In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results: *

Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 − 1 e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1 1+1/e 0.731. This conjecture remained open prior to this paper for over 30 years.In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of 1 1+1/e , this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-iid distributions and discuss its applications in mechanism design.

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.

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that with high probability estimates the size of a maximum matching within a constant factor usingÕ(n 2/3 ) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o(n) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size toÕ( √ n) for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity.Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o(n 1/2 ) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o(n) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

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.