This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts. We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners. The standard reductions from set-disjointness and equality seem unsuitable for this task because no specific edge needs to be removed from the graph. Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity using information theory. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.
In his seminal paper, Pettie @cite_21 presents lower bounds for the number of rounds needed by distributed algorithms in order to construct several families of spanners. Specifically, it is shown that computing an all-pair additive @math -spanner with size @math in expectation, for a constant @math , requires @math rounds of communication. Because this is an indistinguishability-based lower bound, it holds even for the less restricted LOCAL mode, where message lengths can be unbounded.
@cite_21
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This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as ( ( [3]n B ) ) and ( ( [4]n B ) ), respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of ( ( [2]n B) ) for graphs of diameter Ω(log n). This paper addresses the cornerstone family of 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 @math nodes can only send @math -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 @math rounds, where @math is the diameter of the graph. The best known running time in terms of @math alone is @math , which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based @math -round algorithm, where @math is the maximal degree in the graph. On the way to obtaining the above, we show that in the model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in @math rounds. , where @math is the maximum degree. When @math , the bound improves to @math and holds also for @math -coloring. In addition, we deterministically construct a @math -spanner with @math edges in @math rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a @math -spanner with @math edges runs in @math rounds. We show that any randomised Monte Carlo distributed algorithm for the Lov 'asz local lemma requires @math communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of @math , where @math is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lov 'asz local lemma with a running time of @math rounds in bounded-degree graphs, and the best lower bound before our work was @math rounds [ 2014]. The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their @math -bit input strings is large (i.e., at least @math ) or small (i.e., at most @math ); they do not care if it is neither large nor small. This @math gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an @math lower bound on this problem was known only in the one-way setting. We prove an @math lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that @math -approximately counting the number of distinct elements in a data stream requires @math space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight @math lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an @math lower bound on the one-way randomized communication complexity.