We initiate a thorough study of distributed property testing -producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general and sparse models we obtain faster tests for trianglefreeness, cycle-freeness and bipartiteness, respectively. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model.In most cases, aided by parallelism, the distributed algorithms have a much shorter running time as compared to their counterparts from the sequential querying model of traditional property testing. The simplest property testing algorithms allow a relatively smooth transitioning to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest.

This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions.Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send O(log n)-bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results.Our techniques give a deterministic maximal independent set (MIS) algorithm in the CONGEST model, where the communication graph is identical to the input graph, in O(D log 2 n) rounds, where D is the diameter of the graph. The best known running time in terms of n alone is 2 O( √ log n) , which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based O(∆+log * n)-round algorithm, where ∆ is the maximal degree in the graph. To the best of our knowledge, ours is the first deterministic MIS algorithm for the CONGEST model, which for polylogarithmic values of D is only a polylogarithmic factor off compared with its randomized counterparts.On the way to obtaining the above, we show that in the Congested Clique model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in O(log ∆ log n) rounds.When ∆ = O(n 1/3 ), the bound improves to O(log ∆) and holds also for (∆ + 1)-coloring.In addition, we deterministically construct a (2k − 1)-spanner with O(kn 1+1/k log n) edges in O(k log n) rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a (2k − 1)-spanner with O(kn 1+1/k ) edges runs in O(n 1−1/k ) rounds.A cornerstone family of problems in distributed computing are the so-called local problems. These include finding a maximal independent set (MIS), a (∆ + 1)-coloring where ∆ is the maximal degree in the network graph, finding a maximal matching, constructing multiplicative spanners, and more. Intuitively, as opposed to global problems, local problems admit solutions that do not require communication over the entire network graph.One fundamental characteristic of distributed algorithms for local problems is whether they are deterministic or randomized. Currently, there exists a curious gap between the known complexities of randomized and deterministic solutions for local problems. Interestingly, the main indistinguishabilitybased technique used for obtaining the relatively few lower bounds that are known seems unsuitable for separating these cases. Building upon an important new lower bound technique of Brandt et al. [BFH + 16], a beautiful recent work of Chang et al. [CKP16] sheds some light over this question, by proving that the randomized complexity of any local problem is at least its deterministic complexity on instances of size √ log n. In addition, they show an exponential separation between th...

We present a simple deterministic single-pass (2 + )-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5 + ).Our algorithm uses O(n log 2 n) space for constant values of . It relies on a variation of the local-ratio theorem, which may be of independent interest in the semi-streaming model.

We present a simple distributed ∆-approximation algorithm for maximum weight independent set (MaxIS) in the CONGEST model which completes in O(MIS(G) · log W ) rounds, where ∆ is the maximum degree, MIS(G) is the number of rounds needed to compute a maximal independent set (MIS) on G, and W is the maximum weight of a node. Plugging in the best known algorithm for MIS gives a randomized solution in O(log n log W ) rounds, where n is the number of nodes. We also present a deterministic O(∆ + log * n)-round algorithm based on coloring.We then show how to use our MaxIS approximation algorithms to compute a 2-approximation for maximum weight matching without incurring any additional round penalty in the CONGEST model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of local aggregation algorithms for which we describe a mechanism that allows the simulation to run in the CONGEST model without an additional overhead.Next, we show that for maximum weight matching, relaxing the approximation factor to (2 + ε) allows us to devise a distributed algorithm requiring O( log ∆ log log ∆ ) rounds for any constant ε > 0. For the unweighted case, we can even obtain a (1+ε)-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on ∆.

We present a simple deterministic distributed (2 + ϵ)-approximation algorithm for minimum-weight vertex cover, which completes in O (log Δ/ϵlog log Δ) rounds, where Δ is the maximum degree in the graph, for any ϵ > 0 that is at most O (1). For a constant ϵ, this implies a constant approximation in O (log Δ/log log Δ) rounds, which contradicts the lower bound of [KMW10].