We study how to spread k tokens of information to every node on an n-node dynamic network, the edges of which are changing at each round. This basic gossip problem can be completed in O(n+k) rounds in any static network, and determining its complexity in dynamic networks is central to understanding the algorithmic limits and capabilities of various dynamic network models. Our focus is on token-forwarding algorithms, which do not manipulate tokens in any way other than storing, copying and forwarding them.We first consider the strongly adaptive adversary model where in each round, each node first chooses a token to broadcast to all its neighbors (without knowing who they are), and then an adversary chooses an arbitrary connected communication network for that round with the knowledge of the tokens chosen by each node. We show that Ω(nk/ log n + n) rounds are needed for any randomized (centralized or distributed) token-forwarding algorithm to disseminate the k tokens, thus resolving an open problem raised in [KLO10]. The bound applies to a wide class of initial token distributions, including those in which each token is held by exactly one node and well-mixed ones in which each node has each token independently with a constant probability.Our result for the strongly adaptive adversary model motivates us to study the weakly adaptive adversary model where in each round, the adversary is required to lay down the network first, and then each node sends a possibly distinct token to each of its neighbors. We propose a simple randomized distributed algorithm where in each round, along every edge (u, v), a token sampled uniformly at random from the symmetric difference of the sets of tokens held by node u and node v is exchanged. We prove that starting from any well-mixed distribution of tokens where each node has each token independently with a constant probability, this algorithm solves the k-gossip problem in O((n + k) log n log k) rounds with high probability over the initial token distribution and the randomness of the protocol. We then show how the above uniform sampling problem can be solved usingÕ(log k) bits of communication, making the overall algorithm communication-efficient. * College of Computer and Information Science, Northeastern University, Boston, 02115, USA. Email: {chinmoy,rraj,austin,viola}@ccs.neu.edu. Chinmoy Dutta is supported in part by NSF grant CCF-0845003 and a Microsoft grant to Ravi Sundaram; Rajmohan Rajaraman and Zhifeng Sun are supported in part by NSF grant CNS-0915985; Emanuele Viola is supported by NSF grant CCF-0845003.† Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 and Department of Computer Science, Brown University, Providence, RI 02912, USA. Email: gopalpandurangan@gmail.com. Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and a grant from the US-Israel Binational Science Foundation ...

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We study a distributed randomized information propagation mechanism in networks we call the coalescingbranching random walk (cobra walk, for short). A cobra walk is a generalization of the well-studied "standard" random walk, and is useful in modeling and understanding the Susceptible-Infected-Susceptible (SIS)-type of epidemic processes in networks. It can also be helpful in performing lightweight information dissemination in resource-constrained networks. A cobra walk is parameterized by a branching factor k. The process starts from an arbitrary vertex, which is labeled active for step 1. In each step of a cobra walk, each active vertex chooses k random neighbors to become active for the next step ("branching"). A vertex is active for step t + 1 only if it is chosen by an active vertex in step t ("coalescing"). This results in a stochastic process in the underlying network with properties that are quite different from both the standard random walk (which is equivalent to the cobra walk with branching factor 1) as well as other gossip-based rumor spreading mechanisms. We focus on the cover time of the cobra walk, which is the number of steps for the walk to reach all the vertices, and derive almost-tight bounds for various graph classes. We show an O(log 2 n) high probability bound for the cover time of cobra walks on expanders, if either the expansion factor or the branching factor is sufficiently large; we also obtain an O(log n) high probability bound for the partial cover time, which is the number of steps needed for the walk to reach at least a constant fraction of the vertices. We also show that the cover time of the cobra walk is, with high probability, O(n log n) on any n-vertex tree for k ≥ 2,Õ(n 1/d) on a d-dimensional grid for k ≥ 2, and O(log n) on the complete graph. CCS Concepts: r Mathematics of computing → Graph algorithms; r Theory of computation → Distributed algorithms; Random walks and Markov chains; r Computing methodologies → Selforganization;

We study a distributed randomized information propagation mechanism in networks we call the coalescing-branching random walk (cobra walk, for short). A cobra walk is a generalization of the well-studied "standard" random walk, and is useful in modeling and understanding the SusceptibleInfected-Susceptible (SIS)-type of epidemic processes in networks. It can also be helpful in performing light-weight information dissemination in resource-constrained networks. A cobra walk is parameterized by a branching factor k. The process starts from an arbitrary node, which is labeled active for step 1. (For instance, this could be a node that has a piece of data, rumor, or a virus.) In each step of a cobra walk, each active node chooses k random neighbors to become active for the next step ("branching"). A node is active for step t + 1 only if it is chosen by an active node in step t ("coalescing"). This results in a stochastic process in the underlying network with properties that are quite different from both the standard random walk (which is equivalent * Chinmoy ...$15.00.to the cobra walk with branching factor 1) as well as other gossip-based rumor spreading mechanisms.We focus on the cover time of the cobra walk, which is the number of steps for the walk to reach all the nodes, and derive almost-tight bounds for various graph classes. Our main technical result is an O(log 2 n) high probability bound for the cover time of cobra walks on expanders, if either the expansion factor or the branching factor is sufficiently large; we also obtain an O(log n) high probability bound for the partial cover time, which is the number of steps needed for the walk to reach at least a constant fraction of the nodes. We show that the cobra walk takes O(n log n) steps on any n-node tree for k ≥ 2, andÕ(n 1/d ) steps on a d-dimensional grid for k ≥ 2, with high probability.

We show a tight lower bound of Ω(N log log N ) on the number of transmissions required to compute several functions (including the parity function and the majority function) in a network of N randomly placed sensors, communicating using local transmissions, and operating with power near the connectivity threshold. This result considerably simplifies and strengthens an earlier result of Dutta, Kanoria Manjunath and Radhakrishnan (SODA 08) that such networks cannot compute the parity function reliably with significantly fewer than N log log N transmissions, thereby showing that the protocol with O(N log log N ) transmissions due to Ying, Srikant and Dullerud (WiOpt 06) is optimal. We also observe that all the lower bounds shown by Evans and Pippenger (SIAM J. on Computing, 1999) on the average noisy decision tree complexity for several functions can be derived using our technique simply and in a unified way.