Network-Based Vertex Dissolution

We introduce a graph-theoretic vertex dissolution model that applies to a number of redistribution scenarios such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their load to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the knowledge of which vertices should be deleted, and the relation between old and new vertex loads influence the computational complexity of the underlying graph problems. Our results establish a clear borderline between tractable and intractable cases.Comment: Version accepted at SIAM Journal on Discrete Mathematic

Network Coding for Computing: Cut-Set Bounds

The following \textit{network computing} problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function $f$ of the messages. The objective is to maximize the average number of times $f$ can be computed per network usage, i.e., the ``computing capacity''. The \textit{network coding} problem for a single-receiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network \textit{min-cut} upper bound. We extend the definition of min-cut to the network computing problem and show that the min-cut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multi-edge tree networks and for computing linear target functions in any network. We also study the bound's tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the min-cut bound.Comment: Submitted to the IEEE Transactions on Information Theory (Special Issue on Facets of Coding Theory: from Algorithms to Networks); Revised on Aug 9, 201

Function Computation in Networks

Ruutimine, mis kasutab ainuüht parimat teekonda sõnumite edastamiseks, on praegusel hetkel peamine meetod informatsiooni edastamiseks võrgus. Väljapakutud alternatiiviks on võrgukodeerimine, mis lubab kogu võrgul osaleda informatsiooni edastamises, saates kodeeritud infot läbi mitme teekonna ja taastades algse sõnumi vastuvõtjas. Mõningate rakenduste korral on algsete sõnumite taastamise asemel vaja funktsiooni üle nende sõnumite. Nimetame seda funktsiooni arvutuseks võrgus. Selline lähenemine lubab arvutusi teha teekonna jooksul, mil sõnum liigub allikatest saajateni. See töötab hästi näiteks võrkudes, kus ühendatud on palju piiratud arvutusvõimsusega väikseid seadmeid. Situatsioon, mis IoT esiletõusuga ilmneb aina tihedamini. Kuna funktsiooni arvutus võrkudes on suhteliselt uus mõiste, ei ole veel täiesti suudetud mõista võrgu funktsionaalarvutuse rakendatavust ja teoreetilise jõudlikkuse piire.Käesolev töö keskendub kindlale sihtfunktsioonide perekonnale ja tuvastab võrgu omadusi, et funktsionaalarvutus oleks edukas. See töö esitab kodeerimislahendusi, mis lubavad edukalt võrgus funktsionaalarvutusi läbi viia, kus sõnumiteks on üksikud sümbolid. Tulemused on seejärel laiendatud suvalise sümbolite arvuga sõnumitele, kasutades sarnast kodeerimislahendust.Routing, that uses a single best path in the network, is currently the primary method for information transfer in networks. A proposed alternative to routing is called network coding that allows for the whole network to participate in the transmission of information by sending the coded data using multiple paths and then reconstructing the original message at the receiver. In some applications instead of reconstructing the original messages a function of those messages needs to be obtained. The corresponding problem is called a problem of function computation in the network. This approach allows for efficient en-route computing that works especially well with many small connected devices with limited computational capacities, a situation that appears often with the rise of the IoT. Since network function computation is a relatively new concept, the applicability and theoretical performance limits of this approach are not yet fully understood. The current work focuses on a certain family of target functions and identifies properties a network must have for function computation to be feasible. We propose encoding solutions that allow for successful network function computation. The results are then extended to packets with arbitrary number of symbols using a similar encoding scheme

Distributed Detection and Estimation in Wireless Sensor Networks

In this article we consider the problems of distributed detection and estimation in wireless sensor networks. In the first part, we provide a general framework aimed to show how an efficient design of a sensor network requires a joint organization of in-network processing and communication. Then, we recall the basic features of consensus algorithm, which is a basic tool to reach globally optimal decisions through a distributed approach. The main part of the paper starts addressing the distributed estimation problem. We show first an entirely decentralized approach, where observations and estimations are performed without the intervention of a fusion center. Then, we consider the case where the estimation is performed at a fusion center, showing how to allocate quantization bits and transmit powers in the links between the nodes and the fusion center, in order to accommodate the requirement on the maximum estimation variance, under a constraint on the global transmit power. We extend the approach to the detection problem. Also in this case, we consider the distributed approach, where every node can achieve a globally optimal decision, and the case where the decision is taken at a central node. In the latter case, we show how to allocate coding bits and transmit power in order to maximize the detection probability, under constraints on the false alarm rate and the global transmit power. Then, we generalize consensus algorithms illustrating a distributed procedure that converges to the projection of the observation vector onto a signal subspace. We then address the issue of energy consumption in sensor networks, thus showing how to optimize the network topology in order to minimize the energy necessary to achieve a global consensus. Finally, we address the problem of matching the topology of the network to the graph describing the statistical dependencies among the observed variables.Comment: 92 pages, 24 figures. To appear in E-Reference Signal Processing, R. Chellapa and S. Theodoridis, Eds., Elsevier, 201