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24
Rumour spreading and graph conductance
 IN PROCEEDINGS OF THE 21ST ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
, 2010
"... We show that if a connected graph with n nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(log 4 n/φ 6) many steps, with high probability, using the PUSHPULL strategy. An interesting feature of our approach is that it draws ..."
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Cited by 21 (2 self)
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We show that if a connected graph with n nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(log 4 n/φ 6) many steps, with high probability, using the PUSHPULL strategy. An interesting feature of our approach is that it draws a connection between rumour spreading and the spectral sparsification procedure of Spielman and Teng [23].
Network Coding for Computing: CutSet Bounds
, 2011
"... The following network computing problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e. ..."
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Cited by 10 (4 self)
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The following network computing problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e., the “computing capacity”. The network coding problem for a singlereceiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network mincut upper bound. We extend the definition of mincut to the network computing problem and show that the mincut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multiedge tree networks. It is also tight for computing linear target functions in any network. We also study the bound’s tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the mincut bound.
The Complexity of Data Aggregation in Directed Networks
"... Abstract. We study problems of data aggregation, such as approximate counting and computing the minimum input value, in synchronous directed networks with bounded message bandwidthB = Ω(logn). In undirected networks of diameter D, many such problems can easily be solved in O(D) rounds, using O(logn) ..."
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Cited by 6 (2 self)
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Abstract. We study problems of data aggregation, such as approximate counting and computing the minimum input value, in synchronous directed networks with bounded message bandwidthB = Ω(logn). In undirected networks of diameter D, many such problems can easily be solved in O(D) rounds, using O(logn)size messages. We show that for directed networks this is not the case: when the bandwidth B is small, several classical data aggregation problems have a time complexity that depends polynomially on the size of the network, even when the diameter of the network is constant. We show that computing anǫapproximation to the size n of the network requires Ω(min { n,1/ǫ 2} /B) rounds, even in networks of diameter 2. We also show that computing a sensitive function (e.g., minimum and maximum) requires Ω ( √ n/B) rounds in networks of diameter 2, provided that the diameter is not known in advance to be o ( √ n/B). Our lower bounds are established by reduction from several wellknown problems in communication complexity. On the positive side, we give a nearly optimal Õ(D+ √ n/B)round algorithm for computing simple sensitive functions using messages of size B = Ω(logN), where N is a loose upper bound on the size of the network and D is the diameter. 1
Rumor Spreading and Vertex Expansion on Regular Graphs
, 2011
"... We study the relation between the vertex expansion of a graph and the performance of randomized rumor spreading (push model). We prove that randomized rumor spreading takes O((1/α) · polylog(n)) time on any regular nvertex graph with vertex expansion α. This bound extends previously known upper bo ..."
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Cited by 6 (1 self)
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We study the relation between the vertex expansion of a graph and the performance of randomized rumor spreading (push model). We prove that randomized rumor spreading takes O((1/α) · polylog(n)) time on any regular nvertex graph with vertex expansion α. This bound extends previously known upper bounds by replacing conductance by vertex expansion. Our result is almost tight in the sense that the dependency on (1/α) is optimal (up to logarithmic factors) and that on nonregular graphs with constant vertex expansion, the runtime can be polynomial in n. Our upper bound also implies that randomized rumor spreading is “fast” on every vertextransitive graph and yields a new upper bound on the cover time of random walks. We also exhibit a subtle difference between the impact of vertex expansion and conductance on rumor spreading. We show that there are regular graphs with constant vertex expansion for which randomized rumor spreading takes considerably longer than on any regular graph with constant conductance. Finally, we also prove a more general, but weaker result for the push & pull model which also covers nonregular graphs.
Fully Distributed Algorithms for Convex Optimization Problems ∗
, 2007
"... Motivated by resource allocation in ad hoc and mobile networks, we design and analyze a fully distributed algorithm for convex constrained optimization in networks without any consistent naming infrastructure. The algorithm produces an approximately feasible and nearoptimal solution in time polynom ..."
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Cited by 6 (1 self)
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Motivated by resource allocation in ad hoc and mobile networks, we design and analyze a fully distributed algorithm for convex constrained optimization in networks without any consistent naming infrastructure. The algorithm produces an approximately feasible and nearoptimal solution in time polynomial in the network size, the inverse of the permitted error, and a measure of curvature variation in the dual optimization problem. It blends, in a novel way, gossipbased information spreading, iterative gradient ascent, and the barrier method from the design of interiorpoint algorithms. ∗ Regular submission, but can be considered for brief announcement track. Eligible for best student paper.
Ultrafast rumor spreading in social networks
 In SODA
, 2012
"... We analyze the popular pushpull protocol for spreading a rumor in networks. Initially, a single node knows of a rumor. In each succeeding round, every node chooses a random neighbor, and the two nodes share the rumor if one of them is already aware of it. We present the first theoretical analysis o ..."
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Cited by 6 (0 self)
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We analyze the popular pushpull protocol for spreading a rumor in networks. Initially, a single node knows of a rumor. In each succeeding round, every node chooses a random neighbor, and the two nodes share the rumor if one of them is already aware of it. We present the first theoretical analysis of this protocol on random graphs that have a power law degree distribution with an arbitrary exponent β> 2. Our main findings reveal a striking dichotomy in the performance of the protocol that depends on the exponent of the power law. More specifically, we show that if 2 < β < 3, then the rumor spreads to almost all nodes in Θ(log log n) rounds with high probability. On the other hand, if β> 3, then Ω(log n) rounds are necessary. We also investigate the asynchronous version of the pushpull protocol, where the nodes do not operate in rounds, but exchange information according to a Poisson process with rate 1. Surprisingly, we are able to show that, if 2 < β < 3, the rumor spreads even in constant time, which is much smaller than the typical distance of two nodes. To the best of our knowledge, this is the first result that establishes a gap between the synchronous and the asynchronous protocol. 1
Rumor spreading and vertex expansion
 In SODA
, 2012
"... We study the relation between the rate at which rumors spread throughout a graph and the vertex expansion of the graph. We consider the standard rumor spreading protocol where every node chooses a random neighbor in each round and the two nodes exchange the rumors they know. For any nnode graph wit ..."
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Cited by 5 (0 self)
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We study the relation between the rate at which rumors spread throughout a graph and the vertex expansion of the graph. We consider the standard rumor spreading protocol where every node chooses a random neighbor in each round and the two nodes exchange the rumors they know. For any nnode graph with vertex expansion α, we show that this protocol spreads a rumor from a single node to all other nodes in O(α −1 log 2 n √ log n) rounds with high probability. Further, we construct graphs for which Ω(α −1 log 2 n) rounds are needed. Our results complement a long series of works that relate rumor spreading to edgebased notions of expansion, resolving one of the most natural questions on the connection between rumor spreading and expansion. 1
Low Randomness Rumor Spreading via Hashing
"... We consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple pushbased protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independe ..."
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Cited by 3 (1 self)
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We consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple pushbased protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independent hash functions, or a pseudorandom generator, respectively. For several wellstudied topologies our algorithms use exponentially fewer random bits than previous protocols. For example, in complete graphs, expanders, and random graphs only a polylogarithmic number of random bits are needed in total to spread the rumor in O(log n) rounds with high probability. Previous explicit algorithms, e.g., [6, 10, 15, 17], require Ω(n) random bits to achieve the same round complexity. For complete graphs, the amount of randomness used by our hashingbased algorithm is within an O(log n)factor of the theoretical minimum determined by
Fast Averaging
"... Abstract—We are interested in the following question: given n numbers x1,...,xn, what sorts of approximation of average xave = 1 (x1 + ·· · + xn) can be achieved by knowing only r of n these n numbers. Indeed the answer depends on the variation in these n numbers. As the main result, we show that if ..."
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Abstract—We are interested in the following question: given n numbers x1,...,xn, what sorts of approximation of average xave = 1 (x1 + ·· · + xn) can be achieved by knowing only r of n these n numbers. Indeed the answer depends on the variation in these n numbers. As the main result, we show that if the vector of these n numbers satisfies certain regularity properties captured in the form of finiteness of their empirical moments (third or higher), then it is possible to compute approximation of xave that is within 1±ε multiplicative factor with probability at least 1 − δ by choosing, on an average, r = r(ε,δ,σ) of the n numbers at random with r is dependent only on ε,δ and the amount of variation σ in the vector and is independent of n. The task of computing average has a variety of applications such as distributed estimation and optimization, a model for reaching consensus and computing symmetric functions. We discuss implications of the result in the context of two applications: loadbalancing in a computational facility running MapReduce, and fast distributed averaging. Index Terms—Averaging, Probabilistic approximation. I.