and, per update, takes O(log 2 ∆ · log 2 n) expected time. Furthermore, the algorithm can be adjusted to have O(log 2 ∆ · log 4 n) worst-case update-time with high probability. Here, n denotes the number of vertices and ∆ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC'18] who presented an algorithm with O(m 3/4 ) update-time (and thus broke the natural Ω(m) barrier) where m denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had O(min{ √ n, m 1/3 }) update-time [Assadi et al. SODA'19].Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time. * A preliminary version of this paper is

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