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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 33 (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].
Tight bounds for rumor spreading in graphs of a given conductance
 In Proc. 28th STACS
, 2011
"... conductance∗ ..."
Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness
"... Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading ..."
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Cited by 27 (9 self)
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Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom rumor spreading inform all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and ErdősRényi random graphs. For all network topologies, we show that if one of the push or pull model works well, so does the other. We also show that quasirandom rumor spreading is robust against transmission failures. If each message sent out gets lost with probability f, then the runtime increases only by a factor of O(1/(1 − f)).
Fast Information Spreading in Graphs with Large Weak Conductance
"... Gathering data from nodes in a network is at the heart of many distributed applications, most notably, while performing a global task. We consider information spreading among n nodes of a network, where each node v has a message m(v) which must be received by all other nodes. The time required for i ..."
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Cited by 17 (2 self)
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Gathering data from nodes in a network is at the heart of many distributed applications, most notably, while performing a global task. We consider information spreading among n nodes of a network, where each node v has a message m(v) which must be received by all other nodes. The time required for information spreading has been previously upperbounded with an inverse relationship to the conductance of the underlying communication graph. This implies high running times for graphs with small conductance. The main contribution of this paper is an information spreading algorithm which overcomes communication bottlenecks and thus achieves fast information spreading for a wide class of graphs, despite their small conductance. As a key tool in our study we use the recently defined concept of
Deterministic Random Walks on Regular Trees
"... Jim Propp’s rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary p ..."
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Cited by 15 (6 self)
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Jim Propp’s rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid Z d and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips deviates from the expected number the random walk would have gotten there, by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite kary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least k Θ(D) vertices contribute by being occupied by a number of chips not divisible by k in a certain time interval. 1
Tight Bounds for Quasirandom Rumor Spreading
"... This paper addresses the following fundamental problem: Suppose that in a group of n people, where each person knows all other group members, a single person holds a piece of information that must be disseminated to everybody within the group. How should the people propagate the information so that ..."
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Cited by 13 (5 self)
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This paper addresses the following fundamental problem: Suppose that in a group of n people, where each person knows all other group members, a single person holds a piece of information that must be disseminated to everybody within the group. How should the people propagate the information so that after short time everyone is informed? The classical approach, known as the push model, requires that in each round, every informed person selects some other person in the group at random, whom it then informs. In a different model, known as the quasirandom push model, each person maintains a cyclic list, i.e., permutation, of all members in the group (for instance, a contact list of persons). Once a person is informed, it chooses a random member in its own list, and from that point onwards, it informs a new person per round, in the order dictated by the list. In this paper we show that with probability 1 − o(1) the quasirandom protocol informs everybody in (1 ± o(1))log 2 n + ln n rounds; furthermore we also show that this bound is tight. This result, together with previous work on the randomized push model, demonstrates that irrespectively of the choice of lists, quasirandom broadcasting is as fast as broadcasting in the randomized push model, up to lower order terms. At the same time it reduces the number of random bits from O(log 2 n) to only ⌈log 2 n ⌉ per person. 1
Efficient Randomised Broadcasting in Random Regular Networks with Applications in PeertoPeer Systems
"... We consider broadcasting in random dregular graphs by using a simple modification of the socalled random phone call model introduced by Karp et al. [19]. In the phone call model every time step each node calls on a randomly chosen neighbour to establish a communication channel with this node. The ..."
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Cited by 12 (1 self)
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We consider broadcasting in random dregular graphs by using a simple modification of the socalled random phone call model introduced by Karp et al. [19]. In the phone call model every time step each node calls on a randomly chosen neighbour to establish a communication channel with this node. The communication channels can then be used to transmit messages in both directions. We show that, if we allow every node to choose four distinct neighbours instead of one, then the average number of message transmissions per node decreases exponentially. Formally, we present a broadcasting algorithm that has time complexity O(log n) and uses O(n log log n) transmissions per message. In contrast, we show for the standard model that every distributed and addressoblivious algorithm that broadcasts a message in time O(log n) needs Ω(n log n / log d) message transmissions. Our algorithm can efficiently handle limited communication failures, only requires rough estimates of the number of nodes, and is robust against limited changes in the size of the network. Our results have applications in peertopeer networks and replicated databases.
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
Quasirandom Load Balancing
"... We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm approximates the idealized p ..."
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Cited by 11 (7 self)
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We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm approximates the idealized process (where the tokens are divisible) on important network topologies surprisingly closely. On ddimensional torus graphs with n nodes it deviates from the idealized process only by an additive constant. In contrast to that, the randomized rounding approach of Friedrich and Sauerwald [8] can deviate up to Ω(polylogn) and the deterministic algorithm of Rabani, Sinclair and Wanka [23] has a deviation of Ω(n 1/d). This makes our quasirandom algorithm the first known algorithm for this setting which is optimal both in time and achieved smoothness. We further show that also on the hypercubeour algorithm has a smaller deviation from the idealized process than the previous algorithms. To prove these results, we derive several combinatorial andprobabilistic results thatwe believe to beof independent interest. In particular, we show that firstpassage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions. 1
Fast simulation of largescale growth models
"... Abstract. We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a “least action principle ” which characterizes the odometer function of the growth process. Starting from an educated guess for the odometer, we ..."
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Cited by 11 (5 self)
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Abstract. We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a “least action principle ” which characterizes the odometer function of the growth process. Starting from an educated guess for the odometer, we successively correct under and overestimates and provably arrive at the correct final state. The degree of speedup depends on the accuracy of the initial guess. Determining the size of the boundary fluctuations in internal diffusionlimited aggregation is a longstanding open problem in statistical physics. As an application of our method, we calculate the size of fluctuations over two orders of magnitude beyond previous simulations. Our data strongly support the conjecture that the fluctuations are logarithmic in the radius. 1.