There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Impatti, adattamento e vulnerabilita Le cause e le responsabilita dei cambiamenti climatici sono state trattate sul numero di ottobre della rivista Cda. Approfondiamo l’argomento presentando il documento: “Cambiamenti climatici 2007: impatti, adattamento e vulnerabilita” votato ad aprile 2007 dal secondo gruppo di lavoro del Comitato Intergovernativo sui Cambiamenti Climatici (Intergovernmental Panel on Climate Change). Si tratta del secondo di tre documenti che compongono il quarto rapporto sui cambiamenti climatici.

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Climate Change 2007: Impacts, Adaptation and Vulnerability.

This self-contained introduction to the distributed control of robotic networks offers a distinctive blend of computer science and control theory. The book presents a broad set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity; and it analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation. The unifying theme is a formal model for robotic networks that explicitly incorporates their communication, sensing, control, and processing capabilities--a model that in turn leads to a common formal language to describe and analyze coordination algorithms.Written for first- and second-year graduate students in control and robotics, the book will also be useful to researchers in control theory, robotics, distributed algorithms, and automata theory. The book provides explanations of the basic concepts and main results, as well as numerous examples and exercises.Self-contained exposition of graph-theoretic concepts, distributed algorithms, and complexity measures for processor networks with fixed interconnection topology and for robotic networks with position-dependent interconnection topology Detailed treatment of averaging and consensus algorithms interpreted as linear iterations on synchronous networks Introduction of geometric notions such as partitions, proximity graphs, and multicenter functions Detailed treatment of motion coordination algorithms for deployment, rendezvous, connectivity maintenance, and boundary estimation

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Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms

Cryptocurrencies, such as Bitcoin and 250 similar alt-coins, embody at their core a blockchain protocol --- a mechanism for a distributed network of computational nodes to periodically agree on a set of new transactions. Designing a secure blockchain protocol relies on an open challenge in security, that of designing a highly-scalable agreement protocol open to manipulation by byzantine or arbitrarily malicious nodes. Bitcoin's blockchain agreement protocol exhibits security, but does not scale: it processes 3--7 transactions per second at present, irrespective of the available computation capacity at hand. In this paper, we propose a new distributed agreement protocol for permission-less blockchains called ELASTICO. ELASTICO scales transaction rates almost linearly with available computation for mining: the more the computation power in the network, the higher the number of transaction blocks selected per unit time. ELASTICO is efficient in its network messages and tolerates byzantine adversaries of up to one-fourth of the total computational power. Technically, ELASTICO uniformly partitions or parallelizes the mining network (securely) into smaller committees, each of which processes a disjoint set of transactions (or "shards"). While sharding is common in non-byzantine settings, ELASTICO is the first candidate for a secure sharding protocol with presence of byzantine adversaries. Our scalability experiments on Amazon EC2 with up to $1, 600$ nodes confirm ELASTICO's theoretical scaling properties.

A Secure Sharding Protocol For Open Blockchains

The increasing popularity of blockchain-based cryptocurrencies has made scalability a primary and urgent concern. We analyze how fundamental and circumstantial bottlenecks in Bitcoin limit the ability of its current peer-to-peer overlay network to support substantially higher throughputs and lower latencies. Our results suggest that reparameterization of block size and intervals should be viewed only as a first increment toward achieving next-generation, high-load blockchain protocols, and major advances will additionally require a basic rethinking of technical approaches. We offer a structured perspective on the design space for such approaches. Within this perspective, we enumerate and briefly discuss a number of recently proposed protocol ideas and offer several new ideas and open challenges.

On Scaling Decentralized Blockchains

Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is Θ(D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound Ω (n2) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

Deterministic Rendezvous in Graphs

We consider distributed broadcasting in radio networks, modeled as undirected graphs, whose nodes have no information on the topology of the network, nor even on their immediate neighborhood. For randomized broadcasting, we give an algorithm working in expected time O(D log(n/D) + log2 n) in n-node radio networks of diameter D, which is optimal, as it matches the lower bounds of Alon et al. [1] and Kushilevitz and Mansour [16]. Our algorithm improves the best previously known randomized broadcasting algorithm of Bar-Yehuda, Goldreich and Itai [3], running in expected time O(D log n + log2 n). (In fact, our result holds also in the setting of n-node directed radio networks of radius D.) For deterministic broadcasting, we show the lower bound Ω(n log n/log (n/m)) on broadcasting time in n-node radio networks of diameter D. This implies previously known lower bounds of Bar-Yehuda, Goldreich and Itai [3] and Bruschi and Del Pinto [5], and is sharper than any of them in many cases. We also give an algorithm working in time O(n log n), thus shrinking - for the first time - the gap between the upper and the lower bound on deterministic broadcasting time to a logarithmic factor.

Broadcasting in undirected ad hoc radio networks

We consider deterministic broadcasting in radio networks whose nodes have full topological information about the network. The aim is to design a polynomial algorithm, which, given a graph G with source s, produces a fast broadcast scheme in the radio network represented by G. The problem of finding a fastest broadcast scheme for a given graph is NP-hard, hence it is only possible to get an approximation algorithm. We give a deterministic polynomial algorithm which produces a broadcast scheme of length O(D + log2n), for every n-node graph of diameter D, thus improving a result of Gasieniec et al. (PODC 2005) [17] and solving a problem stated there. Unless the inclusion NP ⊆ BPTIME(nO(log log n)) holds, the length O(D + log2n) of a polynomially constructible deterministic broadcast scheme is optimal.

Optimal deterministic broadcasting in known topology radio networks

An n-node tree has to be explored by k mobile agents (robots), starting at its root. Every edge of the tree must be traversed by at least one robot, and exploration must be completed as fast as possible. Even when the tree is known in advance, scheduling optimal collective exploration turns out to be NP-hard. We investigate the problem of how communication among robots influences the time for exploring unknown trees. To this end, we consider two scenarios. In the first scenario, robots can communicate by writing at the currently visited node previously acquired information, and reading information available at this node. In the second scenario robots are oblivious of one another, and each of them knows only the part of the tree previously explored by itself. We show that this difference of communication capability significantly influences time of collective exploration. Under the first scenario, we construct an exploration algorithm whose running time for any tree is only O(ks log k) larger than the optimal exploration time with full knowledge of the tree, while under the second scenario we prove that every algorithm works in time Ω(k) larger than optimal, for some trees. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(3), 166–177 2006

Collective tree exploration

A selection problem is among the basic communication primitives in networks. In this problem at most k participating stations have to broadcast successfully their messages. This problem is especially important in packet radio networks, where simultaneous transmissions of many neighbors result in interference among delivered messages. This work focuses on a single-hop radio networks with n-stations, also called a multiple access channel, and considers both static and dynamic versions of the selection problem. We construct a family of efficient oblivious deterministic protocols based on selectors, one of them with selection time cO(klog(n/k)), and the second explicit construction with selection time cO(k polylog n). The first construction matches the lower bound Ω(klog(n/k)) on deterministic oblivious selection, while the second one is the first known explicit construction better than Θ(k2). In the dynamic case we introduce the model of dynamic requests, called k-streams, which generalizes the static model and the dynamic requests with at most k participants. We prove that each oblivious deterministic protocol has latency Ω(k2/log k), and on the other hand we prove the existence of the oblivious deterministic protocol with latency cO(k2log n). In view of the existence of the randomized oblivious protocol with expected latency cO(klog(n/k)), this shows that randomization is substantially better than determinism for dynamic setting. Selection problem can be applied to implement other communication primitives --- we demonstrate it in the example of broadcast problem in multi-hop ad-hoc radio networks. In particular, we design an adaptive deterministic protocol broadcasting in time cO(nlog D) in every D-hop radio network of n-stations.

Dariusz R. Kowalski

Papers

Deterministic Rendezvous in Graphs

Broadcasting in undirected ad hoc radio networks

Optimal deterministic broadcasting in known topology radio networks

Collective tree exploration

On selection problem in radio networks