Silent MST approximation for tiny memory

In network distributed computing, minimum spanning tree (MST) is one of the key problems, and silent self-stabilization one of the most demanding fault-tolerance properties. For this problem and this model, a polynomial-time algorithm with $O(\log^2\!n)$ memory is known for the state model. This is memory optimal for weights in the classic $[1,\text{poly}(n)]$ range (where $n$ is the size of the network). In this paper, we go below this $O(\log^2\!n)$ memory, using approximation and parametrized complexity. More specifically, our contributions are two-fold. We introduce a second parameter~$s$, which is the space needed to encode a weight, and we design a silent polynomial-time self-stabilizing algorithm, with space $O(\log n \cdot s)$. In turn, this allows us to get an approximation algorithm for the problem, with a trade-off between the approximation ratio of the solution and the space used. For polynomial weights, this trade-off goes smoothly from memory $O(\log n)$ for an $n$-approximation, to memory $O(\log^2\!n)$ for exact solutions, with for example memory $O(\log n\log\log n)$ for a 2-approximation

Computing in Additive Networks with Bounded-Information Codes

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 Edge Connectivity in Sublinear Time

We present the first sublinear-time algorithm for a distributed message-passing network sto compute its edge connectivity $\lambda$ exactly in the CONGEST model, as long as there are no parallel edges. Our algorithm takes $\tilde O(n^{1-1/353}D^{1/353}+n^{1-1/706})$ time to compute $\lambda$ and a cut of cardinality $\lambda$ with high probability, where $n$ and $D$ are the number of nodes and the diameter of the network, respectively, and $\tilde O$ hides polylogarithmic factors. This running time is sublinear in $n$ (i.e. $\tilde O(n^{1-\epsilon})$) whenever $D$ is. Previous sublinear-time distributed algorithms can solve this problem either (i) exactly only when $\lambda=O(n^{1/8-\epsilon})$ [Thurimella PODC'95; Pritchard, Thurimella, ACM Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]. To achieve this we develop and combine several new techniques. First, we design the first distributed algorithm that can compute a $k$-edge connectivity certificate for any $k=O(n^{1-\epsilon})$ in time $\tilde O(\sqrt{nk}+D)$. Second, we show that by combining the recent distributed expander decomposition technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup, STOC'15], we can decompose the network into a sublinear number of clusters with small average diameter and without any mincut separating a cluster (except the `trivial' ones). Finally, by extending the tree packing technique from [Karger STOC'96], we can find the minimum cut in time proportional to the number of components. As a byproduct of this technique, we obtain an $\tilde O(n)$-time algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019

Optimal-time adaptive strong renaming, with applications to counting

10.1145/1993806.1993850Proceedings of the Annual ACM Symposium on Principles of Distributed Computing239-24885LR

The one-out-of-kretrieval problem and linear network coding

In this paper we show how linear network coding can reduce the number of queries needed to retrieve one specific message among k distinct ones replicated across a large number of randomly accessed nodes storing one message each. Without network coding, this would require k queries on average. After proving that no scheme can perform better than a straightforward lower bound of 0:5k average queries, we propose and asymptotically evaluate, using mean field arguments, a few example practical schemes, the best of which attains 0:794k queries on average. The paper opens two complementary challenges: a systematic analysis of practical schemes so as to identify the best performing ones and design guideline strategies, as well as the need to identify tighter, nontrivial, lower bounds

The One-Out-of-k Retrieval Problem and Linear Network Coding

In this paper we show how linear network coding can reduce the number of queries needed to retrieve one specific message among k distinct ones replicated across a large number of randomly accessed nodes storing one message each. Without network coding, this would require k queries on average. After proving that no scheme can perform better than a straightforward lower bound of 0. 5k average queries, we propose and asymptotically evaluate, using mean field arguments, a few example practical schemes, the best of which attains 0. 82k queries on average. The paper opens two complementary challenges: a systematic analysis of practical schemes so as to identify the best performing ones and design guideline strategies, as well as the need to identify tighter, nontrivial, lower bounds

Distributed Distance Approximation

© Bertie Ancona, Keren Censor-Hillel, Mina Dalirrooyfard, Yuval Efron, and Virginia Vassilevska Williams; licensed under Creative Commons License CC-BY 24th International Conference on Principles of Distributed Systems (OPODIS 2020). Diameter, radius and eccentricities are fundamental graph parameters, which are extensively studied in various computational settings. Typically, computing approximate answers can be much more efficient compared with computing exact solutions. In this paper, we give a near complete characterization of the trade-offs between approximation ratios and round complexity of distributed algorithms for approximating these parameters, with a focus on the weighted and directed variants. Furthermore, we study bi-chromatic variants of these parameters defined on a graph whose vertices are colored either red or blue, and one focuses only on distances for pairs of vertices that are colored differently. Motivated by applications in computational geometry, bi-chromatic diameter, radius and eccentricities have been recently studied in the sequential setting [Backurs et al. STOC'18, Dalirrooyfard et al. ICALP'19]. We provide the first distributed upper and lower bounds for such problems. Our technical contributions include introducing the notion of approximate pseudo-center, which extends the pseudo-centers of [Choudhary and Gold SODA'20], and presenting an efficient distributed algorithm for computing approximate pseudo-centers. On the lower bound side, our constructions introduce the usage of new functions into the framework of reductions from 2-party communication complexity to distributed algorithms

Tight bounds for asynchronous renaming

10.1145/2597630Journal of the ACM613-JOAC

Silent MST Approximation for Tiny Memory

International audienceIn this paper we show that approximation can help reduce the space used for self-stabilization. In the classic state model, where the nodes of a network communicate by reading the states of their neighbors, an important measure of efficiency is the space: the number of bits used at each node to encode the state. In this model, a classic requirement is that the algorithm has to be silent, that is, after stabilization the states should not change anymore. We design a silent self-stabilizing algorithm for the problem of minimum spanning tree, that has a trade-off between the quality of the solution and the space needed to compute it