arXiv subjects

Merav Parter

Publications and source records attributed to Merav Parter.

At least 55 records · Page 3Linked to original sources

arXiv2019

Planar Diameter via Metric Compression

We develop a new approach for distributed distance computation in planar graphs that is based on a variant of the metric compression problem recently introduced by Abboud et al. [SODA'18]. One of our key technical contributions is in providing a compression scheme that encodes all $S \times T$ distances using $\widetilde{O}(|S|\cdot poly(D)+|T|)$ bits for unweighted graphs with diameter $D$. This significantly improves the state of the art of $\widetilde{O}(|S|\cdot 2^{D}+|T| \cdot D)$ bits. We also consider an approximate version of the problem for \emph{weighted} graphs, where the goal is to encode $(1+ε)$ approximation of the $S \times T$ distances. At the heart of this compact compression scheme lies a VC-dimension type argument on planar graphs. This efficient compression scheme leads to several improvements and simplifications in the setting of diameter computation, most notably in the distributed setting: - There is an $\widetilde{O}(D^5)$-round randomized distributed algorithm for computing the diameter in planar graphs, w.h.p. - There is an $\widetilde{O}(D^3)+ poly(\log n/ε)\cdot D^2$-round randomized distributed algorithm for computing an $(1+ε)$ approximation of the diameter in weighted graphs with polynomially bounded weights, w.h.p. No sublinear round algorithms were known for these problems before. These distributed constructions are based on a new recursive graph decomposition that preserves the (unweighted) diameter of each of the subgraphs up to a logarithmic term. Using this decomposition, we also get an \emph{exact} SSSP tree computation within $\widetilde{O}(D^2)$ rounds.

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arXiv2019

Counting to Ten with Two Fingers: Compressed Counting with Spiking Neurons

We consider the task of measuring time with probabilistic threshold gates implemented by bio-inspired spiking neurons. In the model of spiking neural networks, network evolves in discrete rounds, where in each round, neurons fire in pulses in response to a sufficiently high membrane potential. This potential is induced by spikes from neighboring neurons that fired in the previous round, which can have either an excitatory or inhibitory effect. We first consider a deterministic implementation of a neural timer and show that $Θ(\log t)$ (deterministic) threshold gates are both sufficient and necessary. This raised the question of whether randomness can be leveraged to reduce the number of neurons. We answer this question in the affirmative by considering neural timers with spiking neurons where the neuron $y$ is required to fire for $t$ consecutive rounds with probability at least $1-δ$, and should stop firing after at most $2t$ rounds with probability $1-δ$ for some input parameter $δ\in (0,1)$. Our key result is a construction of a neural timer with $O(\log\log 1/δ)$ spiking neurons. Interestingly, this construction uses only one spiking neuron, while the remaining neurons can be deterministic threshold gates. We complement this construction with a matching lower bound of $Ω(\min\{\log\log 1/δ, \log t\})$ neurons. This provides the first separation between deterministic and randomized constructions in the setting of spiking neural networks. Finally, we demonstrate the usefulness of compressed counting networks for synchronizing neural networks.

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arXiv2019

Winner-Take-All Computation in Spiking Neural Networks

In this work we study biological neural networks from an algorithmic perspective, focusing on understanding tradeoffs between computation time and network complexity. Our goal is to abstract real neural networks in a way that, while not capturing all interesting features, preserves high-level behavior and allows us to make biologically relevant conclusions. Towards this goal, we consider the implementation of algorithmic primitives in a simple yet biologically plausible model of $stochastic\ spiking\ neural\ networks$. In particular, we show how the stochastic behavior of neurons in this model can be leveraged to solve a basic $symmetry-breaking\ task$ in which we are given neurons with identical firing rates and want to select a distinguished one. In computational neuroscience, this is known as the winner-take-all (WTA) problem, and it is believed to serve as a basic building block in many tasks, e.g., learning, pattern recognition, and clustering. We provide efficient constructions of WTA circuits in our stochastic spiking neural network model, as well as lower bounds in terms of the number of auxiliary neurons required to drive convergence to WTA in a given number of steps. These lower bounds demonstrate that our constructions are near-optimal in some cases. This work covers and gives more in-depth proofs of a subset of results originally published in [LMP17a]. It is adapted from the last chapter of C. Musco's Ph.D. thesis [Mus18].

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arXiv2019

Parallel Balanced Allocations: The Heavily Loaded Case

We study parallel algorithms for the classical balls-into-bins problem, in which $m$ balls acting in parallel as separate agents are placed into $n$ bins. Algorithms operate in synchronous rounds, in each of which balls and bins exchange messages once. The goal is to minimize the maximal load over all bins using a small number of rounds and few messages. While the case of $m=n$ balls has been extensively studied, little is known about the heavily loaded case. In this work, we consider parallel algorithms for this somewhat neglected regime of $m\gg n$. The naive solution of allocating each ball to a bin chosen uniformly and independently at random results in maximal load $m/n+Θ(\sqrt{m/n\cdot \log n})$ (for $m\geq n \log n$) w.h.p. In contrast, for the sequential setting Berenbrink et al (SIAM J. Comput 2006) showed that letting each ball join the least loaded bin of two randomly selected bins reduces the maximal load to $m/n+O(\log\log m)$ w.h.p. To date, no parallel variant of such a result is known. We present a simple parallel threshold algorithm that obtains a maximal load of $m/n+O(1)$ w.h.p. within $O(\log\log (m/n)+\log^* n)$ rounds. The algorithm is symmetric (balls and bins all "look the same"), and balls send $O(1)$ messages in expectation per round. The additive term of $O(\log^* n)$ in the complexity is known to be tight for such algorithms (Lenzen and Wattenhofer Distributed Computing 2016). We also prove that our analysis is tight, i.e., algorithms of the type we provide must run for $Ω(\min\{\log\log (m/n),n\})$ rounds w.h.p. Finally, we give a simple asymmetric algorithm (i.e., balls are aware of a common labeling of the bins) that achieves a maximal load of $m/n + O(1)$ in a constant number of rounds w.h.p. Again, balls send only a single message per round, and bins receive $(1+o(1))m/n+O(\log n)$ messages w.h.p.

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arXiv2019

Local Computation Algorithms for Spanners

A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store. To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge $(u,v) \in E$ belongs to the output spanner. Such LCAs give the user the `illusion' that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present the following results: -For general $n$-vertex graphs and $r \in \{2,3\}$, there exists an LCA for $(2r-1)$-spanners with $\widetilde{O}(n^{1+1/r})$ edges and sublinear probe complexity of $\widetilde{O}(n^{1-1/2r})$. These size/stretch tradeoffs are best possible (up to polylogarithmic factors). -For every $k \geq 1$ and $n$-vertex graph with maximum degree $Δ$, there exists an LCA for $O(k^2)$ spanners with $\widetilde{O}(n^{1+1/k})$ edges, probe complexity of $\widetilde{O}(Δ^4 n^{2/3})$, and random seed of size $\mathrm{polylog}(n)$. This improves upon, and extends the work of [Lenzen-Levi, 2018]. We also complement our results by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with $o(m)$ edges.

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arXiv2019

Distributed Algorithms Made Secure: A Graph Theoretic Approach

In the area of distributed graph algorithms a number of network's entities with local views solve some computational task by exchanging messages with their neighbors. Quite unfortunately, an inherent property of most existing distributed algorithms is that throughout the course of their execution, the nodes get to learn not only their own output but rather learn quite a lot on the inputs or outputs of many other entities. This leakage of information might be a major obstacle in settings where the output (or input) of network's individual is a private information. In this paper, we introduce a new framework for \emph{secure distributed graph algorithms} and provide the first \emph{general compiler} that takes any "natural" non-secure distributed algorithm that runs in $r$ rounds, and turns it into a secure algorithm that runs in $\widetilde{O}(r \cdot D \cdot poly(Δ))$ rounds where $Δ$ is the maximum degree in the graph and $D$ is its diameter. The security of the compiled algorithm is information-theoretic but holds only against a semi-honest adversary that controls a single node in the network. This compiler is made possible due to a new combinatorial structure called \emph{private neighborhood trees}: a collection of $n$ trees $T(u_1),\ldots,T(u_n)$, one for each vertex $u_i \in V(G)$, such that each tree $T(u_i)$ spans the neighbors of $u_i$ {\em without going through $u_i$}. Intuitively, each tree $T(u_i)$ allows all neighbors of $u_i$ to exchange a \emph{secret} that is hidden from $u_i$, which is the basic graph infrastructure of the compiler. In a $(d,c)$-private neighborhood trees each tree $T(u_i)$ has depth at most $d$ and each edge $e \in G$ appears in at most $c$ different trees. We show a construction of private neighborhood trees with $d=\widetilde{O}(Δ\cdot D)$ and $c=\widetilde{O}(D)$.

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arXiv2018

The Power of Distributed Verifiers in Interactive Proofs

We explore the power of interactive proofs with a distributed verifier. In this setting, the verifier consists of $n$ nodes and a graph $G$ that defines their communication pattern. The prover is a single entity that communicates with all nodes by short messages. The goal is to verify that the graph $G$ belongs to some language in a small number of rounds, and with small communication bound, i.e., the proof size. This interactive model was introduced by Kol, Oshman and Saxena (PODC 2018) as a generalization of non-interactive distributed proofs. They demonstrated the power of interaction in this setting by constructing protocols for problems as Graph Symmetry and Graph Non-Isomorphism -- both of which require proofs of $Ω(n^2)$-bits without interaction. In this work, we provide a new general framework for distributed interactive proofs that allows one to translate standard interactive protocols to ones where the verifier is distributed with short proof size. We show the following: * Every (centralized) computation that can be performed in time $O(n)$ can be translated into three-round distributed interactive protocol with $O(\log n)$ proof size. This implies that many graph problems for sparse graphs have succinct proofs. * Every (centralized) computation implemented by either a small space or by uniform NC circuit can be translated into a distributed protocol with $O(1)$ rounds and $O(\log n)$ bits proof size for the low space case and $polylog(n)$ many rounds and proof size for NC. * We show that for Graph Non-Isomorphism, there is a 4-round protocol with $O(\log n)$ proof size, improving upon the $O(n \log n)$ proof size of Kol et al. * For many problems we show how to reduce proof size below the naturally seeming barrier of $\log n$. We get a 5-round protocols with proof size $O(\log \log n)$ for a family of problems.

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arXiv2018

Low Congestion Cycle Covers and their Applications

A cycle cover of a bridgeless graph $G$ is a collection of simple cycles in $G$ such that each edge $e$ appears on at least one cycle. The common objective in cycle cover computation is to minimize the total lengths of all cycles. Motivated by applications to distributed computation, we introduce the notion of \emph{low-congestion} cycle covers, in which all cycles in the cycle collection are both \emph{short} and nearly \emph{edge-disjoint}. Formally, a $(d,c)$-cycle cover of a graph $G$ is a collection of cycles in $G$ in which each cycle is of length at most $d$ and each edge participates in at least one cycle and at most $c$ cycles. A-priori, it is not clear that cycle covers that enjoy both a small overlap and a short cycle length even exist, nor if it is possible to efficiently find them. Perhaps quite surprisingly, we prove the following: Every bridgeless graph of diameter $D$ admits a $(d,c)$-cycle cover where $d = \tilde{O}(D)$ and $c=\tilde{O}(1)$. These parameters are existentially tight up to polylogarithmic terms. Furthermore, we show how to extend our result to achieve universally optimal cycle covers. Let $C_e$ is the length of the shortest cycle that covers $e$, and let $OPT(G)= \max_{e \in G} C_e$. We show that every bridgeless graph admits a $(d,c)$-cycle cover where $d = \tilde{O}(OPT(G))$ and $c=\tilde{O}(1)$. We demonstrate the usefulness of low congestion cycle covers in different settings of resilient computation. For instance, we consider a Byzantine fault model where in each round, the adversary chooses a single message and corrupt in an arbitrarily manner. We provide a compiler that turns any $r$-round distributed algorithm for a graph $G$ with diameter $D$, into an equivalent fault tolerant algorithm with $r\cdot poly(D)$ rounds.

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arXiv2018

Congested Clique Algorithms for Graph Spanners

Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling schemes and routing to solving linear systems and spectral sparsification. A $k$-spanner maintains pairwise distances up to multiplicative factor of $k$. It is a folklore that for every $n$-vertex graph $G$, one can construct a $(2k-1)$ spanner with $O(n^{1+1/k})$ edges. In a distributed setting, such spanners can be constructed in the standard CONGEST model using $O(k^2)$ rounds, when randomization is allowed. In this work, we consider spanner constructions in the congested clique model, and show: (1) A randomized construction of a $(2k-1)$-spanner with $\widetilde{O}(n^{1+1/k})$ edges in $O(\log k)$ rounds. The previous best algorithm runs in $O(k)$ rounds. (2) A deterministic construction of a $(2k-1)$-spanner with $\widetilde{O}(n^{1+1/k})$ edges in $O(\log k +(\log\log n)^3)$ rounds. The previous best algorithm runs in $O(k\log n)$ rounds. This improvement is achieved by a new derandomization theorem for hitting sets which might be of independent interest. (3) A deterministic construction of a $O(k)$-spanner with $O(k \cdot n^{1+1/k})$ edges in $O(\log k)$ rounds.

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arXiv2018

Wireless Expanders

This paper introduces an extended notion of expansion suitable for radio networks. A graph $G=(V,E)$ is called an $(α_w, β_w)$-{wireless expander} if for every subset $S \subseteq V$ s.t. $|S|\leq α_w \cdot |V|$, there exists a subset $S'\subseteq S$ s.t. there are at least $β_w \cdot |S|$ vertices in $V\backslash S$ adjacent in $G$ to exactly one vertex in $S'$. The main question we ask is the following: to what extent are ordinary expanders also good {wireless} expanders? We answer this question in a nearly tight manner. On the positive side, we show that any $(α, β)$-expander with maximum degree $Δ$ and $β\geq 1/Δ$ is also a $(α_w, β_w)$ wireless expander for $β_w = Ω(β/ \log (2 \cdot \min\{Δ/ β, Δ\cdot β\}))$. Thus the wireless expansion is smaller than the ordinary expansion by at most a factor logarithmic in $\min\{Δ/ β, Δ\cdot β\}$, which depends on the graph \emph{average degree} rather than maximum degree; e.g., for low arboricity graphs, the wireless expansion matches the ordinary expansion up to a constant. We complement this positive result by presenting an explicit construction of a "bad" $(α, β)$-expander for which the wireless expansion is $β_w = O(β/ \log (2 \cdot \min\{Δ/ β, Δ\cdot β\})$. We also analyze the theoretical properties of wireless expanders and their connection to unique neighbor expanders, and demonstrate their applicability: Our results yield improved bounds for the {spokesmen election problem} that was introduced in the seminal paper of Chlamtac and Weinstein (1991) to devise efficient broadcasting for multihop radio networks. Our negative result yields a significantly simpler proof than that from the seminal paper of Kushilevitz and Mansour (1998) for a lower bound on the broadcast time in radio networks.

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arXiv2017

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $Ω(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.

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arXiv2017

Neuro-RAM Unit with Applications to Similarity Testing and Compression in Spiking Neural Networks

We study distributed algorithms implemented in a simplified biologically inspired model for stochastic spiking neural networks. We focus on tradeoffs between computation time and network complexity, along with the role of randomness in efficient neural computation. It is widely accepted that neural computation is inherently stochastic. In recent work, we explored how this stochasticity could be leveraged to solve the `winner-take-all' leader election task. Here, we focus on using randomness in neural algorithms for similarity testing and compression. In the most basic setting, given two $n$-length patterns of firing neurons, we wish to distinguish if the patterns are equal or $ε$-far from equal. Randomization allows us to solve this task with a very compact network, using $O \left (\frac{\sqrt{n}\log n}ε\right)$ auxiliary neurons, which is sublinear in the input size. At the heart of our solution is the design of a $t$-round neural random access memory, or indexing network, which we call a neuro-RAM. This module can be implemented with $O(n/t)$ auxiliary neurons and is useful in many applications beyond similarity testing. Using a VC dimension-based argument, we show that the tradeoff between runtime and network size in our neuro-RAM is nearly optimal. Our result has several implications -- since our neuro-RAM can be implemented with deterministic threshold gates, it shows that, in contrast to similarity testing, randomness does not provide significant computational advantages for this problem. It also establishes a separation between feedforward networks whose gates spike with sigmoidal probability functions, and well-studied deterministic sigmoidal networks, whose gates output real number sigmoidal values, and which can implement a neuro-RAM much more efficiently.

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arXiv2017

Improved Deterministic Distributed Construction of Spanners

Graph spanners are fundamental graph structures with a wide range of applications in distributed networks. We consider a standard synchronous message passing model where in each round $O(\log n)$ bits can be transmitted over every edge (the CONGEST model). The state of the art of deterministic distributed spanner constructions suffers from large messages. The only exception is the work of Derbel et al. '10, which computes an optimal-sized $(2k-1)$-spanner but uses $O(n^{1-1/k})$ rounds. In this paper, we significantly improve this bound. We present a deterministic distributed algorithm that given an unweighted $n$-vertex graph $G = (V, E)$ and a parameter $k > 2$, constructs a $(2k-1)$-spanner with $O(k \cdot n^{1+1/k})$ edges within $O(2^{k} \cdot n^{1/2 - 1/k})$ rounds for every even $k$. For odd $k$, the number of rounds is $O(2^{k} \cdot n^{1/2 - 1/(2k)})$. For the weighted case, we provide the first deterministic construction of a $3$-spanner with $O(n^{3/2})$ edges that uses $O(\log n)$-size messages and $\widetilde{O}(1)$ rounds. If the nodes have IDs in $[1, Θ(n)]$, then the algorithm works in only $2$ rounds!

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arXiv2017

Preserving Distances in Very Faulty Graphs

Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain {\em replacement paths} around the faulted set. In this paper we make substantial progress on fault tolerant preservers and additive spanners: (1) We present the first truly sub-quadratic size single-pair preservers in unweighted (possibly directed) graphs for \emph{any} fixed number $f$ of faults. Our result indeed generalizes to the single-source case, and can be used to build new fault-tolerant additive spanners (for all pairs). (2) The size of the above single-pair preservers is $O(n^{2-g(f)})$ for some positive function $g$, and grows to $O(n^2)$ for increasing $f$. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size $O(n^{2-ε})$ for $ε>0$, then the additive error has to be $Ω(\eps f)$. This surprisingly matches known upper bounds in the literature. (3) For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is $Θ(n^2)$ for $f\geq 2$ in both directed and undirected graphs, while for $f=1$ the size is $Θ(n)$ in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound. Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure.

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arXiv2016

Computational Tradeoffs in Biological Neural Networks: Self-Stabilizing Winner-Take-All Networks

We initiate a line of investigation into biological neural networks from an algorithmic perspective. We develop a simplified but biologically plausible model for distributed computation in stochastic spiking neural networks and study tradeoffs between computation time and network complexity in this model. Our aim is to abstract real neural networks in a way that, while not capturing all interesting features, preserves high-level behavior and allows us to make biologically relevant conclusions. In this paper, we focus on the important `winner-take-all' (WTA) problem, which is analogous to a neural leader election unit: a network consisting of $n$ input neurons and $n$ corresponding output neurons must converge to a state in which a single output corresponding to a firing input (the `winner') fires, while all other outputs remain silent. Neural circuits for WTA rely on inhibitory neurons, which suppress the activity of competing outputs and drive the network towards a converged state with a single firing winner. We attempt to understand how the number of inhibitors used affects network convergence time. We show that it is possible to significantly outperform naive WTA constructions through a more refined use of inhibition, solving the problem in $O(θ)$ rounds in expectation with just $O(\log^{1/θ} n)$ inhibitors for any $θ$. An alternative construction gives convergence in $O(\log^{1/θ} n)$ rounds with $O(θ)$ inhibitors. We compliment these upper bounds with our main technical contribution, a nearly matching lower bound for networks using $\ge \log\log n$ inhibitors. Our lower bound uses familiar indistinguishability and locality arguments from distributed computing theory. It lets us derive a number of interesting conclusions about the structure of any network solving WTA with good probability, and the use of randomness and inhibition within such a network.

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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.

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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.

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arXiv2015

Dual Failure Resilient BFS Structure

We study {\em breadth-first search (BFS)} spanning trees, and address the problem of designing a sparse {\em fault-tolerant} BFS structure, or {\em FT-BFS } for short, resilient to the failure of up to two edges in the given undirected unweighted graph $G$, i.e., a sparse subgraph $H$ of $G$ such that subsequent to the failure of up to two edges, the surviving part $H'$ of $H$ still contains a BFS spanning tree for (the surviving part of) $G$. FT-BFS structures, as well as the related notion of replacement paths, have been studied so far for the restricted case of a single failure. It has been noted widely that when concerning shortest-paths in a variety of contexts, there is a sharp qualitative difference between a single failure and two or more failures. Our main results are as follows. We present an algorithm that for every $n$-vertex unweighted undirected graph $G$ and source node $s$ constructs a (two edge failure) FT-BFS structure rooted at $s$ with $O(n^{5/3})$ edges. To provide a useful theory of shortest paths avoiding 2 edges failures, we take a principled approach to classifying the arrangement these paths. We believe that the structural analysis provided in this paper may decrease the barrier for understanding the general case of $f\geq 2$ faults and pave the way to the future design of $f$-fault resilient structures for $f \geq 2$. We also provide a matching lower bound, which in fact holds for the general case of $f \geq 1$ and multiple sources $S \subseteq V$. It shows that for every $f\geq 1$, and integer $1 \leq σ\leq n$, there exist $n$-vertex graphs with a source set $S \subseteq V$ of cardinality $σ$ for which any FT-BFS structure rooted at each $s \in S$, resilient to up to $f$-edge faults has $Ω(σ^{1/(f+1)} \cdot n^{2-1/(f+1)})$ edges.

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