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23
Network Coding for Computing: CutSet Bounds
, 2011
"... The following 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. ..."
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Cited by 19 (4 self)
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The following 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 network coding problem for a singlereceiver 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 mincut upper bound. We extend the definition of mincut to the network computing problem and show that the mincut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multiedge tree networks. It is also tight 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 mincut bound.
Network coding for computing
 Proceedings of the fortysixth Annual Allerton Conference on Communication, Control, and Computing
, 2008
"... Abstract — The following network computation problem is considered. A set of source nodes in an acyclic network generates independent messages and a single receiver node computes a function f of the messages. The objective is to characterize the maximum number of times f can be computed per network ..."
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Cited by 15 (1 self)
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Abstract — The following network computation problem is considered. A set of source nodes in an acyclic network generates independent messages and a single receiver node computes a function f of the messages. The objective is to characterize the maximum number of times f can be computed per network usage. The network coding problem for a single receiver network is a special case of the network computation problem (taking f to be the identity map) in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to be rateoptimal and achieves the network mincut upper bound. We give a generalized mincut upper bound for the network computation problem. We show that the bound reduces to the usual network mincut when f is the identity and the bound is tight for the computation of “divisible functions ” over “tree networks”. We also show that the bound is not tight in general. I.
Some bounds on the capacity of communicating the sum of sources
 IEEE Information Theory Workshop (ITW
, 2010
"... Abstract — We consider directed acyclic networks with multiple sources and multiple terminals where each source generates one i.i.d. random process over an abelian group and all the terminals want to recover the sum of these random processes. The different source processes are assumed to be independ ..."
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Cited by 8 (2 self)
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Abstract — We consider directed acyclic networks with multiple sources and multiple terminals where each source generates one i.i.d. random process over an abelian group and all the terminals want to recover the sum of these random processes. The different source processes are assumed to be independent. The solvability of such networks has been considered in some previous works. In this paper we investigate on the capacity of such networks, referred as sumnetworks, and present some bounds in terms of mincut, and the numbers of sources and terminals. I.
Communicating the sum of sources in a 3sources/3terminals network. Manuscript, 2009, available at http://www.openu.ac.il/home/mikel/ISIT09/ISIT09.pdf
"... Abstract—We consider the network communication scenario in which a number of sources si each holding independent information Xi wish to communicate the sum ∑ Xi to a set of terminals tj. In this work we consider directed acyclic graphs with unit capacity edges and independent sources of unitentropy ..."
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Cited by 8 (1 self)
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Abstract—We consider the network communication scenario in which a number of sources si each holding independent information Xi wish to communicate the sum ∑ Xi to a set of terminals tj. In this work we consider directed acyclic graphs with unit capacity edges and independent sources of unitentropy. The case in which there are only two sources or only two terminals was considered by the work of Ramamoorthy [ISIT 2008] where it was shown that communication is possible if and only if each source terminal pair si/tj is connected by at least a single path. In this work we study the communication problem in general, and show that even for the case of three sources and three terminals, a single path connecting source/terminal pairs does not suffice to communicate ∑ Xi. We then present an efficient encoding scheme which enables the communication of ∑ Xi for the three sources, three terminals case, given that each source terminal pair is connected by two edge disjoint paths. Our encoding scheme includes a structural decomposition of the network at hand which may be found useful for other network coding problems as well. I.
Function computation over linear channels
 in Proc. IEEE Int. Symp. Network Coding (NetCod
, 2010
"... Abstract—We consider multiple noncolocated sources communicating over a network to a common sink. We assume that the network operation is fixed, and its end result is to convey a fixed linear deterministic transformation of the source data to the sink. This linear transformation is known both at t ..."
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Abstract—We consider multiple noncolocated sources communicating over a network to a common sink. We assume that the network operation is fixed, and its end result is to convey a fixed linear deterministic transformation of the source data to the sink. This linear transformation is known both at the sources and at the sink. We are interested in the problem of function computation over such networks. We design communication protocols that can perform computation without modifying the network operation, by appropriately selecting the codebook that the sources employ to map their measurements to the data they send over the network. I.
Linear codes, target function classes, and network computing capacity
, 2011
"... We study the use of linear codes for network computing in singlereceiver networks with various classes of target functions of the source messages. Such classes include reducible, injective, semiinjective, and linear target functions over finite fields. Computing capacity bounds and achievability a ..."
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We study the use of linear codes for network computing in singlereceiver networks with various classes of target functions of the source messages. Such classes include reducible, injective, semiinjective, and linear target functions over finite fields. Computing capacity bounds and achievability are given with respect to these target function classes for network codes that use routing, linear coding, or nonlinear coding.
Computation in Multicast Networks: Function Alignment and Converse Theorems
, 2012
"... The classical problem in network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers which decode the same set of messages. In this work, computation over multicast networks is considered: each receiver decodes an iden ..."
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Cited by 3 (3 self)
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The classical problem in network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers which decode the same set of messages. In this work, computation over multicast networks is considered: each receiver decodes an identical function of the original messages. For a countably infinite class of twotransmitter tworeceiver singlehop linear deterministic networks, the computing capacity is characterized for a linear function (modulo2 sum) of Bernoulli sources. Inspired by the geometric concept of interference alignment in networks, a new achievable coding scheme called function alignment is introduced. A new converse theorem is established that is tighter than cutset based and genieaided bounds. Computation (vs. communication) over multicast networks requires additional analysis to account for multiple receivers sharing a network’s computational resources. We also develop a network decomposition theorem which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality. The decomposition theorem provides a conceptuallys impler algebraic proof of achievability that generalizes to Ltransmitter Lreceiver networks.
Multisession Function Computation and Multicasting in Undirected Graphs
"... In the function computation problem, certain nodes of an undirected graph have access to independent data, while some other nodes of the graph require certain functions of the data; this model, motivated by sensor networks and cloud computing, is the focus of this paper. We study the maximum rates ..."
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Cited by 2 (0 self)
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In the function computation problem, certain nodes of an undirected graph have access to independent data, while some other nodes of the graph require certain functions of the data; this model, motivated by sensor networks and cloud computing, is the focus of this paper. We study the maximum rates at which function computation is possible on a capacitated graph; the capacities on the edges of the graph impose constraints on the communication rate. We consider a simple class of computation strategies based on Steinertree packing (socalled computation trees), which does not involve block coding and has minimal delay. With a single terminal requiring function computation, computation trees are known to be optimal when the underlying graph is itself a directed tree, but have arbitrarily poor performance in general directed graphs. Our main result is that computation trees are near optimal for a wide class of function computation requirements even at multiple terminals in undirected graphs. The key technical contribution involves connecting approximation algorithms for Steiner cuts in undirected graphs to the function computation problem. Furthermore, we show that existing algorithms for Steiner tree packings allow us to compute approximately optimal packings of computation trees in polynomial time. We also show a close connection between the function computation problem and a communication problem involving multiple multicasts.
Network Coding with Computation Alignment
"... Abstract—Determining the capacity of multireceiver networks with arbitrary message demands is an open problem in the network coding literature. In this paper, we consider a multisource, multireceiver symmetric deterministic network model parameterized by channel coefficients (inspired by wireless ..."
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Abstract—Determining the capacity of multireceiver networks with arbitrary message demands is an open problem in the network coding literature. In this paper, we consider a multisource, multireceiver symmetric deterministic network model parameterized by channel coefficients (inspired by wireless network flow) in which the receivers compute a sum of the symbols generated at the sources. Scalar and vector linear coding strategies are analyzed. It is shown that computation alignment over finite field vector spaces is necessary to achieve the computation capacities in the network. To aid in the construction of coding strategies, network equivalence theorems are established for the decomposition of deterministic models into elementary subnetworks. The linear coding capacity for computation is characterized for all channel parameters considered in the model for a countably infinite class of networks. The constructive coding schemes introduced herein for a specific class of networks provide an optimistic viewpoint for the application of structured codes in network communication. Index Terms—Vector linear network coding, computation capacity, computation alignment, structured codes. I.