## About

100

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Introduction

Merav Parter currently works at the Computer Science, Weizmann. Merav does research in Communication Design. Their most recent publication is 'Distributed Algorithms Made Secure: A Graph Theoretic Approach'.

## Publications

Publications (100)

We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 200...

In this paper we present succinct labeling schemes for supporting connectivity queries under vertex faults. For a given $n$-vertex graph $G$, an $f$-VFT (resp., EFT) connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of a vertex pair $u$ and $v$, a...

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain \(\mathrm{poly}(\log \log n) \) -round algorithms for the following problems in unweighted undirected n -vertex graphs: • (1 + ϵ)-approximation of multi-source shortest paths (MSSP) from \(O(\sqrt {n}) \) s...

Theoretical study of optimization problems in wireless communication often deals with tasks that concern a single point. For example, the power control problem requires computing a power assignment guaranteeing that each transmitting station s i is successfully received at a single receiver point r i . This paper aims at addressing communication ap...

We present a deterministic $O(\log \log \log n)$-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of $(\Delta+1)$-coloring on $n$-vertex graphs. In this model, every machine has a sublinear local memory of size $n^{\phi}$ for any arbitrary constant $\phi \in (0,1)$. Our algorithm works under the relaxed setti...

For an $n$-vertex digraph $G=(V,E)$, a \emph{shortcut set} is a (small) subset of edges $H$ taken from the transitive closure of $G$ that, when added to $G$ guarantees that the diameter of $G \cup H$ is small. Shortcut sets, introduced by Thorup in 1993, have a wide range of applications in algorithm design, especially in the context of parallel, d...

Low congestion shortcuts, introduced by Ghaffari and Haeupler (SODA 2016), provide a unified framework for global optimization problems in the congest model of distributed computing. Roughly speaking, for a given graph $G$ and a collection of vertex-disjoint connected subsets $S_1,\ldots, S_\ell \subseteq V(G)$, $(c,d)$ low-congestion shortcuts aug...

We study the power and limitations of component-stable algorithms in the low-space model of Massively Parallel Computation (MPC). Recently Ghaffari, Kuhn and Uitto (FOCS 2019) introduced the class of component-stable low-space MPC algorithms, which are, informally, defined as algorithms for which the outputs reported by the nodes in different conne...

This paper concerns the behavior of an SINR diagram of wireless systems, composed of a set S of n stations embedded in Rd, when restricted to the corresponding Voronoi diagram imposed on S. The diagram obtained by restricting the SINR zones to their corresponding Voronoi cells is referred to hereafter as an SINR+Voronoi diagram.
Uniform SINR diagra...

The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given $n$-vertex graph $G$ and a bound $f$ on the number of faults, an $f$-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the lab...

The Massively Parallel Computation (MPC) model is an emerging model that distills core aspects of distributed and parallel computation, developed as a tool to solve combinatorial (typically graph) problems in systems of many machines with limited space.
Recent work has focused on the regime in which machines have sublinear (in n , the number of nod...

The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical short...

Following the immense recent advances in distributed networks, the explosive growth of the Internet, and our increased dependence on these infrastructures, guaranteeing the uninterrupted operation of communication networks has become a major objective in network algorithms. The modern instantiations of distributed networks, such as, the Bitcoin net...

This paper addresses the problem of designing a \(\beta \)-additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances sati...

Fault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied mainly from a centralized viewpoint. Despite the fact fault tolerant preservers are mainly motivated by the error-prone nature of distribut...

We initiate the study of biological neural networks from the perspective of streaming algorithms. Like computers, human brains suffer from memory limitations which pose a significant obstacle when processing large scale and dynamically changing data. In computer science, these challenges are captured by the well-known streaming model, which can be...

We settle the complexity of the $(\Delta+1)$-coloring and $(\Delta+1)$-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round $(\Delta+1)$-list coloring algorithm due...

In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph $G$, a vertex pair $(s,t) \in V(G)\times V(G)$, and a set of edge faults $F \subseteq E(G)$, a replacement path $P(s,t,F)$ is an $s$-$t$ shortest path in $G \setminus F$. For integer parameters $...

Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of $\lfloor k/2 \rfloor$ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the t...

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 messa...

We present the first round efficient algorithms for several fundamental distributed tasks in the presence of a Byzantine edge. Our algorithms work in the CONGEST model of distributed computing. In the \emph{Byzantine Broadcast} problem, given is a network $G=(V,E)$ with an unknown Byzantine edge $e'$. There is a source node $s$ holding an initial m...

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain $poly(\log\log n)$-round algorithms for the following problems in unweighted undirected $n$-vertex graphs: -- $(1+\epsilon)$-approximation of multi-source shortest paths (MSSP) from $O(\sqrt{n})$ sources. -...

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 $\widetilde{O}(|S|\cdot poly(D)+|T|)$ b...

The Massively Parallel Computation (MPC) model is an emerging model which distills core aspects of distributed and parallel computation. The aim is to solve (typically graph) problems in systems where the input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear memory, with r...

We revisit classical connectivity problems in the CONGEST model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with $\widetilde{O}(1)$ rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of $\Omega(Dia...

An $f(d)$-spanner of an unweighted $n$-vertex graph $G=(V,E)$ is a subgraph $H$ satisfying that $dist_H(u, v)$ is at most $f(dist_G(u, v))$ for every $u,v \in V$. We present new spanner constructions that achieve a nearly optimal stretch of $O(\lceil k /d \rceil)$ for any distance value $d \in [1,k^{1-o(1)}]$, and $d \geq k^{1+o(1)}$. We show the f...

In this paper, we study secure distributed algorithms that are nearly optimal, with respect to running time, for the given input graph G. Roughly speaking, an algorithm is secure if the nodes learn only their final output while gaining no information on the input (or output) of other nodes.
A graph theoretic framework for secure distributed computa...

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]. In our variant of the Planar Graph Metric Compression Problem, one is given an n-vertex planar graph G=(V,E), a set of S ⊆ V source terminals lying on a single f...

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 con...

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 mes...

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 ne...

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 essent...

Merav Parter, Ronitt Rubinfeld, Ali Vakilian, and Anak Yodpinyanee. 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...

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...

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...

This paper introduces an extended notion of expansion suitable for radio networks. A graph G=(V,E) is said to be an (α_w, β_w) -\em wireless expander if for every subset S \subseteq V s.t. |S|łeq 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 that are adjacent in G to exactly one...

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 dist...

In this paper, we present improved algorithms for the $(\Delta+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its incident edges, and per round each two nodes can exchange $O(\log n)$ bits of information. Our key result is a rando...

This paper introduces an extended notion of expansion suitable for radio networks. A graph $G=(V,E)$ is called an $(\alpha_w, \beta_w)$-{wireless expander} if for every subset $S \subseteq V$ s.t. $|S|\leq \alpha_w \cdot |V|$, there exists a subset $S'\subseteq S$ s.t. there are at least $\beta_w \cdot |S|$ vertices in $V\backslash S$ adjacent in $...

A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. This article addresses the problem of designing a fault-tolerant (α , β) approximate BFS structure (or FT-ABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure...

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 bu...

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 $...

A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a fault-tolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H sti...

This note considers a 1-dimensional wireless network consisting of a set of n stations located on a line, in the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. The behavior of a multi-station network is described using the convenient representa...

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 me...

We study distributed algorithms implemented in a simplified but biologically plausible 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 rece...

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 d...

Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, denoted by N[P]. A path is said to be secluded if...

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...

A fault-tolerant structure for a network is required for continued functioning following the failure of some of the network's edges or vertices. This article considers breadth-first search (BFS) spanning trees and addresses the problem of designing a sparse fault-tolerant BFS structure (FT-BFS structure), namely, a sparse subgraph T of the given ne...

In this paper we study the reception zones of a wireless network in the SINR model with receivers that employ interference cancellation (IC), a technique that allows a receiver to decode interfering signals, and cancel them from the received signal in order to decode its intended message. We first derive some important topological properties of the...

We present a randomized algorithm that computes a Minimum Spanning Tree (MST) in O(log* n) rounds, with high probability, in the Congested Clique model of distributed computing. In this model, the input is a graph on n nodes, initially each node knows only its incident edges, and per round each two nodes can exchange O(log n) bits. Our key technica...

Consider n anonymous nodes each initially supporting an opinion in {1, 2, …, k} and suppose that they should all learn the opinion with the largest support. Per round, each node contacts a random other node and exchanges B bits with it, where typically B is at most O(log n).
This basic distributed computing problem is called the plurality consensus...

Tree structures such as breadth-first search (BFS) trees and minimum spanning trees (MST) are among the most fundamental graph structures in distributed network algorithms. However, by definition, these structures are not robust against failures and even a single edge's removal can disrupt their functionality. A well-studied concept which attempts...

A distributed task is local if its time complexity is (nearly) constant, otherwise it is global. Unfortunately, local tasks are relatively scarce, and most distributed tasks require time at least logarithmic in the network size (and often higher than that). In a dynamic setting, i.e., when the network undergoes repeated and frequent topological cha...

This paper concerns the behavior of an SINR diagram of wireless systems, composed of a set S of n stations embedded in \(\mathbb R^d\), when restricted to the corresponding Voronoi diagram imposed on S. The diagram obtained by restricting the SINR zones to their corresponding Voronoi cells is referred to hereafter as an SINR+Voronoi diagram.
While...

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...

In this review, we illustrate the relations between wireless communication and computational geometry. As a concrete example, we consider a fundamental geometric object from each field: SINR diagrams and Voronoi diagrams. We discuss the relations between these representations, which appear in several distinct settings of wireless communication, as...

This paper initiates the study of fault resilient network structures that mix two orthogonal protection mechanisms:(a) backup, namely, augmenting the structure with many (redundant) low-cost but fault-prone components, and (b) reinforcement, namely, acquiring high-cost but fault-resistant components. To study the trade-off between these two mechani...

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, t...

This paper initiates the study of fault resilient network structures that mix
two orthogonal protection mechanisms: (a) {\em backup}, namely, augmenting the
structure with many (redundant) low-cost but fault-prone components, and (b)
{\em reinforcement}, namely, acquiring high-cost but fault-resistant
components. To study the trade-off between thes...

The Perron–Frobenius (PF) theorem provides a simple characterization of the eigenvectors and eigenvalues of irreducible nonnegative square matrices. A generalization of the PF theorem to nonsquare matrices, which can be interpreted as representing systems with additional degrees of freedom, was recently presented in [1]. This generalized theorem re...

This paper addresses the problem of designing a {\em fault-tolerant}
$(\alpha, \beta)$ approximate BFS structure (or {\em FT-ABFS structure} for
short), namely, a subgraph $H$ of the network $G$ such that subsequent to the
failure of some subset $F$ of edges or vertices, the surviving part of $H$
still contains an \emph{approximate} BFS spanning tr...

An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n
1 + 1/k
). T...

In this paper we study the reception zones of a wireless network in the SINR
model with receivers that employ \emph{interference cancellation} (IC), a
technique that allows a receiver to decode interfering signals, and
\emph{cancel} them from the received signal in order to decode its intended
message. We first derive some important topological pro...

Consider a setting where possibly sensitive information sent over a path in a
network is visible to every {neighbor} of the path, i.e., every neighbor of
some node on the path, thus including the nodes on the path itself. The
exposure of a path $P$ can be measured as the number of nodes adjacent to it,
denoted by $N[P]$. A path is said to be seclud...

The celebrated Perron--Frobenius (PF) theorem is stated for irreducible
nonnegative square matrices, and provides a simple characterization of their
eigenvectors and eigenvalues. The importance of this theorem stems from the
fact that eigenvalue problems on such matrices arise in many fields of science
and engineering, including dynamical systems t...

When comparing new wireless technologies, it is common to consider the effect that they have on the capacity of the network (defined as the maximum number of simultaneously satisfiable links). For example, it has been shown that giving receivers the ability to do interference cancellation, or allowing transmitters to use power control, never decrea...

This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph $T$ of the
given network $G$ such that subsequent to the failure of a single edge or
vertex, the surviving part $T'$ of $T$ still contains a BFS spanning tree for
(the surviving part of) $G$. Our main resul...

The celebrated Perron-Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems th...

The paper tackles the power of randomization in the context of locality by
analyzing the ability to`boost' the success probability of deciding a
distributed language. The main outcome of this analysis is that the distributed
computing setting contrasts significantly with the sequential one as far as
randomization is concerned. Indeed, we prove that...

The power control problem for wireless networks in the SINR model requires determining the optimal power assignment for a set of communication requests such that the SINR threshold is met for all receivers. If the network topology is known to all participants, then it is possible to compute an optimal power assignment in polynomial time. In realist...

In this paper we study the topological properties of wireless communication
maps and their usability in algorithmic design. We consider the SINR model,
which compares the received power of a signal at a receiver against the sum of
strengths of other interfering signals plus background noise. To describe the
behavior of a multi-station network, we u...