arXiv subjects

Mohsen Ghaffari

Publications and source records attributed to Mohsen Ghaffari.

88 records · Page 5Linked to original sources

arXiv2015

Near-Optimal Distributed Maximum Flow

We present a near-optimal distributed algorithm for $(1+o(1))$-approximation of single-commodity maximum flow in undirected weighted networks that runs in $(D+ \sqrt{n})\cdot n^{o(1)}$ communication rounds in the \Congest model. Here, $n$ and $D$ denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of $O(n^2)$, and it nearly matches the $\tildeΩ(D+ \sqrt{n})$ round complexity lower bound. The development of the algorithm contains two results of independent interest: (i) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of a spanning tree of average stretch $n^{o(1)}$. (ii) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of an $n^{o(1)}$-congestion approximator consisting of the cuts induced by $O(\log n)$ virtual trees. The distributed representation of the cut approximator allows for evaluation in $(D+\sqrt{n})\cdot n^{o(1)}$ rounds. All our algorithms make use of randomization and succeed with high probability.

cs.DS↗

arXiv2015

An Improved Distributed Algorithm for Maximal Independent Set

The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some recent techniques, also leads to a near-optimal global complexity. Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability, all nodes terminate after $O(\log n)$ rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node $v$ terminates after $O(\log \mathsf{deg}(v)+\log 1/ε)$ rounds, with probability at least $1-ε$. The guarantee holds even if the randomness outside $2$-hops neighborhood of $v$ is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high probability global complexity of $O(\log Δ) + 2^{O(\sqrt{\log \log n})}$, where $Δ$ denotes the maximum degree. This improves over the $O(\log^2 Δ) + 2^{O(\sqrt{\log \log n})}$ result of Barenboim et al., and gets close to the $Ω(\min\{\log Δ, \sqrt{\log n}\})$ lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lovász Local Lemma.

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arXiv2015

Distributed House-Hunting in Ant Colonies

We introduce the study of the ant colony house-hunting problem from a distributed computing perspective. When an ant colony's nest becomes unsuitable due to size constraints or damage, the colony must relocate to a new nest. The task of identifying and evaluating the quality of potential new nests is distributed among all ants. The ants must additionally reach consensus on a final nest choice and the full colony must be transported to this single new nest. Our goal is to use tools and techniques from distributed computing theory in order to gain insight into the house-hunting process. We develop a formal model for the house-hunting problem inspired by the behavior of the Temnothorax genus of ants. We then show a Ω(log n) lower bound on the time for all n ants to agree on one of k candidate nests. We also present two algorithms that solve the house-hunting problem in our model. The first algorithm solves the problem in optimal O(log n) time but exhibits some features not characteristic of natural ant behavior. The second algorithm runs in O(k log n) time and uses an extremely simple and natural rule for each ant to decide on the new nest.

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arXiv2014

On the Importance of Registers for Computability

All consensus hierarchies in the literature assume that we have, in addition to copies of a given object, an unbounded number of registers. But why do we really need these registers? This paper considers what would happen if one attempts to solve consensus using various objects but without any registers. We show that under a reasonable assumption, objects like queues and stacks cannot emulate the missing registers. We also show that, perhaps surprisingly, initialization, shown to have no computational consequences when registers are readily available, is crucial in determining the synchronization power of objects when no registers are allowed. Finally, we show that without registers, the number of available objects affects the level of consensus that can be solved. Our work thus raises the question of whether consensus hierarchies which assume an unbounded number of registers truly capture synchronization power, and begins a line of research aimed at better understanding the interaction between read-write memory and the powerful synchronization operations available on modern architectures.

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arXiv2014

Broadcast Throughput in Radio Networks: Routing vs. Network Coding

The broadcast throughput in a network is defined as the average number of messages that can be transmitted per unit time from a given source to all other nodes when time goes to infinity. Classical broadcast algorithms treat messages as atomic tokens and route them from the source to the receivers by making intermediate nodes store and forward messages. The more recent network coding approach, in contrast, prompts intermediate nodes to mix and code together messages. It has been shown that certain wired networks have an asymptotic network coding gap, that is, they have asymptotically higher broadcast throughput when using network coding compared to routing. Whether such a gap exists for wireless networks has been an open question of great interest. We approach this question by studying the broadcast throughput of the radio network model which has been a standard mathematical model to study wireless communication. We show that there is a family of radio networks with a tight $Θ(\log \log n)$ network coding gap, that is, networks in which the asymptotic throughput achievable via routing messages is a $Θ(\log \log n)$ factor smaller than that of the optimal network coding algorithm. We also provide new tight upper and lower bounds that show that the asymptotic worst-case broadcast throughput over all networks with $n$ nodes is $Θ(1 / \log n)$ messages-per-round for both routing and network coding.

cs.DS↗

arXiv2014

Multi-Message Broadcast with Abstract MAC Layers and Unreliable Links

We study the multi-message broadcast problem using abstract MAC layer models of wireless networks. These models capture the key guarantees of existing MAC layers while abstracting away low-level details such as signal propagation and contention. We begin by studying upper and lower bounds for this problem in a {\em standard abstract MAC layer model}---identifying an interesting dependence between the structure of unreliable links and achievable time complexity. In more detail, given a restriction that devices connected directly by an unreliable link are not too far from each other in the reliable link topology, we can (almost) match the efficiency of the reliable case. For the related restriction, however, that two devices connected by an unreliable link are not too far from each other in geographic distance, we prove a new lower bound that shows that this efficiency is impossible. We then investigate how much extra power must be added to the model to enable a new order of magnitude of efficiency. In more detail, we consider an {\em enhanced abstract MAC layer model} and present a new multi-message broadcast algorithm that (under certain natural assumptions) solves the problem in this model faster than any known solutions in an abstract MAC layer setting.

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arXiv2014

Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set

This paper presents a near-optimal distributed approximation algorithm for the minimum-weight connected dominating set (MCDS) problem. The presented algorithm finds an $O(\log n)$ approximation in $\tilde{O}(D+\sqrt{n})$ rounds, where $D$ is the network diameter and $n$ is the number of nodes. MCDS is a classical NP-hard problem and the achieved approximation factor $O(\log n)$ is known to be optimal up to a constant factor, unless P=NP. Furthermore, the $\tilde{O}(D+\sqrt{n})$ round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.---STOC'11].

cs.DS↗

arXiv2014

Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding

We study coding schemes for error correction in interactive communications. Such interactive coding schemes simulate any $n$-round interactive protocol using $N$ rounds over an adversarial channel that corrupts up to $ρN$ transmissions. Important performance measures for a coding scheme are its maximum tolerable error rate $ρ$, communication complexity $N$, and computational complexity. We give the first coding scheme for the standard setting which performs optimally in all three measures: Our randomized non-adaptive coding scheme has a near-linear computational complexity and tolerates any error rate $δ< 1/4$ with a linear $N = Θ(n)$ communication complexity. This improves over prior results which each performed well in two of these measures. We also give results for other settings of interest, namely, the first computationally and communication efficient schemes that tolerate $ρ< \frac{2}{7}$ adaptively, $ρ< \frac{1}{3}$ if only one party is required to decode, and $ρ< \frac{1}{2}$ if list decoding is allowed. These are the optimal tolerable error rates for the respective settings. These coding schemes also have near linear computational and communication complexity. These results are obtained via two techniques: We give a general black-box reduction which reduces unique decoding, in various settings, to list decoding. We also show how to boost the computational and communication efficiency of any list decoder to become near linear.

cs.DS↗

arXiv2014

Fast Structuring of Radio Networks for Multi-Message Communications

We introduce collision free layerings as a powerful way to structure radio networks. These layerings can replace hard-to-compute BFS-trees in many contexts while having an efficient randomized distributed construction. We demonstrate their versatility by using them to provide near optimal distributed algorithms for several multi-message communication primitives. Designing efficient communication primitives for radio networks has a rich history that began 25 years ago when Bar-Yehuda et al. introduced fast randomized algorithms for broadcasting and for constructing BFS-trees. Their BFS-tree construction time was $O(D \log^2 n)$ rounds, where $D$ is the network diameter and $n$ is the number of nodes. Since then, the complexity of a broadcast has been resolved to be $T_{BC} = Θ(D \log \frac{n}{D} + \log^2 n)$ rounds. On the other hand, BFS-trees have been used as a crucial building block for many communication primitives and their construction time remained a bottleneck for these primitives. We introduce collision free layerings that can be used in place of BFS-trees and we give a randomized construction of these layerings that runs in nearly broadcast time, that is, w.h.p. in $T_{Lay} = O(D \log \frac{n}{D} + \log^{2+ε} n)$ rounds for any constant $ε>0$. We then use these layerings to obtain: (1) A randomized algorithm for gathering $k$ messages running w.h.p. in $O(T_{Lay} + k)$ rounds. (2) A randomized $k$-message broadcast algorithm running w.h.p. in $O(T_{Lay} + k \log n)$ rounds. These algorithms are optimal up to the small difference in the additive poly-logarithmic term between $T_{BC}$ and $T_{Lay}$. Moreover, they imply the first optimal $O(n \log n)$ round randomized gossip algorithm.

cs.NI↗

arXiv2014

Near Optimal Leader Election in Multi-Hop Radio Networks

We present distributed randomized leader election protocols for multi-hop radio networks that elect a leader in almost the same time $T_{BC}$ required for broadcasting a message. For the setting without collision detection, our algorithm runs with high probability in $O(D \log \frac{n}{D} + \log^3 n) \min\{\log\log n,\log \frac{n}{D}\}$ rounds on any $n$-node network with diameter $D$. Since $T_{BC} = Θ(D \log \frac{n}{D} + \log^2 n)$ is a lower bound, our upper bound is optimal up to a factor of at most $\log \log n$ and the extra $\log n$ factor on the additive term. This algorithm is furthermore the first $O(n)$ time algorithm for this setting. Our algorithms improve over a 25 year old simulation approach of Bar-Yehuda, Goldreich and Itai with a $O(T_{BC} \log n)$ running time: In 1987 they designed a fast broadcast protocol and subsequently in 1989 they showed how it can be used to simulate one round of a single-hop network that has collision detection in $T_{BC}$ time. The prime application of this simulation was to simulate Willards single-hop leader election protocol, which elects a leader in $O(\log n)$ rounds with high probability and $O(\log \log n)$ rounds in expectation. While it was subsequently shown that Willards bounds are tight, it was unclear whether the simulation approach is optimal. Our results break this barrier and essentially remove the logarithmic slowdown over the broadcast time $T_{BC}$ by going away from the simulation approach. We also give a distributed randomized leader election algorithm for the setting with collision detection that runs in $O(D + \log n \log \log n) \cdot \min\{\log \log n, \log \frac{n}{D}\}$ rounds. This round complexity is optimal up to $O(\log \log n)$ factors and improves over a deterministic algorithm that requires $Θ(n)$ rounds independently of the diameter $D$.

cs.DS↗

arXiv2014

Randomized Broadcast in Radio Networks with Collision Detection

We present a randomized distributed algorithm that in radio networks with collision detection broadcasts a single message in $O(D + \log^6 n)$ rounds, with high probability. This time complexity is most interesting because of its optimal additive dependence on the network diameter $D$. It improves over the currently best known $O(D\log\frac{n}{D}\,+\,\log^2 n)$ algorithms, due to Czumaj and Rytter [FOCS 2003], and Kowalski and Pelc [PODC 2003]. These algorithms where designed for the model without collision detection and are optimal in that model. However, as explicitly stated by Peleg in his 2007 survey on broadcast in radio networks, it had remained an open question whether the bound can be improved with collision detection. We also study distributed algorithms for broadcasting $k$ messages from a single source to all nodes. This problem is a natural and important generalization of the single-message broadcast problem, but is in fact considerably more challenging and less understood. We show the following results: If the network topology is known to all nodes, then a $k$-message broadcast can be performed in $O(D + k\log n + \log^2 n)$ rounds, with high probability. If the topology is not known, but collision detection is available, then a $k$-message broadcast can be performed in $O(D + k\log n + \log^6 n)$ rounds, with high probability. The first bound is optimal and the second is optimal modulo the additive $O(\log^6 n)$ term.

cs.DS↗

arXiv2013

Distributed Connectivity Decomposition

We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity $k$ into (fractionally) vertex-disjoint weighted dominating trees with total weight $Ω(\frac{k}{\log n})$, in $\widetilde{O}(D+\sqrt{n})$ rounds. (II) A decomposition of each undirected graph with edge-connectivity $λ$ into (fractionally) edge-disjoint weighted spanning trees with total weight $\lceil\frac{λ-1}{2}\rceil(1-\varepsilon)$, in $\widetilde{O}(D+\sqrt{nλ})$ rounds. We also show round complexity lower bounds of $\tildeΩ(D+\sqrt{\frac{n}{k}})$ and $\tildeΩ(D+\sqrt{\frac{n}λ})$ for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from $O(n^3)$ to near-optimal $\tilde{O}(m)$. As corollaries, we also get distributed oblivious routing broadcast with $O(1)$-competitive edge-congestion and $O(\log n)$-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal $O(\log n)$-approximation of vertex connectivity: centralized $\widetilde{O}(m)$ and distributed $\tilde{O}(D+\sqrt{n})$. The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an $O(m)$ centralized exact algorithm while the latter is the first distributed vertex connectivity approximation.

cs.DS↗

arXiv2013

Distributed Minimum Cut Approximation

We study the problem of computing approximate minimum edge cuts by distributed algorithms. We use a standard synchronous message passing model where in each round, $O(\log n)$ bits can be transmitted over each edge (a.k.a. the CONGEST model). We present a distributed algorithm that, for any weighted graph and any $ε\in (0, 1)$, with high probability finds a cut of size at most $O(ε^{-1}λ)$ in $O(D) + \tilde{O}(n^{1/2 + ε})$ rounds, where $λ$ is the size of the minimum cut. This algorithm is based on a simple approach for analyzing random edge sampling, which we call the random layering technique. In addition, we also present another distributed algorithm, which is based on a centralized algorithm due to Matula [SODA '93], that with high probability computes a cut of size at most $(2+ε)λ$ in $\tilde{O}((D+\sqrt{n})/ε^5)$ rounds for any $ε>0$. The time complexities of both of these algorithms almost match the $\tildeΩ(D + \sqrt{n})$ lower bound of Das Sarma et al. [STOC '11], thus leading to an answer to an open question raised by Elkin [SIGACT-News '04] and Das Sarma et al. [STOC '11]. Furthermore, we also strengthen the lower bound of Das Sarma et al. by extending it to unweighted graphs. We show that the same lower bound also holds for unweighted multigraphs (or equivalently for weighted graphs in which $O(w\log n)$ bits can be transmitted in each round over an edge of weight $w$), even if the diameter is $D=O(\log n)$. For unweighted simple graphs, we show that even for networks of diameter $\tilde{O}(\frac{1}λ\cdot \sqrt{\frac{n}{αλ}})$, finding an $α$-approximate minimum cut in networks of edge connectivity $λ$ or computing an $α$-approximation of the edge connectivity requires $\tildeΩ(D + \sqrt{\frac{n}{αλ}})$ rounds.

cs.DS↗

arXiv2013

A New Perspective on Vertex Connectivity

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoint spanning trees and that help us to better grasp the structure of graphs based on their vertex connectivity. The objective of the CDS partition problem is to partition the nodes of a graph into as many connected dominating sets as possible. The CDS packing problem is the corresponding fractional relaxation, where CDSs are allowed to overlap as long as this is compensated by assigning appropriate weights. CDS partition and CDS packing can be viewed as the counterparts of the well-studied edge-disjoint spanning trees, focusing on vertex disjointedness rather than edge disjointness. We constructively show that every $k$-vertex-connected graph with $n$ nodes has a CDS packing of size $Ω(k/\log n)$ and a CDS partition of size $Ω(k/\log^5 n)$. We prove that the $Ω(k/\log n)$ CDS packing bound is existentially optimal. Using CDS packing, we show that if vertices of a $k$-vertex-connected graph are independently sampled with probability $p$, then the graph induced by the sampled vertices has vertex connectivity $\tildeΩ(kp^2)$. Moreover, using our $Ω(k/\log n)$ CDS packing, we get a store-and-forward broadcast algorithm with optimal throughput in the networking model where in each round, each node can send one bounded-size message to all its neighbors.

cs.DM↗

arXiv2013

A Bound on the Throughput of Radio Networks

We consider the well-studied radio network model: a synchronous model with a graph G=(V,E) with |V|=n where in each round, each node either transmits a packet, with length B=Omega(log n) bits, or listens. Each node receives a packet iff it is listening and exactly one of its neighbors is transmitting. We consider the problem of k-message broadcast, where k messages, each with Theta(B) bits, are placed in an arbitrary nodes of the graph and the goal is to deliver all messages to all the nodes. We present a simple proof showing that there exist a radio network with radius 2 where for any k, broadcasting k messages requires at least Omega(k log n) rounds. That is, in this network, regardless of the algorithm, the maximum achievable broadcast throughput is O(1/log n).

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arXiv2012

Bounds on Contention Management in Radio Networks

The local broadcast problem assumes that processes in a wireless network are provided messages, one by one, that must be delivered to their neighbors. In this paper, we prove tight bounds for this problem in two well-studied wireless network models: the classical model, in which links are reliable and collisions consistent, and the more recent dual graph model, which introduces unreliable edges. Our results prove that the Decay strategy, commonly used for local broadcast in the classical setting, is optimal. They also establish a separation between the two models, proving that the dual graph setting is strictly harder than the classical setting, with respect to this primitive.

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