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Local Distributed Decision
 In FOCS 2011
"... A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired ..."
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Cited by 17 (11 self)
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A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD(t) (for local decision) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD(t, p, q), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p and rejects incorrect ones with probability at least q. We show that p 2 +q = 1 is a threshold for the containment of LD(t) in BPLD(t, p, q). More precisely, we show that there exists a language that does not belong to LD(t) for any t = o(n) but does belong to BPLD(0, p, q) for any p, q ∈ (0, 1] such that p 2 +q ≤ 1. On the other hand, we show that, restricted to
Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance
, 2012
"... In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each r ..."
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Cited by 12 (3 self)
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In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is wellconnected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ −1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog(n)) rounds in a network of diameter D, with no dependence on the conductance. This is
Distributed approximation algorithms for weighted shortest paths
 In Proc. of the Symp. on Theory of Comp. (STOC
"... A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O(log n) bandwidth restriction on edges (the standard synchronous CONGEST model). In this problem ..."
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Cited by 9 (1 self)
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A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O(log n) bandwidth restriction on edges (the standard synchronous CONGEST model). In this problem every node wants to know its distance to some others nodes. The question whether approximation algorithms help speed up the shortest paths computation (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is wellunderstood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both singlesource shortest paths (SSSP) and allpairs shortest paths (APSP) in the weighted case. Our main result is an algorithm for SSSP. Previous results are the classic O(n)time BellmanFord algorithm and an Õ(n1/2+1/2k + D)time (8kdlog(k + 1)e − 1)approximation algorithm, for any integer k ≥ 1, which follows from the result of Lenzen and PattShamir (STOC 2013). (Note that Lenzen and PattShamir in fact solve a harder problem, and we use Õ(·) to hide the
Local Computation of Nearly Additive Spanners
"... An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β) ..."
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Cited by 6 (3 self)
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An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every nnode graph and integer k ≥ 1, an (α, β)spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic
Fully Dynamic Randomized Algorithms for Graph Spanners
, 2008
"... Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. ..."
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Cited by 4 (0 self)
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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a tspanner of itself, the research as well as applications of spanners invariably deal with a tspanner which has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a tspanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a tspanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
Multipath Spanners
"... This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We s ..."
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Cited by 3 (3 self)
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This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We study in this paper spanners that approximate short cycles, and more generally p edgedisjoint paths with p> 1, between any pair of vertices. For every unweighted graph G, we construct a 2multipath 3spanner of O(n 3/2) edges. In other words, for any two vertices u, v of G, the length of the shortest cycle (with no edge replication) traversing u, v in the spanner is at most thrice the length of the shortest one in G. This construction is shown to be optimal in term of stretch and of size. In a second construction, we produce a 2multipath (2, 8)spanner of O(n 3/2) edges, i.e., the length of the shortest cycle traversing any two vertices have length at most twice the shortest length in G plus eight. For arbitrary p, we observe that, for each integer k � 1, every weighted graph has a pmultipath p(2k−1)spanner with O(pn 1+1/k) edges, leaving open the question whether, with similar size, the stretch of the spanner can be reduced to 2k − 1 for all p> 1.
NearOptimal Distributed Approximation of MinimumWeight Connected Dominating Set
"... This paper presents a nearoptimal distributed approximation algorithm for the minimumweight connected dominating set (MCDS) problem. We use the standard distributed message passing model called the CONGEST model in which in each round each node can send O(log n) bits to each neighbor. The presente ..."
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This paper presents a nearoptimal distributed approximation algorithm for the minimumweight connected dominating set (MCDS) problem. We use the standard distributed message passing model called the CONGEST model in which in each round each node can send O(log n) bits to each neighbor. The presented algorithm finds an O(log n) approximation in Õ(D+√n) rounds, where D is the network diameter and n is the number of nodes. MCDS is a classical NPhard problem and the achieved approximation factor O(log n) is known to be optimal up to a constant factor, unless P = NP. Furthermore, the Õ(D +√n) round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.—STOC’11]. 1 Introduction and Related Work Connected dominating set (CDS) is one of the classical structures studied in graph optimization problems which also has deep roots in networked computation. For instance, CDSs have been used rather extensively in distributed algorithms for wireless networks (see e.g. [2, 3, 5–10, 30, 38, 39]), typically as a globalconnectivity backbone.
Towards a Complexity Theory for Local Distributed Computing
, 2013
"... A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Insp ..."
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Cited by 2 (0 self)
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A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD(t) (for local decision) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD(t, p, q), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p, and rejects incorrect ones with probability at least q. We
Simple, Fast and Deterministic Gossip and Rumor Spreading
"... We study gossip algorithms for the rumor spreading problem which asks each node to deliver a rumor to all nodes in an unknown network. Gossip algorithms allow nodes only to call one neighbor per round and have recently attracted attention as message efficient, simple and robust solutions to the rumo ..."
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We study gossip algorithms for the rumor spreading problem which asks each node to deliver a rumor to all nodes in an unknown network. Gossip algorithms allow nodes only to call one neighbor per round and have recently attracted attention as message efficient, simple and robust solutions to the rumor spreading problem. A long series of papers analyzed the performance of uniform random gossip in which nodes repeatedly call a random neighbor to exchange all rumors with. A main result of this investigation was that uniform gossip comlog n pletes in O( Φ) rounds where Φ is the conductance of the network. More recently, nonuniform random gossip schemes were devised to allow efficient rumor spreading in networks with bottlenecks. In particular, [CensorHillel et al., STOC’12] gave an O(log 3 n) algorithm to solve the 1local broadcast problem in which each node wants to exchange rumors locally with its 1neighborhood. By repeatedly applying this protocol one can solve the global rumor spreading quickly for all networks with small diameter, independently of the conductance. All these algorithms are inherently randomized in their design and analysis. A parallel research direction has been to reduce and determine the amount of randomness needed for efficient rumor spreading. This has been done via lower bounds for restricted models and by designing gossip algorithms with a reduced need for randomness, e.g., by using pseudorandom generators with short random seeds. The general intuition and consensus of these results has been that randomization plays a important role in effectively spreading rumors and that at least a polylogarithmic number of random bit are crucially needed. In this paper we improves over this state of the art in several ways by presenting a deterministic gossip algorithm that solves the the klocal broadcast problem in 2(k + log n) log n rounds1. Besides being the first efficient deterministic solution to the rumor spreading problem this algorithm is interesting in many aspects: It is simpler, more natural, more robust and faster than