arXiv subjects

Andrea E. F. Clementi

Publications and source records attributed to Andrea E. F. Clementi.

4 recordsLinked to original sources

arXiv2018

A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors

The \emph{Undecided-State Dynamics} is a well-known protocol for distributed consensus. We analyze it in the parallel \pull\ communication model on the complete graph for the \emph{binary} case (every node can either support one of \emph{two} possible colors, or be in the undecided state). An interesting open question is whether this dynamics \emph{always} (i.e., starting from an arbitrary initial configuration) reaches consensus \emph{quickly} (i.e., within a polylogarithmic number of rounds) in a complete graph with $n$ nodes. Previous work in this setting only considers initial color configurations with no undecided nodes and a large \emph{bias} (i.e., $Θ(n)$) towards the majority color. In this paper we present an \textit{unconditional} analysis of the Undecided-State Dynamics that answers to the above question in the affirmative. We prove that, starting from \textit{any} initial configuration, the process reaches a monochromatic configuration within $O(\log n)$ rounds, with high probability. This bound turns out to be tight. Our analysis also shows that, if the initial configuration has bias $Ω(\sqrt{n\log n})$, then the dynamics converges toward the initial majority color, with high probability.

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arXiv2009

MANETS: High mobility can make up for low transmission power

We consider a Mobile Ad-hoc NETworks (MANET) formed by "n" nodes that move independently at random over a finite square region of the plane. Nodes exchange data if they are at distance at most "r" within each other, where r>0 is the node transmission radius. The "flooding time" is the number of time steps required to broadcast a message from a source node to every node of the network. Flooding time is an important measure of the speed of information spreading in dynamic networks. We derive a nearly-tight upper bound on the flooding time which is a decreasing function of the maximal "velocity" of the nodes. It turns out that, when the node velocity is sufficiently high, even if the node transmission radius "r" is far below the "connectivity threshold", the flooding time does not asymptotically depend on "r". This implies that flooding can be very fast even though every "snapshot" (i.e. the static random geometric graph at any fixed time) of the MANET is fully disconnected. Data reach all nodes quickly despite these ones use very low transmission power. Our result is the first analytical evidence of the fact that high, random node mobility strongly speed-up information spreading and, at the same time, let nodes save energy.

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arXiv2008

Minimum-energy broadcast in random-grid ad-hoc networks: approximation and distributed algorithms

The Min Energy broadcast problem consists in assigning transmission ranges to the nodes of an ad-hoc network in order to guarantee a directed spanning tree from a given source node and, at the same time, to minimize the energy consumption (i.e. the energy cost) yielded by the range assignment. Min energy broadcast is known to be NP-hard. We consider random-grid networks where nodes are chosen independently at random from the $n$ points of a $\sqrt n \times \sqrt n$ square grid in the plane. The probability of the existence of a node at a given point of the grid does depend on that point, that is, the probability distribution can be non-uniform. By using information-theoretic arguments, we prove a lower bound $(1-ε) \frac nπ$ on the energy cost of any feasible solution for this problem. Then, we provide an efficient solution of energy cost not larger than $1.1204 \frac nπ$. Finally, we present a fully-distributed protocol that constructs a broadcast range assignment of energy cost not larger than $8n$,thus still yielding constant approximation. The energy load is well balanced and, at the same time, the work complexity (i.e. the energy due to all message transmissions of the protocol) is asymptotically optimal. The completion time of the protocol is only an $O(\log n)$ factor slower than the optimum. The approximation quality of our distributed solution is also experimentally evaluated. All bounds hold with probability at least $1-1/n^{Θ(1)}$.

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arXiv2001

Distributed Broadcast in Wireless Networks with Unknown Topology

A multi-hop synchronous wirelss network is said to be unknown if the nodes have no knowledge of the topology. A basic task in wireless network is that of broadcasting a message (created by a fixed source node) to all nodes of the network. The multi-broadcast that consists in performing a set of r independent broadcasts. In this paper, we study the completion and the termination time of distributed protocols for both the (single) broadcast and the multi-broadcast operations on unknown networks as functions of the number of nodes n, the maximum eccentricity D, the maximum in-degree Delta, and the congestion c of the networks. We establish new connections between these operations and some combinatorial concepts, such as selective families, strongly-selective families (also known as superimposed codes), and pairwise r-different families. Such connections, combined with a set of new lower and upper bounds on the size of the above families, allow us to derive new lower bounds and new distributed protocols for the broadcast and multi-broadcast operations. In particular, our upper bounds are almost tight and improve exponentially over the previous bounds when D and Delta are polylogarithmic in n. Network topologies having ``small'' eccentricity and ``small'' degree (such as bounded-degree expanders) are often used in practice to achieve efficient communication.

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