arXiv subjects

Keren Censor-Hillel

Publications and source records attributed to Keren Censor-Hillel.

At least 55 records · Page 3Linked to original sources

arXiv2019

Distributed Reconfiguration of Maximal Independent Sets

In this paper, we investigate a distributed maximal independent set (MIS) reconfiguration problem, in which there are two maximal independent sets for which every node is given its membership status, and the nodes need to communicate with their neighbors in order to find a reconfiguration schedule that switches from the first MIS to the second. Such a schedule is a list of independent sets that is restricted by forbidding two neighbors to change their membership status at the same step. In addition, these independent sets should provide some covering guarantee. We show that obtaining an actual MIS (and even a 3-dominating set) in each intermediate step is impossible. However, we provide efficient solutions when the intermediate sets are only required to be independent and 4-dominating, which is almost always possible, as we fully characterize. Consequently, our goal is to pin down the tradeoff between the possible length of the schedule and the number of communication rounds. We prove that a constant length schedule can be found in $O(\texttt{MIS}+\texttt{R32})$ rounds, where $\texttt{MIS}$ is the complexity of finding an MIS in a worst-case graph and $\texttt{R32}$ is the complexity of finding a $(3,2)$-ruling set. For bounded degree graphs, this is $O(\log^*n)$ rounds and we show that it is necessary. On the other extreme, we show that with a constant number of rounds we can find a linear length schedule.

cs.DC↗

arXiv2019

Smaller Cuts, Higher Lower Bounds

This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for CONGEST lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

cs.DC↗

arXiv2018

Distributed Spanner Approximation

We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results. Our main hardness result shows that an $α$-approximation for the minimum directed $k$-spanner problem for $k \geq 5$ requires $Ω(n /\sqrtα\log{n})$ rounds using deterministic algorithms or $Ω(\sqrt{n }/\sqrtα\log{n})$ rounds using randomized ones, in the CONGEST model of distributed computing. Combined with the constant-round $O(n^ε)$-approximation algorithm in the LOCAL model of [Barenboim, Elkin and Gavoille, 2016], as well as a polylog-round $(1+ε)$-approximation algorithm in the LOCAL model that we show here, our lower bounds for the CONGEST model imply a strict separation between the LOCAL and CONGEST models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem. Similarly, a separation between the directed and undirected cases is implied. We also prove a nearly-linear lower bound for the minimum weighted $k$-spanner problem for $k \geq 4$, and we show lower bounds for the weighted 2-spanner problem. On the algorithmic side, apart from the aforementioned $(1+ε)$-approximation algorithm for minimum $k$-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of $O(\log(m/n))$ for a graph with $n$ vertices and $m$ edges, which matches the best known ratio for polynomial time sequential algorithms [Kortsarz and Peleg, 1994], and is tight if we restrict ourselves to polynomial local computations. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem.

cs.DS↗

arXiv2017

Fooling Views: A New Lower Bound Technique for Distributed Computations under Congestion

We introduce a novel lower bound technique for distributed graph algorithms under bandwidth limitations. We define the notion of \emph{fooling views} and exemplify its strength by proving two new lower bounds for triangle membership in the CONGEST(B) model: (i) Any $1$-round algorithm requires $B\geq cΔ\log n$ for a constant $c>0$. (ii) If $B=1$, even in constant-degree graphs any algorithm must take $Ω(\log^* n)$ rounds. The implication of the former is the first proven separation between the LOCAL and the CONGEST models for deterministic triangle membership. The latter result is the first non-trivial lower bound on the number of rounds required, even for \emph{triangle detection}, under limited bandwidth. All previous known techniques are provably incapable of giving these bounds. We hope that our approach may pave the way for proving lower bounds for additional problems in various settings of distributed computing for which previous techniques do not suffice.

cs.DS↗

arXiv2017

Distributed Approximation of Maximum Independent Set and Maximum Matching

We present a simple distributed $Δ$-approximation algorithm for maximum weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes in $O(\texttt{MIS}(G)\cdot \log W)$ rounds, where $Δ$ is the maximum degree, $\texttt{MIS}(G)$ is the number of rounds needed to compute a maximal independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the \texttt{MIS} algorithm used as a black-box. 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 $O(Δ+\log^* n)$-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 $\mathsf{CONGEST}$ 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 \emph{local aggregation algorithms} for which we describe a mechanism that allows the simulation to run in the $\mathsf{CONGEST}$ model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ($2+\varepsilon$) allows us to devise a distributed algorithm requiring $O(\frac{\log Δ}{\log\logΔ})$ rounds for any constant $\varepsilon>0$. For the unweighted case, we can even obtain a $(1+\varepsilon)$-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on $Δ$.

cs.DC↗

arXiv2017

Fast Distributed Approximation for Max-Cut

Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $ε> 0$, we develop randomized approximation algorithms achieving a ratio of $(1-ε)$ to the optimum for Max-Cut on bipartite graphs in the $\mathcal{CONGEST}$ model, and on general graphs in the $\mathcal{LOCAL}$ model. We further present efficient deterministic algorithms, including a $1/3$-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of $1/4$. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest.

cs.DS↗

arXiv2017

Broadcasting in Noisy Radio Networks

The widely-studied radio network model [Chlamtac and Kutten, 1985] is a graph-based description that captures the inherent impact of collisions in wireless communication. In this model, the strong assumption is made that node $v$ receives a message from a neighbor if and only if exactly one of its neighbors broadcasts. We relax this assumption by introducing a new noisy radio network model in which random faults occur at senders or receivers. Specifically, for a constant noise parameter $p \in [0,1)$, either every sender has probability $p$ of transmitting noise or every receiver of a single transmission in its neighborhood has probability $p$ of receiving noise. We first study single-message broadcast algorithms in noisy radio networks and show that the Decay algorithm [Bar-Yehuda et al., 1992] remains robust in the noisy model while the diameter-linear algorithm of Gasieniec et al., 2007 does not. We give a modified version of the algorithm of Gasieniec et al., 2007 that is robust to sender and receiver faults, and extend both this modified algorithm and the Decay algorithm to robust multi-message broadcast algorithms. We next investigate the extent to which (network) coding improves throughput in noisy radio networks. We address the previously perplexing result of Alon et al. 2014 that worst case coding throughput is no better than worst case routing throughput up to constants: we show that the worst case throughput performance of coding is, in fact, superior to that of routing -- by a $Θ(\log(n))$ gap -- provided receiver faults are introduced. However, we show that any coding or routing scheme for the noiseless setting can be transformed to be robust to sender faults with only a constant throughput overhead. These transformations imply that the results of Alon et al., 2014 carry over to noisy radio networks with sender faults.

cs.DC↗

arXiv2017

Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model

We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires $Ω(n^2/\log^2{n})$ rounds in the worst case in the CONGEST model, as well as any algorithm for $χ$-coloring a graph, where $χ$ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple $Ω(n)$ lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a complexity of $Θ(n/\log{n})$. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.

cs.DC↗

arXiv2017

Making Asynchronous Distributed Computations Robust to Noise

We consider the problem of making distributed computations robust to noise, in particular to worst-case (adversarial) corruptions of messages. We give a general distributed interactive coding scheme which simulates any asynchronous distributed protocol while tolerating an optimal corruption of a $Θ(1/n)$ fraction of all messages while incurring a moderate blowup of $O(n\log^2 n)$ in the communication complexity. Our result is the first fully distributed interactive coding scheme in which the topology of the communication network is not known in advance. Prior work required either a coordinating node to be connected to all other nodes in the network or assumed a synchronous network in which all nodes already know the complete topology of the network.

cs.DS↗

arXiv2016

Derandomizing Local Distributed Algorithms under Bandwidth Restrictions

This paper addresses the cornerstone family of \emph{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 $2^{O(\sqrt{\log n})}$, which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based $O(Δ+ \log^* n)$-round algorithm, where $Δ$ is the maximal degree in the graph. On the way to obtaining the above, we show that in the \emph{Congested Clique} model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in $O(\log Δ\log n)$ rounds.%, where $Δ$ is the maximum degree. When $Δ=O(n^{1/3})$, the bound improves to $O(\log Δ)$ and holds also for $(Δ+1)$-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.

cs.DC↗

arXiv2016

Distributed Construction of Purely Additive Spanners

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.

cs.DS↗

arXiv2016

Near-Linear Lower Bounds for Distributed Distance Computations, Even in Sparse Networks

We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an $\widetildeΩ(n)$ lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [Frishknecht et al., SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show $\widetildeΩ(n)$ lower bounds for computing $(\frac{3}{2}-\varepsilon)$-approximations of the diameter or the radius, and for computing a $(\frac{5}{3}-\varepsilon)$-approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [Frishknecht et al., SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of $(α,β)$-spanners, for $α< β+1$.

cs.DC↗

arXiv2016

Fast Distributed Algorithms for Testing Graph Properties

We initiate a thorough study of \emph{distributed property testing} -- producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called \emph{dense} testing model we emulate sequential tests for nearly all graph properties having $1$-sided tests, while in the \emph{sparse} and \emph{general} models we obtain faster tests for triangle-freeness and bipartiteness respectively. 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 is of independent interest. This includes a method for distributed maintenance of multiple random walks.

cs.DC↗

arXiv2016

A Distributed $(2+ε)$-Approximation for Vertex Cover in $O(\logΔ/ε\log\logΔ)$ Rounds

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

cs.DC↗

arXiv2015

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.

cs.DC↗

arXiv2015

Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.

cs.DC↗

arXiv2015

On Fast and Robust Information Spreading in the Vertex-Congest Model

This paper initiates the study of the impact of failures on the fundamental problem of \emph{information spreading} in the Vertex-Congest model, in which in every round, each of the $n$ nodes sends the same $O(\log{n})$-bit message to all of its neighbors. Our contribution to coping with failures is twofold. First, we prove that the randomized algorithm which chooses uniformly at random the next message to forward is slow, requiring $Ω(n/\sqrt{k})$ rounds on some graphs, which we denote by $G_{n,k}$, where $k$ is the vertex-connectivity. Second, we design a randomized algorithm that makes dynamic message choices, with probabilities that change over the execution. We prove that for $G_{n,k}$ it requires only a near-optimal number of $O(n\log^3{n}/k)$ rounds, despite a rate of $q=O(k/n\log^3{n})$ failures per round. Our technique of choosing probabilities that change according to the execution is of independent interest.

cs.DC↗

arXiv2015

Computing Exact Distances in the Congested Clique

This paper gives simple distributed algorithms for the fundamental problem of computing graph distances in the Congested Clique model. One of the main components of our algorithms is fast matrix multiplication, for which we show an $O(n^{1/3})$-round algorithm when the multiplication needs to be performed over a semi-ring, and an $O(n^{0.157})$-round algorithm when the computation can be performed over a field. We propose to denote by $κ$ the exponent of matrix multiplication in this model, which gives $κ< 0.157$. We show how to compute all-pairs-shortest-paths (APSP) in $O(n^{1/3}\log{n})$ rounds in weighted graphs of $n$ nodes, implying also the computation of the graph diameter $D$. In unweighted graphs, APSP can be computed in $O(\min\{n^{1/3}\log{D},n^κ D\})$ rounds, and the diameter can be computed in $O(n^κ\log{D})$ rounds. Furthermore, we show how to compute the girth of a graph in $O(n^{1/3})$ rounds, and provide triangle detection and 4-cycle detection algorithms that complete in $O(n^κ)$ rounds. All our algorithms are deterministic. Our triangle detection and 4-cycle detection algorithms improve upon the previously best known algorithms in this model, and refute a conjecture that $\tilde Ω(n^{1/3})$ rounds are required for detecting triangles by any deterministic oblivious algorithm. Our distance computation algorithms are exact, and improve upon the previously best known $\tilde O(n^{1/2})$ algorithm of Nanongkai [STOC 2014] for computing a $(2+o(1))$-approximation of APSP. Finally, we give lower bounds that match the above for natural families of algorithms. For the Congested Clique Broadcast model, we derive unconditioned lower bounds for matrix multiplication and APSP. The matrix multiplication algorithms and lower bounds are adapted from parallel computations, which is a connection of independent interest.

cs.DC↗