13,415 research outputs found

    Time vs. Information Tradeoffs for Leader Election in Anonymous Trees

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    The leader election task calls for all nodes of a network to agree on a single node. If the nodes of the network are anonymous, the task of leader election is formulated as follows: every node vv of the network must output a simple path, coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this paper, we study deterministic leader election in anonymous trees. Our aim is to establish tradeoffs between the allocated time τ\tau and the amount of information that has to be given a priori\textit{a priori} to the nodes to enable leader election in time τ\tau in all trees for which leader election in this time is at all possible. Following the framework of algorithms with advice\textit{algorithms with advice}, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire tree. The length of this string is called the size of advice\textit{size of advice}. For an allocated time τ\tau, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time τ\tau. We consider nn-node trees of diameter diamDdiam \leq D. While leader election in time diamdiam can be performed without any advice, for time diam1diam-1 we give tight upper and lower bounds of Θ(logD)\Theta (\log D). For time diam2diam-2 we give tight upper and lower bounds of Θ(logD)\Theta (\log D) for even values of diamdiam, and tight upper and lower bounds of Θ(logn)\Theta (\log n) for odd values of diamdiam. For the time interval [βdiam,diam3][\beta \cdot diam, diam-3] for constant β>1/2\beta >1/2, we prove an upper bound of O(nlognD)O(\frac{n\log n}{D}) and a lower bound of Ω(nD)\Omega(\frac{n}{D}), the latter being valid whenever diamdiam is odd or when the time is at most diam4diam-4. Finally, for time αdiam\alpha \cdot diam for any constant α<1/2\alpha <1/2 (except for the case of very small diameters), we give tight upper and lower bounds of Θ(n)\Theta (n)

    Leader Election in Anonymous Rings: Franklin Goes Probabilistic

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    We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size

    Breaking Symmetries

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    A well-known result by Palamidessi tells us that {\pi}mix (the {\pi}-calculus with mixed choice) is more expressive than {\pi}sep (its subset with only separate choice). The proof of this result argues with their different expressive power concerning leader election in symmetric networks. Later on, Gorla of- fered an arguably simpler proof that, instead of leader election in symmetric networks, employed the reducibility of "incestual" processes (mixed choices that include both enabled senders and receivers for the same channel) when running two copies in parallel. In both proofs, the role of breaking (ini- tial) symmetries is more or less apparent. In this paper, we shed more light on this role by re-proving the above result-based on a proper formalization of what it means to break symmetries-without referring to another layer of the distinguishing problem domain of leader election. Both Palamidessi and Gorla rephrased their results by stating that there is no uniform and reason- able encoding from {\pi}mix into {\pi}sep . We indicate how the respective proofs can be adapted and exhibit the consequences of varying notions of uniformity and reasonableness. In each case, the ability to break initial symmetries turns out to be essential

    Computing in Additive Networks with Bounded-Information Codes

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    This paper studies the theory of the additive wireless network model, in which the received signal is abstracted as an addition of the transmitted signals. Our central observation is that the crucial challenge for computing in this model is not high contention, as assumed previously, but rather guaranteeing a bounded amount of \emph{information} in each neighborhood per round, a property that we show is achievable using a new random coding technique. Technically, we provide efficient algorithms for fundamental distributed tasks in additive networks, such as solving various symmetry breaking problems, approximating network parameters, and solving an \emph{asymmetry revealing} problem such as computing a maximal input. The key method used is a novel random coding technique that allows a node to successfully decode the received information, as long as it does not contain too many distinct values. We then design our algorithms to produce a limited amount of information in each neighborhood in order to leverage our enriched toolbox for computing in additive networks

    Distributed Symmetry Breaking in Hypergraphs

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    Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ+1\Delta +1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log2n)O(\log^2 n) rounds (nn is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an O(Δϵpolylog(n))O(\Delta^{\epsilon}\text{polylog}(n))-round hypergraph MIS algorithm in the CONGEST model where Δ\Delta is the maximum node degree of the hypergraph and ϵ>0\epsilon > 0 is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

    How Many Cooks Spoil the Soup?

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    In this work, we study the following basic question: "How much parallelism does a distributed task permit?" Our definition of parallelism (or symmetry) here is not in terms of speed, but in terms of identical roles that processes have at the same time in the execution. We initiate this study in population protocols, a very simple model that not only allows for a straightforward definition of what a role is, but also encloses the challenge of isolating the properties that are due to the protocol from those that are due to the adversary scheduler, who controls the interactions between the processes. We (i) give a partial characterization of the set of predicates on input assignments that can be stably computed with maximum symmetry, i.e., Θ(Nmin)\Theta(N_{min}), where NminN_{min} is the minimum multiplicity of a state in the initial configuration, and (ii) we turn our attention to the remaining predicates and prove a strong impossibility result for the parity predicate: the inherent symmetry of any protocol that stably computes it is upper bounded by a constant that depends on the size of the protocol.Comment: 19 page
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