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Emanuele Natale

Publications and source records attributed to Emanuele Natale.

44 records · Page 3Linked to original sources
arXiv2016

Noisy Rumor Spreading and Plurality Consensus

Error-correcting codes are efficient methods for handling \emph{noisy} communication channels in the context of technological networks. However, such elaborate methods differ a lot from the unsophisticated way biological entities are supposed to communicate. Yet, it has been recently shown by Feinerman, Haeupler, and Korman {[}PODC 2014{]} that complex coordination tasks such as \emph{rumor spreading} and \emph{majority consensus} can plausibly be achieved in biological systems subject to noisy communication channels, where every message transferred through a channel remains intact with small probability $\frac{1}{2}+ε$, without using coding techniques. This result is a considerable step towards a better understanding of the way biological entities may cooperate. It has been nevertheless be established only in the case of 2-valued \emph{opinions}: rumor spreading aims at broadcasting a single-bit opinion to all nodes, and majority consensus aims at leading all nodes to adopt the single-bit opinion that was initially present in the system with (relative) majority. In this paper, we extend this previous work to $k$-valued opinions, for any $k\geq2$. Our extension requires to address a series of important issues, some conceptual, others technical. We had to entirely revisit the notion of noise, for handling channels carrying $k$-\emph{valued} messages. In fact, we precisely characterize the type of noise patterns for which plurality consensus is solvable. Also, a key result employed in the bivalued case by Feinerman et al. is an estimate of the probability of observing the most frequent opinion from observing the mode of a small sample. We generalize this result to the multivalued case by providing a new analytical proof for the bivalued case that is amenable to be extended, by induction, and that is of independent interest.

cs.DC↗
arXiv2016

Large Peg-Army Maneuvers

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csákány and Juhász (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided at http://solitairearmy.isnphard.com.

cs.DM↗
arXiv2015

Stabilizing Consensus with Many Opinions

We consider the following distributed consensus problem: Each node in a complete communication network of size $n$ initially holds an \emph{opinion}, which is chosen arbitrarily from a finite set $Σ$. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be \emph{valid}, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of an adaptive, malicious adversary who can modify the opinions of a bounded number of nodes in every round. We consider the \emph{3-majority dynamics}: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let $k$ be the number of valid opinions. We show that, if $k \leqslant n^α$, where $α$ is a suitable positive constant, the 3-majority dynamics converges in time polynomial in $k$ and $\log n$ with high probability even in the presence of an adversary who can affect up to $o(\sqrt{n})$ nodes at each round. Previously, the convergence of the 3-majority protocol was known for $|Σ| = 2$ only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for $|Σ| > 2$.

cs.DC↗
arXiv2015

On the Voting Time of the Deterministic Majority Process

In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, every node adopts the majority opinion among its neighbors. By using a potential argument first discovered by Goles and Olivos (1980), it is known that this process always converges in $O(|E|)$ rounds to a two-periodic state in which every node either keeps its opinion or changes it in every round. It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the $O(|E|)$ bound on the convergence time of the deterministic binary majority process is indeed tight even for dense graphs. However, in many graphs such as the complete graph, from any initial opinion assignment, the process converges in just a constant number of rounds. By carefully exploiting the structure of the potential function by Goles and Olivos (1980), we derive a new upper bound on the convergence time of the deterministic binary majority process that accounts for such exceptional cases. We show that it is possible to identify certain modules of a graph $G$ in order to obtain a new graph $G^Δ$ with the property that the worst-case convergence time of $G^Δ$ is an upper bound on that of $G$. Moreover, even though our upper bound can be computed in linear time, we show that, given an integer $k$, it is NP-hard to decide whether there exists an initial opinion assignment for which it takes more than $k$ rounds to converge to the two-periodic state.

cs.DC↗
arXiv2015

Simple Dynamics for Plurality Consensus

We study a \emph{Plurality-Consensus} process in which each of $n$ anonymous agents of a communication network initially supports an opinion (a color chosen from a finite set $[k]$). Then, in every (synchronous) round, each agent can revise his color according to the opinions currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large \emph{bias} $s$ towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by $s$ additional nodes. The goal is having the process to converge to the \emph{stable} configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the \emph{3-majority dynamics}) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time $\mathcal{O}( \min\{ k, (n/\log n)^{1/3} \} \, \log n )$ with high probability, provided that $s \geqslant c \sqrt{ \min\{ 2k, (n/\log n)^{1/3} \}\, n \log n}$. We then prove that our upper bound above is tight as long as $k \leqslant (n/\log n)^{1/4}$. This fact implies an exponential time-gap between the plurality-consensus process and the \emph{median} process studied by Doerr et al. in [ACM SPAA'11]. A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: In particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only.

cs.DM↗
arXiv2014

Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard

The twentieth century has seen the rise of a new type of video games targeted at a mass audience of "casual" gamers. Many of these games require the player to swap items in order to form matches of three and are collectively known as \emph{tile-matching match-three games}. Among these, the most influential one is arguably \emph{Bejeweled} in which the matched items (gems) pop and the above gems fall in their place. Bejeweled has been ported to many different platforms and influenced an incredible number of similar games. Very recently one of them, named \emph{Candy Crush Saga} enjoyed a huge popularity and quickly went viral on social networks. We generalize this kind of games by only parameterizing the size of the board, while all the other elements (such as the rules or the number of gems) remain unchanged. Then, we prove that answering many natural questions regarding such games is actually \NP-Hard. These questions include determining if the player can reach a certain score, play for a certain number of turns, and others. We also \href{http://candycrush.isnphard.com}{provide} a playable web-based implementation of our reduction.

cs.CC↗
arXiv2013

Distributed Community Detection in Dynamic Graphs

Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic version of the well-studied \emph{Planted Bisection Model} $\sdG(n,p,q)$ where the node set $[n]$ of the network is partitioned into two unknown communities and, at every time step, each possible edge $(u,v)$ is active with probability $p$ if both nodes belong to the same community, while it is active with probability $q$ (with $q< 0$), the protocol finds the right "planted" partition in $O(\log n)$ time even when the snapshots of the dynamic graph are sparse and disconnected (i.e. in the case $p=Θ(1/n)$).

cs.SI↗