The Experts below are selected from a list of 116394 Experts worldwide ranked by ideXlab platform
Slawomir Stanczak - One of the best experts on this subject based on the ideXlab platform.
-
on the channel estimation effort for analog Computation over wireless multiple access channels
IEEE Wireless Communications Letters, 2014Co-Authors: Mario Goldenbaum, Slawomir StanczakAbstract:This letter studies a multiple-access transmission scheme that exploits interference for an efficient Function Computation in sensor networks. The central question is how much channel knowledge is generally needed and how the channel estimation effort can significantly be reduced. It is first shown that the channel magnitude at the transmitters is sufficient to achieve the same performance as with full channel state information. It is further shown that for a wide range of fading distributions, no channel state information is needed at the transmitters, provided that the receiver has access to some statistical channel knowledge and is equipped with multiple antennas.
-
harnessing interference for analog Function Computation in wireless sensor networks
IEEE Transactions on Signal Processing, 2013Co-Authors: Mario Goldenbaum, Holger Boche, Slawomir StanczakAbstract:It is known that if the objective of a wireless sensor network is not to reconstruct individual sensor readings at a fusion center but rather to compute a linear Function of them, then the interference property of the wireless channel can be beneficially harnessed by letting nodes transmit simultaneously. Recently, an analog Computation scheme was proposed to show that it is possible to take the advantage of the interference property even if nonlinear Functions are to be computed. The scheme involves some pre-processing on the sensor readings and post-processing on the superimposed signals observed by the fusion center. Correspondingly, this paper provides a thorough base for a theory of analog-computing Functions over wireless channels by specifying what is the maximum achievable. This means it is determined for networks of arbitrary topology which Functions are generally analog-computable over the channel and how many wireless resources are needed. It turns out that the considerations are closely related to the famous 13th Hilbert problem and that analog-Computations can be universally performed in the sense that the pre-processing at sensor nodes is independent of the Function to be computed. Universality reduces the complexity of transmitters and the signaling overhead, and it is shown that this property is preserved if nodes leave or join the network. Analog-computability is therefore of high practical relevance as it allows for an efficient Computation of Functions in sensor networks.
-
robust analog Function Computation via wireless multiple access channels
IEEE Transactions on Communications, 2013Co-Authors: Mario Goldenbaum, Slawomir StanczakAbstract:Wireless sensor network applications often involve the Computation of pre-defined Functions of the measurements such as for example the arithmetic mean or maximum value. Standard approaches to this problem separate communication from Computation: digitized sensor readings are transmitted interference-free to a fusion center that reconstructs each sensor reading and subsequently computes the sought Function value. Such separation-based Computation schemes are generally highly inefficient as a complete reconstruction of individual sensor readings at the fusion center is not necessary to compute a Function of them. In particular, if the mathematical structure of the channel is suitably matched (in some sense) to the Function of interest, then channel collisions induced by concurrent transmissions of different nodes can be beneficially exploited for Computation purposes. This paper proposes an analog Computation scheme that allows for an efficient estimate of linear and nonlinear Functions over the wireless multiple-access channel. A match between the channel and the Function being evaluated is thereby achieved via some pre-processing on the sensor readings and post-processing on the superimposed signals observed by the fusion center. After analyzing the estimation error for two Function examples, simulations are presented to show the potential for huge performance gains over time- and code-division multiple-access based Computation schemes.
-
robust analog Function Computation via wireless multiple access channels
arXiv: Information Theory, 2012Co-Authors: Mario Goldenbaum, Slawomir StanczakAbstract:Various wireless sensor network applications involve the Computation of a pre-defined Function of the measurements without the need for reconstructing each individual sensor reading. Widely-considered examples of such Functions include the arithmetic mean and the maximum value. Standard approaches to the Computation problem separate Computation from communication: quantized sensor readings are transmitted interference-free to a fusion center that reconstructs each sensor reading and subsequently computes the sought Function value. Such separation-based Computation schemes are generally highly inefficient as a complete reconstruction of individual sensor readings is not necessary for the fusion center to compute a Function of them. In particular, if the mathematical structure of the wireless channel is suitably matched (in some sense) to the Function, then channel collisions induced by concurrent transmissions of different nodes can be beneficially exploited for Computation purposes. Therefore, in this paper a practically relevant analog Computation scheme is proposed that allows for an efficient estimate of linear and nonlinear Functions over the wireless multiple-access channel. After analyzing the asymptotic properties of the estimation error, numerical simulations are presented to show the potential for huge performance gains when compared with time-division multiple-access based Computation schemes.
-
on Function Computation via wireless sensor multiple access channels
Wireless Communications and Networking Conference, 2009Co-Authors: Mario Goldenbaum, Slawomir Stanczak, Michal KaliszanAbstract:In wireless sensor networks, the identity of a particular sensor node and a complete reconstruction of sensed data at a designated sink node may be not needed. Indeed, the objective is often to compute a certain Function of the sensed data. Such desired Functions can be for example the arithmetic mean, the geometric mean, polynomials and other Functions that adequately match the mathematical characteristic of the underlying multiple-access channel. In this paper, we propose a simple practical approach to compute desired Functions of sensor network data, exploiting explicitly the mathematical characteristic of the wireless sensor multiple-access channel (WSMAC). In contrast to traditional schemes that are designed to combat interference caused by other connections, we exploit this interference with the goal of computing the desired Functions, which is in a sense a paradigm shift. This leads directly to a higher data rate in terms of Function Computation or a higher SNR in comparison to other schemes like time division multiple access (TDMA). Our approach needs no extensive symbol or phase synchronization, since the measured values are converted into the tranmit power of a specific random transmit sequence with unit norm. Only a coarse block synchronization is necessary so that the proposed scheme is easy to implement.
Mario Goldenbaum - One of the best experts on this subject based on the ideXlab platform.
-
on the channel estimation effort for analog Computation over wireless multiple access channels
IEEE Wireless Communications Letters, 2014Co-Authors: Mario Goldenbaum, Slawomir StanczakAbstract:This letter studies a multiple-access transmission scheme that exploits interference for an efficient Function Computation in sensor networks. The central question is how much channel knowledge is generally needed and how the channel estimation effort can significantly be reduced. It is first shown that the channel magnitude at the transmitters is sufficient to achieve the same performance as with full channel state information. It is further shown that for a wide range of fading distributions, no channel state information is needed at the transmitters, provided that the receiver has access to some statistical channel knowledge and is equipped with multiple antennas.
-
harnessing interference for analog Function Computation in wireless sensor networks
IEEE Transactions on Signal Processing, 2013Co-Authors: Mario Goldenbaum, Holger Boche, Slawomir StanczakAbstract:It is known that if the objective of a wireless sensor network is not to reconstruct individual sensor readings at a fusion center but rather to compute a linear Function of them, then the interference property of the wireless channel can be beneficially harnessed by letting nodes transmit simultaneously. Recently, an analog Computation scheme was proposed to show that it is possible to take the advantage of the interference property even if nonlinear Functions are to be computed. The scheme involves some pre-processing on the sensor readings and post-processing on the superimposed signals observed by the fusion center. Correspondingly, this paper provides a thorough base for a theory of analog-computing Functions over wireless channels by specifying what is the maximum achievable. This means it is determined for networks of arbitrary topology which Functions are generally analog-computable over the channel and how many wireless resources are needed. It turns out that the considerations are closely related to the famous 13th Hilbert problem and that analog-Computations can be universally performed in the sense that the pre-processing at sensor nodes is independent of the Function to be computed. Universality reduces the complexity of transmitters and the signaling overhead, and it is shown that this property is preserved if nodes leave or join the network. Analog-computability is therefore of high practical relevance as it allows for an efficient Computation of Functions in sensor networks.
-
robust analog Function Computation via wireless multiple access channels
IEEE Transactions on Communications, 2013Co-Authors: Mario Goldenbaum, Slawomir StanczakAbstract:Wireless sensor network applications often involve the Computation of pre-defined Functions of the measurements such as for example the arithmetic mean or maximum value. Standard approaches to this problem separate communication from Computation: digitized sensor readings are transmitted interference-free to a fusion center that reconstructs each sensor reading and subsequently computes the sought Function value. Such separation-based Computation schemes are generally highly inefficient as a complete reconstruction of individual sensor readings at the fusion center is not necessary to compute a Function of them. In particular, if the mathematical structure of the channel is suitably matched (in some sense) to the Function of interest, then channel collisions induced by concurrent transmissions of different nodes can be beneficially exploited for Computation purposes. This paper proposes an analog Computation scheme that allows for an efficient estimate of linear and nonlinear Functions over the wireless multiple-access channel. A match between the channel and the Function being evaluated is thereby achieved via some pre-processing on the sensor readings and post-processing on the superimposed signals observed by the fusion center. After analyzing the estimation error for two Function examples, simulations are presented to show the potential for huge performance gains over time- and code-division multiple-access based Computation schemes.
-
robust analog Function Computation via wireless multiple access channels
arXiv: Information Theory, 2012Co-Authors: Mario Goldenbaum, Slawomir StanczakAbstract:Various wireless sensor network applications involve the Computation of a pre-defined Function of the measurements without the need for reconstructing each individual sensor reading. Widely-considered examples of such Functions include the arithmetic mean and the maximum value. Standard approaches to the Computation problem separate Computation from communication: quantized sensor readings are transmitted interference-free to a fusion center that reconstructs each sensor reading and subsequently computes the sought Function value. Such separation-based Computation schemes are generally highly inefficient as a complete reconstruction of individual sensor readings is not necessary for the fusion center to compute a Function of them. In particular, if the mathematical structure of the wireless channel is suitably matched (in some sense) to the Function, then channel collisions induced by concurrent transmissions of different nodes can be beneficially exploited for Computation purposes. Therefore, in this paper a practically relevant analog Computation scheme is proposed that allows for an efficient estimate of linear and nonlinear Functions over the wireless multiple-access channel. After analyzing the asymptotic properties of the estimation error, numerical simulations are presented to show the potential for huge performance gains when compared with time-division multiple-access based Computation schemes.
-
on Function Computation via wireless sensor multiple access channels
Wireless Communications and Networking Conference, 2009Co-Authors: Mario Goldenbaum, Slawomir Stanczak, Michal KaliszanAbstract:In wireless sensor networks, the identity of a particular sensor node and a complete reconstruction of sensed data at a designated sink node may be not needed. Indeed, the objective is often to compute a certain Function of the sensed data. Such desired Functions can be for example the arithmetic mean, the geometric mean, polynomials and other Functions that adequately match the mathematical characteristic of the underlying multiple-access channel. In this paper, we propose a simple practical approach to compute desired Functions of sensor network data, exploiting explicitly the mathematical characteristic of the wireless sensor multiple-access channel (WSMAC). In contrast to traditional schemes that are designed to combat interference caused by other connections, we exploit this interference with the goal of computing the desired Functions, which is in a sense a paradigm shift. This leads directly to a higher data rate in terms of Function Computation or a higher SNR in comparison to other schemes like time division multiple access (TDMA). Our approach needs no extensive symbol or phase synchronization, since the measured values are converted into the tranmit power of a specific random transmit sequence with unit norm. Only a coarse block synchronization is necessary so that the proposed scheme is easy to implement.
P J Morrison - One of the best experts on this subject based on the ideXlab platform.
-
experimental determination of radiated internal wave power without pressure field data
Physics of Fluids, 2014Co-Authors: Frank M Lee, Matthew S Paoletti, Harry L Swinney, P J MorrisonAbstract:We present a method to determine, using only velocity field data, the time-averaged energy flux J and total radiated power P for two-dimensional internal gravity waves. Both J and P are determined from expressions involving only a scalar Function, the stream Function ψ. We test the method using data from a direct numerical simulation for tidal flow of a stratified fluid past a knife edge. The results for the radiated internal wave power given by the stream Function method agree to within 0.5% with results obtained using pressure and velocity data from the numerical simulation. The results for the radiated power computed from the stream Function agree well with power computed from the velocity and pressure if the starting point for the stream Function Computation is on a solid boundary, but if a boundary point is not available, care must be taken to choose an appropriate starting point. We also test the stream Function method by applying it to laboratory data for tidal flow past a knife edge, and the resul...
-
experimental determination of radiated internal wave power without pressure field data
Physics of Fluids, 2014Co-Authors: Frank M Lee, Matthew S Paoletti, Harry L Swinney, P J MorrisonAbstract:We present a method to determine, using only velocity field data, the time-averaged energy flux J and total radiated power P for two-dimensional internal gravity waves. Both J and P are determined from expressions involving only a scalar Function, the stream Function ψ. We test the method using data from a direct numerical simulation for tidal flow of a stratified fluid past a knife edge. The results for the radiated internal wave power given by the stream Function method agree to within 0.5% with results obtained using pressure and velocity data from the numerical simulation. The results for the radiated power computed from the stream Function agree well with power computed from the velocity and pressure if the starting point for the stream Function Computation is on a solid boundary, but if a boundary point is not available, care must be taken to choose an appropriate starting point. We also test the stream Function method by applying it to laboratory data for tidal flow past a knife edge, and the results are found to agree with the direct numerical simulation. The supplementary material includes a Matlab code with a graphical user interface that can be used to compute the energy flux and power from two-dimensional velocity field data.
-
experimental determination of radiated internal wave power without pressure field data
arXiv: Fluid Dynamics, 2014Co-Authors: Frank M Lee, Matthew S Paoletti, Harry L Swinney, P J MorrisonAbstract:We present a method to determine, using only velocity field data, the time-averaged energy flux $\left $ and total radiated power $P$ for two-dimensional internal gravity waves. Both $\left $ and $P$ are determined from expressions involving only a scalar Function, the stream Function $\psi$. We test the method using data from a direct numerical simulation for tidal flow of a stratified fluid past a knife edge. The results for the radiated internal wave power given by the stream Function method agree to within 0.5% with results obtained using pressure and velocity data from the numerical simulation. The results for the radiated power computed from the stream Function agree well with power computed from the velocity and pressure if the starting point for the stream Function Computation is on a solid boundary, but if a boundary point is not available, care must be taken to choose an appropriate starting point. We also test the stream Function method by applying it to laboratory data for tidal flow past a knife edge, and the results are found to agree with the direct numerical simulation. Supplementary Material includes a Matlab code with a graphical user interface (GUI) that can be used to compute the energy flux and power from any two-dimensional velocity field data.
Pulkit Grover - One of the best experts on this subject based on the ideXlab platform.
-
rate distortion for lossy in network linear Function Computation and consensus distortion accumulation and sequential reverse water filling
IEEE Transactions on Information Theory, 2017Co-Authors: Yaoqing Yang, Pulkit Grover, Soummya KarAbstract:We consider the problem of distributed lossy linear Function Computation in a tree network. We examine two cases: 1) data aggregation (only one sink node computes) and 2) consensus (all nodes compute the same Function). By quantifying the accumulation of information loss in distributed computing, we obtain fundamental limits on network Computation rate as a Function of incremental distortions (and hence incremental loss of information) along the edges of the network. The above characterization, based on quantifying distortion accumulation, offers an improvement over classical cut-set type techniques, which are based on overall distortions instead of incremental distortions. This quantification of information loss qualitatively resembles information dissipation in cascaded channels [2] . Surprisingly, this accumulation effect of distortion happens even at infinite blocklength. Combining this observation with an inequality on the dominance of mean-square quantities over relative-entropy quantities, we obtain outer bounds on the rate distortion Function that are tighter than classical cut-set bounds by a difference, which can be arbitrarily large in both data aggregation and consensus. We also obtain inner bounds on the optimal rate using random Gaussian coding, which differ from the outer bounds by $\mathcal {O}(\sqrt {D})$ , where $D$ is the overall distortion. The obtained inner and outer bounds can provide insights on rate (bit) allocations for both the data aggregation problem and the consensus problem. We show that for tree networks, the rate allocation results have a mathematical structure similar to classical reverse water-filling for parallel Gaussian sources. Apart from data aggregation and distributed consensus, the distortion accumulation analysis framework is also applicable in large-scale data summarization through histograms and linear sketching, e.g., word counting tasks for document summarization.
-
graph codes for distributed instant message collection in an arbitrary noisy broadcast network
IEEE Transactions on Information Theory, 2017Co-Authors: Yaoqing Yang, Soummya Kar, Pulkit GroverAbstract:We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide: 1) fundamental limits on the required number of broadcasts of data gathering and 2) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, arbitrary geometric networks and random Erdos-Renyi networks, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight in scaling sense. Our main techniques are three distributed encoding techniques, called graph codes, which are designed, respectively, for the above-mentioned three scenarios. Our work, thus, extends and unifies previous works such as those of Gallager and Karamchandani on the number of broadcasts for distributed Function Computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.
-
rate distortion for lossy in network Function Computation information dissipation and sequential reverse water filling
arXiv: Information Theory, 2016Co-Authors: Yaoqing Yang, Pulkit Grover, Soummya KarAbstract:We consider the problem of distributed lossy linear Function Computation in a tree network. We examine two cases: (i) data aggregation (only one sink node computes) and (ii) consensus (all nodes compute the same Function). By quantifying the accumulation of information loss in distributed computing, we obtain fundamental limits on network Computation rate as a Function of incremental distortions (and hence incremental loss of information) along the edges of the network. The above characterization, based on quantifying distortion accumulation, offers an improvement over classical cut-set type techniques which are based on overall distortions instead of incremental distortions. This quantification of information loss qualitatively resembles information dissipation in cascaded channels [1]. Surprisingly, this accumulation effect of distortion happens even at infinite blocklength. Combining this observation with an inequality on the dominance of mean-square quantities over relative-entropy quantities, we obtain outer bounds on the rate distortion Function that are tighter than classical cut-set bounds by a difference which can be arbitrarily large in both data aggregation and consensus. We also obtain inner bounds on the optimal rate using random Gaussian coding, which differ from the outer bounds by $\mathcal{O}(\sqrt{D})$, where $D$ is the overall distortion. The obtained inner and outer bounds can provide insights on rate (bit) allocations for both the data aggregation problem and the consensus problem. We show that for tree networks, the rate allocation results have a mathematical structure similar to classical reverse water-filling for parallel Gaussian sources.
-
graph codes for distributed instant message collection in an arbitrary noisy broadcast network
arXiv: Information Theory, 2015Co-Authors: Yaoqing Yang, Soummya Kar, Pulkit GroverAbstract:We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide (i) fundamental limits on the required number of broadcasts of data gathering, and (ii) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, \textcolor{black}{arbitrary geometric networks and random Erd$\ddot{o}$s-R$\acute{e}$nyi networks}, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight \textcolor{black}{in order sense}. Our main techniques are three distributed encoding techniques, called graph codes, which are designed respectively for the above-mentioned three scenarios. Our work thus extends and unifies previous works such as those of Gallager [1] and Karamchandani~\emph{et. al.} [2] on number of broadcasts for distributed Function Computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.
Yaoqing Yang - One of the best experts on this subject based on the ideXlab platform.
-
rate distortion for lossy in network linear Function Computation and consensus distortion accumulation and sequential reverse water filling
IEEE Transactions on Information Theory, 2017Co-Authors: Yaoqing Yang, Pulkit Grover, Soummya KarAbstract:We consider the problem of distributed lossy linear Function Computation in a tree network. We examine two cases: 1) data aggregation (only one sink node computes) and 2) consensus (all nodes compute the same Function). By quantifying the accumulation of information loss in distributed computing, we obtain fundamental limits on network Computation rate as a Function of incremental distortions (and hence incremental loss of information) along the edges of the network. The above characterization, based on quantifying distortion accumulation, offers an improvement over classical cut-set type techniques, which are based on overall distortions instead of incremental distortions. This quantification of information loss qualitatively resembles information dissipation in cascaded channels [2] . Surprisingly, this accumulation effect of distortion happens even at infinite blocklength. Combining this observation with an inequality on the dominance of mean-square quantities over relative-entropy quantities, we obtain outer bounds on the rate distortion Function that are tighter than classical cut-set bounds by a difference, which can be arbitrarily large in both data aggregation and consensus. We also obtain inner bounds on the optimal rate using random Gaussian coding, which differ from the outer bounds by $\mathcal {O}(\sqrt {D})$ , where $D$ is the overall distortion. The obtained inner and outer bounds can provide insights on rate (bit) allocations for both the data aggregation problem and the consensus problem. We show that for tree networks, the rate allocation results have a mathematical structure similar to classical reverse water-filling for parallel Gaussian sources. Apart from data aggregation and distributed consensus, the distortion accumulation analysis framework is also applicable in large-scale data summarization through histograms and linear sketching, e.g., word counting tasks for document summarization.
-
graph codes for distributed instant message collection in an arbitrary noisy broadcast network
IEEE Transactions on Information Theory, 2017Co-Authors: Yaoqing Yang, Soummya Kar, Pulkit GroverAbstract:We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide: 1) fundamental limits on the required number of broadcasts of data gathering and 2) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, arbitrary geometric networks and random Erdos-Renyi networks, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight in scaling sense. Our main techniques are three distributed encoding techniques, called graph codes, which are designed, respectively, for the above-mentioned three scenarios. Our work, thus, extends and unifies previous works such as those of Gallager and Karamchandani on the number of broadcasts for distributed Function Computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.
-
rate distortion for lossy in network Function Computation information dissipation and sequential reverse water filling
arXiv: Information Theory, 2016Co-Authors: Yaoqing Yang, Pulkit Grover, Soummya KarAbstract:We consider the problem of distributed lossy linear Function Computation in a tree network. We examine two cases: (i) data aggregation (only one sink node computes) and (ii) consensus (all nodes compute the same Function). By quantifying the accumulation of information loss in distributed computing, we obtain fundamental limits on network Computation rate as a Function of incremental distortions (and hence incremental loss of information) along the edges of the network. The above characterization, based on quantifying distortion accumulation, offers an improvement over classical cut-set type techniques which are based on overall distortions instead of incremental distortions. This quantification of information loss qualitatively resembles information dissipation in cascaded channels [1]. Surprisingly, this accumulation effect of distortion happens even at infinite blocklength. Combining this observation with an inequality on the dominance of mean-square quantities over relative-entropy quantities, we obtain outer bounds on the rate distortion Function that are tighter than classical cut-set bounds by a difference which can be arbitrarily large in both data aggregation and consensus. We also obtain inner bounds on the optimal rate using random Gaussian coding, which differ from the outer bounds by $\mathcal{O}(\sqrt{D})$, where $D$ is the overall distortion. The obtained inner and outer bounds can provide insights on rate (bit) allocations for both the data aggregation problem and the consensus problem. We show that for tree networks, the rate allocation results have a mathematical structure similar to classical reverse water-filling for parallel Gaussian sources.
-
graph codes for distributed instant message collection in an arbitrary noisy broadcast network
arXiv: Information Theory, 2015Co-Authors: Yaoqing Yang, Soummya Kar, Pulkit GroverAbstract:We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide (i) fundamental limits on the required number of broadcasts of data gathering, and (ii) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, \textcolor{black}{arbitrary geometric networks and random Erd$\ddot{o}$s-R$\acute{e}$nyi networks}, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight \textcolor{black}{in order sense}. Our main techniques are three distributed encoding techniques, called graph codes, which are designed respectively for the above-mentioned three scenarios. Our work thus extends and unifies previous works such as those of Gallager [1] and Karamchandani~\emph{et. al.} [2] on number of broadcasts for distributed Function Computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.
