Breaking the log n barrier on rumor spreading

$O(\log n)$ rounds has been a well known upper bound for rumor spreading using push&pull in the random phone call model (i.e., uniform gossip in the complete graph). A matching lower bound of $\Omega(\log n)$ is also known for this special case. Under the assumption of this model and with a natural addition that nodes can call a partner once they learn its address (e.g., its IP address) we present a new distributed, address-oblivious and robust algorithm that uses push&pull with pointer jumping to spread a rumor to all nodes in only $O(\sqrt{\log n})$ rounds, w.h.p. This algorithm can also cope with $F= O(n/2^{\sqrt{\log n}})$ node failures, in which case all but $O(F)$ nodes become informed within $O(\sqrt{\log n})$ rounds, w.h.p

Optimal Gossip with Direct Addressing

Gossip algorithms spread information by having nodes repeatedly forward information to a few random contacts. By their very nature, gossip algorithms tend to be distributed and fault tolerant. If done right, they can also be fast and message-efficient. A common model for gossip communication is the random phone call model, in which in each synchronous round each node can PUSH or PULL information to or from a random other node. For example, Karp et al. [FOCS 2000] gave algorithms in this model that spread a message to all nodes in $\Theta(\log n)$ rounds while sending only $O(\log \log n)$ messages per node on average. Recently, Avin and Els\"asser [DISC 2013], studied the random phone call model with the natural and commonly used assumption of direct addressing. Direct addressing allows nodes to directly contact nodes whose ID (e.g., IP address) was learned before. They show that in this setting, one can "break the $\log n$ barrier" and achieve a gossip algorithm running in $O(\sqrt{\log n})$ rounds, albeit while using $O(\sqrt{\log n})$ messages per node. We study the same model and give a simple gossip algorithm which spreads a message in only $O(\log \log n)$ rounds. We also prove a matching $\Omega(\log \log n)$ lower bound which shows that this running time is best possible. In particular we show that any gossip algorithm takes with high probability at least $0.99 \log \log n$ rounds to terminate. Lastly, our algorithm can be tweaked to send only $O(1)$ messages per node on average with only $O(\log n)$ bits per message. Our algorithm therefore simultaneously achieves the optimal round-, message-, and bit-complexity for this setting. As all prior gossip algorithms, our algorithm is also robust against failures. In particular, if in the beginning an oblivious adversary fails any $F$ nodes our algorithm still, with high probability, informs all but $o(F)$ surviving nodes

Fast Distributed Algorithms for Computing Separable Functions

The problem of computing functions of values at the nodes in a network in a totally distributed manner, where nodes do not have unique identities and make decisions based only on local information, has applications in sensor, peer-to-peer, and ad-hoc networks. The task of computing separable functions, which can be written as linear combinations of functions of individual variables, is studied in this context. Known iterative algorithms for averaging can be used to compute the normalized values of such functions, but these algorithms do not extend in general to the computation of the actual values of separable functions. The main contribution of this paper is the design of a distributed randomized algorithm for computing separable functions. The running time of the algorithm is shown to depend on the running time of a minimum computation algorithm used as a subroutine. Using a randomized gossip mechanism for minimum computation as the subroutine yields a complete totally distributed algorithm for computing separable functions. For a class of graphs with small spectral gap, such as grid graphs, the time used by the algorithm to compute averages is of a smaller order than the time required by a known iterative averaging scheme.Comment: 15 page

Gossip in a Smartphone Peer-to-Peer Network

In this paper, we study the fundamental problem of gossip in the mobile telephone model: a recently introduced variation of the classical telephone model modified to better describe the local peer-to-peer communication services implemented in many popular smartphone operating systems. In more detail, the mobile telephone model differs from the classical telephone model in three ways: (1) each device can participate in at most one connection per round; (2) the network topology can undergo a parameterized rate of change; and (3) devices can advertise a parameterized number of bits about their state to their neighbors in each round before connection attempts are initiated. We begin by describing and analyzing new randomized gossip algorithms in this model under the harsh assumption of a network topology that can change completely in every round. We prove a significant time complexity gap between the case where nodes can advertise $0$ bits to their neighbors in each round, and the case where nodes can advertise $1$ bit. For the latter assumption, we present two solutions: the first depends on a shared randomness source, while the second eliminates this assumption using a pseudorandomness generator we prove to exist with a novel generalization of a classical result from the study of two-party communication complexity. We then turn our attention to the easier case where the topology graph is stable, and describe and analyze a new gossip algorithm that provides a substantial performance improvement for many parameters. We conclude by studying a relaxed version of gossip in which it is only necessary for nodes to each learn a specified fraction of the messages in the system.Comment: Extended Abstract to Appear in the Proceedings of the ACM Conference on the Principles of Distributed Computing (PODC 2017

Optimal epidemic dissemination

We consider the problem of reliable epidemic dissemination of a rumor in a fully connected network of~$n$ processes using push and pull operations. We revisit the random phone call model and show that it is possible to disseminate a rumor to all processes with high probability using $\Theta(\ln n)$ rounds of communication and only $n+o(n)$ messages of size $b$, all of which are asymptotically optimal and achievable with pull and push-then-pull algorithms. This contradicts two highly-cited lower bounds of Karp et al. stating that any algorithm in the random phone call model running in $\mathcal{O}(\ln n)$ rounds with communication peers chosen uniformly at random requires at least $\omega(n)$ messages to disseminate a rumor with high probability, and that any address-oblivious algorithm needs $\Omega(n \ln \ln n)$ messages regardless of the number of communication rounds. The reason for this contradiction is that in the original work, processes do not have to share the rumor once the communication is established. However, it is implicitly assumed that they always do so in the proofs of their lower bounds, which, it turns out, is not optimal. Our algorithms are strikingly simple, address-oblivious, and robust against $\epsilon n$ adversarial failures and stochastic failures occurring with probability $\delta$ for any $0 \leq \{\epsilon,\delta\} < 1$. Furthermore, they can handle multiple rumors of size $b \in \omega(\ln n \ln \ln n)$ with $nb + o(nb)$ bits of communication per rumor.Comment: A brief announcement of this work was presented at PODC 201

Information Spreading in Dynamic Networks

We study the fundamental problem of information spreading (also known as gossip) in dynamic networks. In gossip, or more generally, $k$-gossip, there are $k$ pieces of information (or tokens) that are initially present in some nodes and the problem is to disseminate the $k$ tokens to all nodes. The goal is to accomplish the task in as few rounds of distributed computation as possible. The problem is especially challenging in dynamic networks where the network topology can change from round to round and can be controlled by an on-line adversary. The focus of this paper is on the power of token-forwarding algorithms, which do not manipulate tokens in any way other than storing and forwarding them. We first consider a worst-case adversarial model first studied by Kuhn, Lynch, and Oshman~\cite{kuhn+lo:dynamic} in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. Our main result is an $\Omega(nk/\log n)$ lower bound on the number of rounds needed for any deterministic token-forwarding algorithm to solve $k$-gossip. This resolves an open problem raised in~\cite{kuhn+lo:dynamic}, improving their lower bound of $\Omega(n \log k)$, and matching their upper bound of $O(nk)$ to within a logarithmic factor. We next show that token-forwarding algorithms can achieve subquadratic time in the offline version of the problem where the adversary has to commit all the topology changes in advance at the beginning of the computation, and present two polynomial-time offline token-forwarding algorithms. Our results are a step towards understanding the power and limitation of token-forwarding algorithms in dynamic networks.Comment: 18 page

Algorithm for Achieving Consensus Over Conflicting Rumors: Convergence Analysis and Applications

Motivated by the large expansion in the study of social networks, this paper deals with the problem of multiple messages spreading over the same network using gossip algorithms. Given two messages distributed over some nodes of the graph, we first investigate the final distribution of the messages given an initial state. Then, an algorithm is presented to achieve consensus over one of the messages. Finally, a game theoretical application and an analogy with word-of-mouth marketing are outlined.Comment: IEEE Student Paper Contes

Analyzing Network Coding Gossip Made Easy

We give a new technique to analyze the stopping time of gossip protocols that are based on random linear network coding (RLNC). Our analysis drastically simplifies, extends and strengthens previous results. We analyze RLNC gossip in a general framework for network and communication models that encompasses and unifies the models used previously in this context. We show, in most settings for the first time, that it converges with high probability in the information-theoretically optimal time. Most stopping times are of the form O(k + T) where k is the number of messages to be distributed and T is the time it takes to disseminate one message. This means RLNC gossip achieves "perfect pipelining". Our analysis directly extends to highly dynamic networks in which the topology can change completely at any time. This remains true even if the network dynamics are controlled by a fully adaptive adversary that knows the complete network state. Virtually nothing besides simple O(kT) sequential flooding protocols was previously known for such a setting. While RLNC gossip works in this wide variety of networks its analysis remains the same and extremely simple. This contrasts with more complex proofs that were put forward to give less strong results for various special cases

In-Network Estimation of Frequency Moments

We consider the problem of estimating functions of distributed data using a distributed algorithm over a network. The extant literature on computing functions in distributed networks such as wired and wireless sensor networks and peer-to-peer networks deals with computing linear functions of the distributed data when the alphabet size of the data values is small, O(1). We describe a distributed randomized algorithm to estimate a class of non-linear functions of the distributed data which is over a large alphabet. We consider three types of networks: point-to-point networks with gossip based communication, random planar networks in the connectivity regime and random planar networks in the percolating regime both of which use the slotted Aloha communication protocol. For each network type, we estimate the scaled $k$-th frequency moments, for $k \geq 2$. Specifically, for every $k \geq 2,$ we give a distributed randomized algorithm that computes, with probability $(1-\delta),$ an $\epsilon$-approximation of the scaled $k$-th frequency moment, $F_k/N^k$, using time $O(M^{1-\frac{1}{k-1}} T)$ and $O(M^{1-\frac{1}{k-1}} \log N \log (\delta^{-1})/\epsilon^2)$ bits of transmission per communication step. Here, $N$ is the number of nodes in the network, $T$ is the information spreading time and $M=o(N)$ is the alphabet size

Bounds for Algebraic Gossip on Graphs

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(D*n) where D is the maximum degree of the graph. This leads to a tight bound of Theta(n) for bounded degree graphs and an upper bound of O(n^2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Omega(n^2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1101.437