Computing Linear Functions by Linear Coding Over Networks

Transcription

1 422 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 1, JANUARY 2014 Computing Linear Functions by Linear Coding Over Networks Rathinakumar Appuswamy, Member, IEEE, and Massimo Franceschetti Abstract We consider the scenario in which a set of sources generates messages in a network and a receiver node demands an arbitrary linear function of these messages We formulate an algebraic test to determine whether an arbitrary network can compute linear functions using linear codes We identify a class of linear functions that can be computed using linear codes in every network that satisfies a natural cut-based condition Conversely, for another class of linear functions, we show that the cut-based condition does not guarantee the existence of a linear coding solution For linear functions over the binary field, the two classes are complements of each other Index Terms Function computation, linear coding, network coding, network computing I INTRODUCTION I N many practical networks, including sensor networks and vehicular networks, receivers demand a function of the messages generated by the sources that are distributed across the network, rather than the generated messages Several information-theoretic variations of this problem have been studied in the framework of network computing [3] [6], [8], [10], [11], [16] [19] The classic network coding model of Ahlswede et al [1] is a special case of network computing in which the function to be computed at the receivers corresponds to the set of the source messages The works [16], [17] introduced the notions of linear, nonlinear, and routing computing capacities andusedthemtostudy network computing over single-receiver networks with noiseless links These notions of capacity refer to the ability to perform routing, linear, or nonlinear coding operations at the nodes in the network The nonlinear computing capacity is also simply referred to as the computing capacity of the network The focus of [16] was to derive upper and lower bounds on the computing capacity and to study the tightness of these bounds for a number of example functions The focus of [17] was to compare the linear, routing, and computing capacities Manuscript received February 23, 2011; revised March 10, 2013; accepted June 06, 2013 Date of publication September 27, 2013; date of current version December 20, 2013 This work was supported in part by the National Science Foundation under Award CNS and in part by the UCSD Center for Wireless Communications This paper was presented in part at the Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA, September 2010 R Appuswamy is with IBM Research Almaden, San Jose, CA USA ( rathnam@ieeeorg) M Franceschetti is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA USA ( massimo@eceucsdedu) Communicated by M Gastpar, Associate Editor for Shannon Theory Digital Object Identifier /TIT Following the works [16], [17], this paper considers the scenario in which a set of source nodes generates messages over a finite field, a single receiver node computes a linear function of these messages, and communication occurs over a network with noiseless links We ask whether this linear function can be computed by performing linear coding operations at the intermediate nodes In the special case, when the linear function is scalar valued, it was already shown in [17] that linear codes are sufficient to achieve the computing capacity The same result has been independently obtained in [6] using a different technique, and we shall deal with the vector case here Different from [16], [17], we focus on thesolvability of networks rather than capacity Since, bydefinition, a computing solution achieves a rate of one, the problem of solvability is equivalent to the problem of finding rate-1 solutions This was the original setup where network coding was developed We also restrict to scalar-linear solvability as opposed to vector-linear solvability; namely the nodes in the network are restricted to perform scalar linear encoding operations These codes were introduced and studied in [2] A more general class of linear codes was defined and studied in [14] and [15] Clearly, a network that does not have a scalar-linear solution may permit a vector-linear solution for the class of problems under consideration On the other hand, our limited consideration is motivated by our demonstration that the existence of scalar-linear solutions is computationally decidable whereas, the existence of a vector-linear solution may be undecidable asnotedin[13] In multiple-receiver networks, if each receiver node demands a subset of the source messages (which is an example of a linear function), then Dougherty, et al [14] showed that linear codes are not sufficient to recover the source messages Similarly, if each receiver node demands the sum of the source messages, then Rai and Dey [6] showed that linear codes are also not sufficient to recover the source messages In contrast, in single-receiver networks linear codes are sufficient for both of the aforementioned problems and a simple cut-based condition can be used to test whether a linear solution exists In the former case, when the receiver demands a subset of source messages, if the network satisfies the appropriate min-cut condition, then the sources can simply route the source messages to the receiver as argued in [13] On the other hand, if the source demands a linear sum of the sources, then [8] argued that as long as all the sources are connected to the receiver one can construct a spanning tree with the receiver as the root and sum the source message along the leaves We investigate if a similar cut-based condition guarantees the existence of a linear solution when the receiver node demands an arbitrary linear function of the source messages Our contributions are the following We identify two classes of functions, one for which the cut-based condition is sufficient IEEE

2 APPUSWAMY AND FRANCESCHETTI: COMPUTING LINEAR FUNCTIONS BY LINEAR CODING OVER NETWORKS 423 for solvability and the other for which it is not These classes are complements of each other when the source messages are over the binary field Along the way, we develop an algebraic framework to study linear codes and provide an algebraic condition to test whether a linear solution exists, similar to the one given by Koetter and Médard [2] for classical network coding This paper is organized as follows We formally introduce the network computation model in Section I-A In Section II, we develop the necessary algebraic tools to study linear codes and introduce the cut-based condition In Section III, we show the main results for the two classes of functions Section IV concludes the paper, mentioning some open problems A Network Model and Preliminaries In this paper, a network consists of a finite, directed acyclic multigraph, a set of source nodes,andareceiver Such a network is denoted by We use the word graph to mean a multigraph, and network to mean a single-receiver network We assume that, and that the graph contains a directed path from every node in to the receiver For each node,let and denote the in-edges and out-edges of, respectively We also assume (without loss of generality) that if a network node has no in-edges, then it is a source node We use to denote the number of sources in the network An alphabet is a nonzero finite field with For any positive integer, any vector,andany,let denote the th component of For any index set with, let denote the vector The network computing problem consists of a network,a source alphabet,andatarget function where is the decoding alphabet Atarget function is linear if there exists a matrix over such that 1) Every edge carries an element of and this element is denoted by For any node and any out-edge, the network code specifies an encoding function of the form: if otherwise where for all 2) The decoding function outputs a vector of length whose th component is of the form where for all The arithmetic in (1) and (2) is performed over Note that by (1), In this paper, by a network code, we always mean a linear network code In the literature, the class of network codes we define here is referred to as scalar linear codes These codes were introduced and studied in [2] A more general class of linear codes over were defined and studied in [14] and [15] Depending on the context, we may view as a vector of length- over or as an element of Without explicit mention, we use the fact that the addition of as elements of a finite field coincides with their sum as elements of a vector space over Furthermore, we also view as a subfield of without explicitly stating the inclusion map Let denote the vectors carried by the in-edges of the receiver Definition I2: A linear network code over is called a linear solution for computing in (or simply a linear solution if and are clear from the context) if the decoding function is such that for every choice of message vectors, (1) (2) (3) where denotes matrix transposition For linear target functions the decoding alphabet is of the form,with Without loss of generality, we assume that is full rank (over ) and has no zero columns For example, if is the identity matrix, then the receiver demands the complete set of source messages, and this corresponds to the classical network coding problem On the other hand, if is the row vector of 1 s, then the receiver demands a sum (over ) of the source values Let be a positive integer Given a network with source set and alphabet,amessage vector of length for source will be denoted by 1 We may frequently view as an element of Definition I1: A linear network code in a network consists of the following: 1 We assume that each source node has access to only one message However, the model can be directly generalized to allow source nodes to access more than one source message Remark I3: Each source generates symbols over (viewing as a vector space over ) and the decoder computes the target function for each set of source symbols This simple extension of over the larger finite field is motivated by our intention to keep the function along with the network topology fixed as we explore the coding problem A set of edges is said to separate sources from the receiver, if for each, every path from to contains at least one edge in Aset is said to be a cut if it separates at least one source from the receiver Let denote the set of all cuts in network For any matrix,let denote its th column For an index set,let denote the submatrix of obtained by choosing the columns of indexed by If is a cut in a network,wedefine the set

3 424 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 1, JANUARY 2014 Finally, for any network and matrix,wedefine (4) The aforementioned definition is motivated by the fact that the subspace as varies over all possible values for,while fixing the remaining elements of will have a dimension equal to 2 II ALGEBRAIC FRAMEWORK A Algebraic Test for the Existence of a Linear Solution Linear solvability for the classical network coding problem was shown in [2] to be equivalent to the existence of a nonempty algebraic variety In the following, we present an analogous characterization for computing linear functions, providing an algebraic test to determine whether a linear solution for computing a linear function exists The reverse problem of constructing a multiple-receiver network coding (respectively, network computing) problem given an arbitrary set of polynomials, which is solvable if and only if the corresponding set of polynomials is simultaneously solvable is considered in [15] (respectively, [6]) Webeginbygivingsomedefinitions and stating a technical lemma, followed by the main theorem below For any edge,let and Associated with a linear code over,wedefine the following three types of matrices: 1) For each source,define the matrix as follows: if otherwise 2) Similarly define the matrix as follows: if otherwise 3) Define the matrix as follows: if otherwise Since the graph associated with the network is acyclic, we can assume that the edges are ordered such that the matrix is strictly upper-triangular Let denote the identity matrix of suitable dimension Consider a network with alphabet and consider a linear code over with associated matrices,and For every,define the matrix 2 Our definition here also coincides with the one given in [16] for the class of functions considered in this paper (5) (6) (7) (8) Fig 1 Single-receiver network with three sources Now let be a vector containing all the nonzero entries of the matrices,andlet (respectively, ) be a vector containing all the nonzero entries of the matrix (respectively, ) By a slight abuse of notation, depending on the context we may view,, as elements of or as indeterminates Thus, each of the matrices defined previously may either be a matrix over or a matrix over the polynomial ring The context should make it clear which of these two notions is being referred to at any given point Lemma II1: The following two statements hold: 1) The matrix has a polynomial inverse with coefficients in, the ring of polynomials in the variables constituting 2) The decoding function can be written as Proof: The first assertion is a restatement of [2, Lemma 2] and the second assertion follows from [2, Th 3] Definition II2: Let be a polynomial ring The ideal generated by a subset and denoted by is the smallest ideal in containing Let be a network with alphabet Let and Consider a linear network code for computing the linear function corresponding to in and the associated matrices over Let denote the ideal generated by the elements of the vectors over the ring We denote the Gröbner basis of an ideal generated by subset of a polynomial ring by The following theorem is a consequence of Hilbert Nullstellensatz (see [21, Lemma VIII72] and the remark after [21, Proposition VIII74]) The matrices being central to many of the arguments to follow, we illustrate their computation for the network shown in Fig 1 in the following example

4 APPUSWAMY AND FRANCESCHETTI: COMPUTING LINEAR FUNCTIONS BY LINEAR CODING OVER NETWORKS 425 Example II3: Consider the network shown in Fig 1 and choose Wehave and the matrices defined in (5) (7) can be written as By utilizing the identity and the observation that for we also get Substituting these matrices in (8), we obtain B Minimum Cut Condition It is clear that the set of linear functions that can be solved in a network depends on the network topology It is easily seen that a linear solution for computing a linear target function corresponding to exists only if the network is such that for every, the value of the cut is at least the rank of the submatrix (recall that is the index set of the sources separated by the cut ) This observation is stated in the following lemma which is an immediate consequence of the cut-based bound in [16, Th 21] Lemma II5: For a network, a necessary condition for the existence of a linear solution for computing the target function corresponding to is We now consider two special cases First, consider the case in which the receiver demands all the source messages The corresponding is given by the identity matrix and the condition reduces to Theorem II4: Consider a network with alphabet and the linear target function corresponding to a matrix There exists an and a linear solution over for computing in if and only if Proof: From Lemma II1, the vector computed at the receiver can be written as (9) On the other hand, to compute the linear function corresponding to, the decoding function must satisfy (10) It follows that the encoding coefficients in a linear solution must be such that (11) If we view the coding coefficients as variables, then it follows that a solution must simultaneously solve the generating polynomials of the corresponding ideal By [21, Lemma VIII72], such a solution exists over the algebraic closure of if and only if Furthermore, if and only if Moreover, a solution exists over the algebraic closure of if and only if it exists over some extension field of and the proof is now complete ie, the number of edges in the cut be at least equal to the number of sources separated by the cut Second, consider the case in which the receiver demands the sum of the source messages The corresponding matrix is an row vector and the requirement that reduces to ie, all the sources have a directed path to the receiver For both of the aforementioned cases, the cut condition in Lemma II5 is also sufficient for the existence of a solution This is shown in [7] and [12, Th 42], and is reported in the following Lemma: Lemma II6: Let Foranetwork with the linear target function corresponding to a matrix, a linear solution exists if and only if The focus in the rest of this paper is to extend aforementioned results to the case by using the algebraic test of Theorem II4 III COMPUTING LINEAR FUNCTIONS In the following, we first define an equivalence relation among matrices and then use it to identify a set of functions that are linearly solvable in every network satisfying the condition We then construct a linear function outside this set, and a corresponding network with, on which such a function cannot be computed with linear codes Finally, we use this example as a building block to identify a set of linear functions for which there exist networks satisfying the min-cut condition and on which these functions are not linearly solvable Notice that for a linear function with matrix, each column of corresponds to a single-source node Hence, for

5 426 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 1, JANUARY 2014 every permutation matrix, computing is equivalent to computing after appropriately renaming the source nodes, ie, a solution exists for computing in a network if and only if a solution exists for computing in network which is obtained from by renaming the sources appropriately Furthermore, for every full-rank matrix over, computing is equivalent to computing These observations motivate the following definition: Definition III1: Let and Wesay if there exist an invertible matrix of size and a permutation matrix of size such that,and if such and do not exist Since is assumed to be a full-rank matrix, can be chosen such that the first columns of are linearly independent Let denote the first columns of By choosing,we have where is an matrix So for an arbitrary linear target function and an associated matrix, there exists an matrix such that Without loss of generality, we assume that each column of associated with a target function is nonzero Recall that sarebydefinition row vectors of the length We show that as long as the constraints of the following theorem are satisfied, the network variables can be chosen such that any set of vectors from are linearly independent This part of the proof is similar in flavor to the proof of the multicast theorem in [2] Once such a choice of is made, we then carefully realign the encoding matrices s and the decoding matrix to guarantee the computation of the linear function of choice The fact that the receiver does not require to recover all the source messages is exploited in this last step Theorem III2: Consider a network with a linear target function corresponding to a matrix (ie, ) If where is a column vector whose elements are nonzero elements of the finite field, then, a necessary and sufficient condition for the existence of a linear solution is Proof: Let The necessary part is clear from Lemma II5 We now focus on the sufficiency part Notice that for each, the matrix [computed as in (8)] is a row vector of length Stack these row vectors to form an matrix as follows: Let denote the submatrix of obtained by deleting its th row Claim 1: The matrix such that are full rank over Define two diagonal matrices and such that for (12) Now define the following realigned encoding and decoding matrices over : (13) By Claim 2 it follows that exists If the matrices, and define a linear network code, then by Lemma II1, the vector received by can be written as where We have where (a) follows from from of (14) (15) (16), (b) follows, and (c) follows from the definition Substituting (13) into (16), we get has a nonzero determinant over the ring Claim 2: For each, we have over By Claim 1 and the Schwartz--Zippel lemma [2], [20], it follows that there exists an,andachoiceof from (17)

6 APPUSWAMY AND FRANCESCHETTI: COMPUTING LINEAR FUNCTIONS BY LINEAR CODING OVER NETWORKS 427 Putting together (15) and (17), we now have ProofofClaim2: Let If there exists a such that ;then, (18) By substituting (17) into (14), we conclude that the receiver computes the desired linear function by employing the network code defined by the encoding matrices,, and The proof of the theorem is now complete for the case when If ; then, there exists a full-rank matrix and a column vector of nonzero elements over such that Since a full-rank linear operator preserves linear-independence among vectors, for every such full-rank matrix,wehave (19) Equation (19) implies that Since, from the first part of the proof, there exist an and coding matrices,,and over such that the receiver can compute the linear target function corresponding to ifandonlyif It immediately follows that by utilizing a code corresponding to the coding matrices,,and, the receiver can compute the target function corresponding to All that remains to be done is to provide proofs of claims 1 and 2 ProofofClaim1: If a cut is such that, then Thus, as argued in [13], there exists a routing solution to compute the identity function of the sources at the receiver Let and let for some (arbitrary) By Lemma II1, after fixing, the vector received by can be written as The existence of a routing solution for computing the identity function guarantees that there exist such that the matrix has a nonzero determinant over Itfollows that the determinant of is nonzero over Since was arbitrary in the previous argument, it follows that the determinant of each is nonzero over and the claim follows which implies that is not full rank---a contradiction Remark III3: We provide the following communication-theoretic interpretation of our method of proof above We may view the computation problem as a multiple-input multiple-output channel, where the multiple inputs are given by the vector of symbols generated by the sources, the output is the vector decoded by the receiver, and the channel is characterized by the network topology and the choice of s Our objective is to choose, encoding and decoding matrices to guarantee the desired output The channel gain from source to the receiver is given by the vector of length Thefirstpartofthe proof utilizes the Schwartz--Zippel lemma to establish that there exists a choice of s such that the channel between every set of sources and the receiver is invertible This is similar to the proof of the multicast theorem in [2] In the second part of the proof, we recognize that the sources must combine (or interfere ) coherently for the receiver to be able to compute the function of interest Accordingly, we first rotate the decoding matrix to point to a favorable subspace The encoding matrices are tweaked subsequently to achieve the necessary coherent combination We now show the existence of a linear function that cannot be computed on a network satisfying the min-cut condition This network will then be used as a building block to show an analogous result for a larger class of functions Let denote the matrix (20) and let denote the corresponding linear function It is possible to show with some algebra that for any column vector each of whose elements are nonzero elements of the finite field, so that the conclusion of Theorem III2 does not hold Indeed, for the function, the opposite conclusion is true, namely cannot be computed over using linear codes This is shown by the following Lemma Lemma III4: Let be the network shown in Fig 1 with alphabet Wehave 1) 2) There does not exist a linear solution for computing corresponding to the matrix in Proof: That is easily verified by considering the cut which attains the minimum We now proceed to show, using Theorem II4, that a linear solution does not exist We may assume, without loss of generality, that the node sends its message directly to nodes and (ie, ) The matrices,and were computed in

7 428 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 1, JANUARY 2014 Fig 2 Network with min-cut 1 that does not have an -linear solution for computing Example II3 Substituting and subtracting sfromthecolumnsof,weget Since the elements of these matrices generate the ideal,we have one element of is zero, then there exists a network such that 1) 2) There does not exist a linear solution for computing in Proof: It is clear from Definition III1 and the comment preceding the definition that we can assume without loss in generality Let function Since Define denote the corresponding linear target has at least one zero element, there exists a such that has a zero in th column Denote the elements of by Furthermore, implies that ByTheoremII4, a linear solution does not exist for computing in We now identify a much larger class of linear functions for which there exist networks satisfying the min-cut condition but for which linear solutions do not exist Let be an matrix with at least one zero element and For each in this equivalence class, we show that there exist a network that does not have a solution for computing the linear target function corresponding to but satisfies the cut condition in Lemma II5 The main idea of the proof is to establish that a solution for computing such a function in network implies a solution for computing the function corresponding to in, andthentouselemmaiii4 Theorem III5: Consider a linear target function corresponding to a matrix If such that at least Let be an element of (such a exists from the fact that the th column contains at least one zero) and define and denote the elements of by Since does not contain an all zero column, Now, let denote the network shown in Fig 2 where, denotes a relay node It follows from the construction that which is equal to the transfer matrix definedin(20) (21)

8 APPUSWAMY AND FRANCESCHETTI: COMPUTING LINEAR FUNCTIONS BY LINEAR CODING OVER NETWORKS 429 1) If, then it is easy to see that Similarly, if and,thenagain Consequently, we have (25) 2) If and,thenfromfig3, and Moreover, the index set was constructed such that (26) Fig 3 Subnetwork of used to show the equivalence between solving network and solving network Consequently, we have We now show that if the network has a solution for computing, then such a solution induces a solution for computing in network, contradicting Lemma III4 Let there exist an for which there is a linear solution for computing over over In any such solution, for each,the encoding function on the edge must be of the form From (25) and (27), we conclude that if (27),then (28) (22) for some Since istheonlypathfrom source to the receiver, it is obvious that We choose the following message vectors: Let be arbitrary elements of and let For an arbitrary cut,let denote the number of sources in that are separated from the receiver by (ie, ) We have for for (23) (29) Note that has been chosen such that for any choice of, and, every edge carries the zero vector Furthermore, for the above choice of, the target function associated with reduces to Since each source in is directly connected to the receiver, is equal to the number of edges in separating the sources in from the receiver Consequently, from (28), it follows that: (30) (24) Substituting (30) into (29), we conclude that for all Substituting and into (24), it follows that the receiver can compute from the vectors received on edges and,which implies a solution over for computing the linear target function in (21) for the network shown in Fig 3 It is easy to see that the existence of such a code implies a scalar linear solution for computing in This establishes the desired contradiction Finally, we show that Let be a cut such that (ie, separates sources from only the top and middle rows in the network ) We have the following two cases: Since the edge disconnects the source from the receiver, is immediate and the proof of the theorem is now complete We now consider the case in which the source alphabet is the binary field In this case, we have the two function classes identified by Theorems III2 and III5 are complements of each other, namely either or with containing at least one zero element In particular, if such that is a matrix of 1s, then we show that with containing at least one zero element Theorem III6: Let and let If, then there exists an matrix such that has at least one zero element and

9 430 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 1, JANUARY 2014 Proof: Since is assumed to have a full-row rank, for some matrix over If has 0 s, then we are done Assume to the contrary that is a matrix of 1s We only need to consider the case when (since ) For,let denote the th column vector of the identity matrix Define and let be a permutation matrix that interchanges the th and th columns and leaves the remaining columns unchanged It is now easy to verify that (31) where is an matrix with at least one zero element: for Thus, and by transitivity we conclude that which proves the claim Remark III7: It is possible to extend Theorem II4 to the more general scenario where there are more than one receivers each computing its own linear function By simply stacking the polynomials corresponding to every receiver as in (11) and defining a larger ideal, the arguments in the proof of Theorem II4 can be generalized However, the remaining part of the story falls apart in a multireceiver setting Particularly, the achievability proof of Theorem III2 breaks down for the following reason: The encoding matrices are constructed in (13) after a specific choice of the decoding matrix at the receiver If there are multiple receivers, then this step cannot be carried out except in trivial cases When the receiver demands just the sum of the source messages, [9] treats networks with up to three receivers IV CONCLUSION We wish to mention the following open problems arising from this study 1) Is there a graph-theoretic condition that allows to determine whether a given network is solvable with reference to a given linear function? We have provided an algebraic test in terms of the Gröbner basis of a corresponding ideal, but we wish to know whether there is an algorithmically more efficient test 2) We showed that is not sufficient to guarantee solvability for a certain class of linear functions It is interesting to ask whether there is a constant such that guarantees solvability Conversely, one could ask whether for every constant there exist a network and a matrix such that and does not have a linear solution for computing the linear target function associated with APPENDIX Lemma A1: Let If is a column vector of nonzero elements and, then there exists a full-rank matrix and a column vector of nonzero elements over such that Proof: Let denote the matrix obtained by collecting the first columns of Wewillfirst show that the matrix is full rank After factoring out, we then prove that the last column must have nonzero entries Since, there exists a full-rank matrix and a permutation matrix such that (32) From (32), the columns of are constituted by the columns of in which case is full rank, or columns of contains columns of and We will now show that the vector cannot be written as a linear combination of any set of column vectors of Assume to the contrary that there exist for such that (33) where denotes the th column of Let denote the vector such that,and Wehave (34) Equation (34) contradicts the fact that is full rank Hence, s satisfying (33) do not exist and consequently, is a full-rank matrix We now have where, and hence, Furthermore, and implies that Thus, there exists a full-rank matrix and a permutation matrix such that (35) Let denote the th column of It follows from (35) that either and itself is an permutation matrix, or For some, th column of is, and the remaining columns must constitute the columns of for some If is true, then and the elements of are nonzero since is another permutation matrix If is true, then anditmustbethat (if,then which contradicts ) Let Wemusthave (36)

10 APPUSWAMY AND FRANCESCHETTI: COMPUTING LINEAR FUNCTIONS BY LINEAR CODING OVER NETWORKS 431 If we denote the number of nonzero entries in a vector by, then we have (37) From (37), it follows that and consequently that every element of is nonzero The proof of the lemma is now complete REFERENCES [1] RAhlswede,NCai,S-YRLi,andRWYeung, Network information flow, IEEE Trans Inf Theory, vol 46, no 4, pp , Jul 2000 [2] R Koetter and M Médard, An algebraic approachtonetworkcoding, IEEE/ACM Trans Netw, vol 11, no 5, pp , Oct 2003 [3] HKowshikandPRKumar, Optimalfunction computation in directed and undirected graphs, IEEE TransInfTheory,vol58,no6, pp , Jun 2012 [4] A Giridhar and P R Kumar, Towards a theory of in-network computation in wireless sensor networks, IEEE Commun Mag, vol44, no 4, pp , Apr 2006 [5] A Giridhar and P R Kumar, Computing and communicating functions over sensor networks, IEEE J Sel Areas Commun, vol 23, no 4, pp , Apr 2005 [6] BKRaiandBKDey, Onnetworkcodingforsum-networks, IEEE Trans Inf Theory, vol 58, no 1, pp 50 63, Jan 2012 [7] BKRai,BKDey,andSShenvi, Somebounds on the capacity of communicating the sum of sources, presented at the presented at the IEEE Inf Theory Workshop, Cairo, Egypt, 2010 [8] A Ramamoorthy, Communicating the sum of sources over a network, presented at the presented at the IEEE Int Symp Inf Theory, Toronto, ON, Canada, 2008 [9] A Ramamoorthy and M Langberg, Communicating the sum of sources over a network, IEEE J Sel Areas Commun, vol 31, no 4, pp , Apr 2013, to be published [10] NMa,PIshwar,andPGupta, Information-theoretic bounds for multiround function computation in collocated networks, in Proc IEEE Int Symp Inf Theory, 2009, pp [11] B Nazer and M Gastpar, Computing over multiple-access channels, IEEE Trans Inf Theory, vol53,no 10, pp , Oct 2007 [12] A R Lehman and E Lehman, Complexity classification of network information flow problems, in Proc 15th Annu ACM-SIAM Symp Discrete Algorithms, 2003, pp [13] A Rasala Lehman, Network Coding, PhD dissertation, Dept Elect Eng Comput Sci, Massachusetts Inst Technol, Cambridge, MA,USA,2005 [14] R Dougherty, C Freiling, and K Zeger, Insufficiency of linear coding in network information flow, IEEE Trans Inf Theory, vol 51, no 8, pp , Aug 2005 [15] R Dougherty, C Freiling, and K Zeger, Linear network codes and systems of polynomial equations, IEEE Trans Inf Theory, vol 54, no 5, pp , May 2008 [16] R Appuswamy, M Franceschetti, N Karamchandani, and K Zeger, Network coding for computing: Cut-set bounds, IEEE Trans Inf Theory, vol 57, no 2, pp , Feb 2011, to be published [17] R Appuswamy, M Franceschetti, N Karamchandani, and K Zeger, Network coding for computing: Linear codes, IEEE Trans Inf Theory, tobepublished [18] N Karamchandani, R Appuswamy, and M Franceschetti, Time and energy complexity of function computation over networks, IEEE Trans Inf Theory, vol 57, no 12, pp , Dec 2011 [19] L Ying, R Srikant, and G E Dullerud, Distributed symmetric function computation in noisy wireless sensor networks, IEEE Trans Inf Theory, vol 53, no 12, pp , Dec 2007 [20] J T Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J ACM, vol 27, pp , 1980 [21] T W Hungerford, Algebra New York, NY, USA: Springer-Verlag, 1997 Rathinakumar Appuswamy (S 05 M 13) received the BTech degree from Anna University, Chennai, India, and the MTech degree from the Indian Institute of Technology, Kanpur, India, both in electrical engineering in 2002, and 2004, respectively He received the MA degree in mathematics and the PhD in Electrical and Computer Engineering both from the University of California, San Diego in 2008, and 2011, respectively He was a postdoctoral researcher at IBM Research Almaden from July 2011 to March 2012 and a Research Staff Member there since April 2012 His research interests include multi-modal learning, network coding, communication for computing, and network information theory Massimo Franceschetti is associate professor in the Department of Electrical and Computer Engineering of University of California at San Diego He received the Laurea degree, magna cum laude, in Computer Engineering from the University of Naples in 1997, and the MS and PhD degrees in Electrical Engineering from the California Institute of Technology in 1999, and 2003 Before joining UCSD, he was a post-doctoral scholar at University of California at Berkeley for two years Prof Franceschetti was awarded the C H Wilts Prize in 2003 for best doctoral thesis in Electrical Engineering at Caltech; the S A Schelkunoff award in 2005 for best paper in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION; an NSF CAREER award in 2006, an ONR Young Investigator award in 2007; the IEEE Communications society best tutorial paper award in 2010; and the IEEE Control theory society Ruberti young researcher award in 2012 He has held visiting positions at the Vrije Universiteit Amsterdam in the Netherlands, the Ecole Polytechnique Federale de Lausanne in Switzerland, and the University of Trento in Italy He was associate editor for communication networks for of the IEEE TRANSACTIONS ON INFORMATION THEORY and has served as guest editor for two issues of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATION He is currently serving as associate editor for the IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS His research interests are in communication systems theory and include random networks, wave propagation in random media, wireless communication, and control over networks