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

On network coding for sum-networks

A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear network coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under non-linear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.Comment: Accepted to IEEE Transactions on Information Theor