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...cal Engineering and Computer Sciences, University of California, Berkeley, Berkeley, USA (Email: {ngoela, gastpar}@eecs.berkeley.edu) to the classical problem of communication over multicast networks =-=[5]-=-. As a consequence, it was shown that the cut-set based bound is tight in the single-receiver case. Several results over the past decade have contributed to the understanding of classical communicatio...

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...und is tight for multicast communication. Subsequent research developed practical linear network coding strategies ranging from random linear codes to deterministic polynomial-time code constructions =-=[6]-=-, [7], [8], [9]. The success of traditional multicast communication motivates us to explore the fundamental limits of multicasting a linear function in multi-receiver networks as a natural next step. ...

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...s tight for multicast communication. Subsequent research developed practical linear network coding strategies ranging from random linear codes to deterministic polynomial-time code constructions [6], =-=[7]-=-, [8], [9]. The success of traditional multicast communication motivates us to explore the fundamental limits of multicasting a linear function in multi-receiver networks as a natural next step. For t...

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...ht for multicast communication. Subsequent research developed practical linear network coding strategies ranging from random linear codes to deterministic polynomial-time code constructions [6], [7], =-=[8]-=-, [9]. The success of traditional multicast communication motivates us to explore the fundamental limits of multicasting a linear function in multi-receiver networks as a natural next step. For this o...

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...t, Network Decomposition Theorem I. INTRODUCTION Recently coding for computation in networks has received considerable attention with applications in sensor networks [1] and cloud computing scenarios =-=[2]-=-, [3]. In a sensor network, a fusion node may be interested in computing a relevant function of the measurements from various data nodes. In a cloud computing scenario, a client may download a functio...

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...r multicast communication. Subsequent research developed practical linear network coding strategies ranging from random linear codes to deterministic polynomial-time code constructions [6], [7], [8], =-=[9]-=-. The success of traditional multicast communication motivates us to explore the fundamental limits of multicasting a linear function in multi-receiver networks as a natural next step. For this open p...

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... a linear function (modulo-2 sum) of two independent Bernoulli sources generated at the transmitters. Specifically, we consider the Avestimehr-Diggavi-Tse (ADT) deterministic single-hop network model =-=[13]-=- which captures superposition and broadcast properties of wireless Gaussian networks and is a generalization of networks of orthogonal links. We develop a new achievable coding scheme termed function ...

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...ss Gaussian networks and is a generalization of networks of orthogonal links. We develop a new achievable coding scheme termed function alignment 1 , inspired by the concept of interference alignment =-=[15]-=-, [16]. We also derive a new converse theorem that is tighter than cutset based bounds and genie-aided bounds. As a consequence of this capacity result, we find that unlike the single-receiver case, t...

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...ssian networks and is a generalization of networks of orthogonal links. We develop a new achievable coding scheme termed function alignment 1 , inspired by the concept of interference alignment [15], =-=[16]-=-. We also derive a new converse theorem that is tighter than cutset based bounds and genie-aided bounds. As a consequence of this capacity result, we find that unlike the single-receiver case, the cut...

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... in achieving capacity limits, and structured codes achieve higher computation rates [10]; (b) Linear codes are insufficient in general for computation over multi-receiver networks (cf. both [11] and =-=[12]-=-) and non-linear codes may achieve higher computation rates. To make progress on the problem of multicasting a function in multi-receiver networks, we consider the simplest twotransmitter two-receiver...

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...case, the total number of these N2 computable bits is N1+N2+{m−(N1+N2)} = m. Similarly receiver 2 can compute m bits. Therefore, we can achieve Rcomp = m. Remark 3 (Exploiting Channel Structure [10], =-=[19]-=-): In the regime of 1 2 2 ≤ α ≤ 3 , more-than-half of signal levels at receivers naturally form the mod-sum function. This enables us to create a shared linear subspace. Note in the above example that...

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...puting Capacity, Function Alignment, Network Decomposition Theorem I. INTRODUCTION Recently coding for computation in networks has received considerable attention with applications in sensor networks =-=[1]-=- and cloud computing scenarios [2], [3]. In a sensor network, a fusion node may be interested in computing a relevant function of the measurements from various data nodes. In a cloud computing scenari...

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...next step. For this open problem, some facts are known based on example networks: (a) Random codes are insufficient in achieving capacity limits, and structured codes achieve higher computation rates =-=[10]-=-; (b) Linear codes are insufficient in general for computation over multi-receiver networks (cf. both [11] and [12]) and non-linear codes may achieve higher computation rates. To make progress on the ...

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...twork Decomposition Theorem I. INTRODUCTION Recently coding for computation in networks has received considerable attention with applications in sensor networks [1] and cloud computing scenarios [2], =-=[3]-=-. In a sensor network, a fusion node may be interested in computing a relevant function of the measurements from various data nodes. In a cloud computing scenario, a client may download a function or ...

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...ncept of interference alignment [15], [16] which has shown great potential for a variety of applications such as interference channels [16], cellular networks [20], [21], distributed storage networks =-=[22]-=-, [23], [24] and multiple unicast networks [25], [26]. But the distinction w.r.t our problem comes from the purpose of alignment. In our problem, the aim of alignment is to form a desired function whi...

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...it is shown that feedback can increase the computing rate. Interestingly the feedback gain is shown to be significant - qualitatively similar to the gain in the two-user Gaussian interference channel =-=[28]-=-, which revealed an unbounded feedback gain: the gap between nonfeedback and feedback capacities can be arbitrarily large as the signalto-noise ratio of each link increases. However, the result of [27...

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... is an instance of the important concept of interference alignment [15], [16] which has shown great potential for a variety of applications such as interference channels [16], cellular networks [20], =-=[21]-=-, distributed storage networks [22], [23], [24] and multiple unicast networks [25], [26]. But the distinction w.r.t our problem comes from the purpose of alignment. In our problem, the aim of alignmen...

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...of interference alignment [15], [16] which has shown great potential for a variety of applications such as interference channels [16], cellular networks [20], [21], distributed storage networks [22], =-=[23]-=-, [24] and multiple unicast networks [25], [26]. But the distinction w.r.t our problem comes from the purpose of alignment. In our problem, the aim of alignment is to form a desired function while min...

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...erference alignment [15], [16] which has shown great potential for a variety of applications such as interference channels [16], cellular networks [20], [21], distributed storage networks [22], [23], =-=[24]-=- and multiple unicast networks [25], [26]. But the distinction w.r.t our problem comes from the purpose of alignment. In our problem, the aim of alignment is to form a desired function while minimizin...

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...h has shown great potential for a variety of applications such as interference channels [16], cellular networks [20], [21], distributed storage networks [22], [23], [24] and multiple unicast networks =-=[25]-=-, [26]. But the distinction w.r.t our problem comes from the purpose of alignment. In our problem, the aim of alignment is to form a desired function while minimizing the signal subspace occupied by t...

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...understanding of the feedback gain. Moreover it would be more interesting to explore this feedback gain under more realistic scenarios where feedback is , offered through rate-limited bit-piped links =-=[29]-=- or through the corresponding backward communication network [30]. Furthermore, we are interested in extending to more general multi-hop networks [31]. VIII. CONCLUSION We have established the computi...

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...tter two-receiver Avestimehr-Diggavi-Tse (ADT) deterministic network addition, the approach has potential for the design of structured computation codes in larger multi-hop networks. Related Work: In =-=[17]-=-, [11], [18], the computing capacity for multicasting a sum of sources is explored for arbitrary multiple-source multiple-destination networks. Rai and Dey [11] proved that there exists a linear solva...

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...shown great potential for a variety of applications such as interference channels [16], cellular networks [20], [21], distributed storage networks [22], [23], [24] and multiple unicast networks [25], =-=[26]-=-. But the distinction w.r.t our problem comes from the purpose of alignment. In our problem, the aim of alignment is to form a desired function while minimizing the signal subspace occupied by the sou...

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...gle receiver which computes a function of the original sources. Appuswamy et al. study the fundamental limits of computation for linear and general target function classes for singlereceiver networks =-=[4]-=-. While limited progress has been made for general target functions, the problem of linear function computation in single-receiver networks has been solved in part due to a duality theorem establishin...

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...sition approach offers a conceptually simpler proof of achievability which we use to establish the linear computing capacity of L-transmitter L-receiver single-hop networks. In 1 Niesen-Nazer-Whiting =-=[14]-=- introduced a similar scheme in a different context called computation alignment.2 S1 Enc 1 ∼ Ber 1 2 S2 ∼ Ber 1 2 X1 Tx 1 X2 Enc 2 Tx 2 n11 n22 n12 n21 Y1 Rx 1 Dec 1 Y2 Rx 2 Dec 2 ˆ S1 ⊕ S2 ˆ S1 ⊕ S...

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...eresting to explore this feedback gain under more realistic scenarios where feedback is , offered through rate-limited bit-piped links [29] or through the corresponding backward communication network =-=[30]-=-. Furthermore, we are interested in extending to more general multi-hop networks [31]. VIII. CONCLUSION We have established the computing capacity of a twotransmitter two-receiver ADT symmetric networ...

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...ufficient in achieving capacity limits, and structured codes achieve higher computation rates [10]; (b) Linear codes are insufficient in general for computation over multi-receiver networks (cf. both =-=[11]-=- and [12]) and non-linear codes may achieve higher computation rates. To make progress on the problem of multicasting a function in multi-receiver networks, we consider the simplest twotransmitter two...

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...eiver Avestimehr-Diggavi-Tse (ADT) deterministic network addition, the approach has potential for the design of structured computation codes in larger multi-hop networks. Related Work: In [17], [11], =-=[18]-=-, the computing capacity for multicasting a sum of sources is explored for arbitrary multiple-source multiple-destination networks. Rai and Dey [11] proved that there exists a linear solvably equivale...

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...is , offered through rate-limited bit-piped links [29] or through the corresponding backward communication network [30]. Furthermore, we are interested in extending to more general multi-hop networks =-=[31]-=-. VIII. CONCLUSION We have established the computing capacity of a twotransmitter two-receiver ADT symmetric network where each receiver wishes to compute a modulo-2-sum function of two Bernoulli sour...

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...rce two-destination multi-hop networks when the entropy of each source is limited by 2. B. Role of Feedback for Computation The role of feedback for function computation has initially been studied in =-=[27]-=- where it is shown that feedback can increase the computing rate. Interestingly the feedback gain is shown to be significant - qualitatively similar to the gain in the two-user Gaussian interference c...