We consider the multiple unicast problem with three source-terminal pairs over directed acyclic networks with unit-capacity edges. The three $s_i-t_i$ pairs wish to communicate at unit-rate via network coding. The connectivity between the $s_i - t_i$ pairs is quantified by means of a connectivity level vector, $[k_1 k_2 k_3]$ such that there exist $k_i$ edge-disjoint paths between $s_i$ and $t_i$. In this work we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if $k_i \geq 3$, for all $i = 1, \dots, 3$. In this work, we consider, connectivity level vectors such that $\min_{i = 1, \dots, 3} k_i < 3$. We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unit-rate requirement, for all such connectivity level vectors except the vector $[1~2~4]$ (and its permutations). The benefits of our schemes extend to networks with higher and potentially different edge capacities. Specifically, our experimental results indicate that for networks where the different source-terminal paths have a significant overlap, our constructive unit-rate schemes can be packed along with routing to provide higher throughput as compared to a pure routing approach.
We consider the multiple unicast problem under network coding over directed acyclic networks with unit capacity edges. There is a set of n source-terminal (s_i - t_i) pairs that wish to communicate at unit rate over this network. The connectivity between the s_i - t_i pairs is quantified by means of a connectivity level vector, [k_1 k_2 ... k_n] such that there exist k_i edge-disjoint paths between s_i and t_i. Our main aim is to characterize the feasibility of achieving this for different values of n and [k_1 ... k_n]. For 3 unicast connections (n = 3), we characterize several achievable and …
In this paper, we study the wireline two-unicast-Z communication network over directed acyclic graphs. The two-unicast-Z network is a two-unicast network where the destination intending to decode the second message has apriori side information of the first message. We make three contributions in this paper: 1. We describe a new linear network coding algorithm for two-unicast-Z networks over directed acyclic graphs. Our approach includes the idea of interference alignment as one of its key ingredients. For graphs of a bounded degree, our algorithm has linear complexity in terms of the number of…
This paper studies energy-efficient joint transmit and receive beamforming in multi-cell multi-user multiple-input multiple-output systems. We consider conventional network energy efficiency metric where the users can receive unicasting streams in addition to the group-specific common multicasting streams which have certain rate constraints. The goal is to use the transmission resources more efficiently to improve the energy efficiency, when the users are equipped with multiple antennas. Numerical results show the achieved energy efficiency gains by using the additional degrees of freedom of t…
We consider a directed acyclic network where there are two source-terminal pairs and the terminals need to receive the symbols generated at the respective sources. Each source independently generates an i.i.d. random process over the same alphabet. Each edge in the network is error-free, delay-free, and can carry one symbol from the alphabet per use. We give a simple necessary and sufficient condition for being able to simultaneously satisfy the unicast requirements of the two source-terminal pairs at rate-pair $(1,1)$ using vector network coding. The condition is also sufficient for doing thi…
This paper considers the problem of network coding for multiple unicast connections in networks represented by directed acyclic graphs. The concept of interference alignment, traditionally used in interference networks, is extended to analyze the performance of linear network coding in this setup and to provide a systematic code design approach. It is shown that, for a broad class of three-source three-destination unicast networks, a rate corresponding to half the individual source-destination min-cut is achievable via alignment strategies.
We study the problem of communicating over a single-source single-terminal network in the presence of an adversary that may jam a single link of the network. If any one of the edges can be jammed, the capacity of such networks is well understood and follows directly from the connection between the minimum cut and maximum flow in single-source single- terminal networks. In this work we consider networks in which some edges cannot be jammed, and show that determining the network communication capacity is at least as hard as solving the multiple-unicast network coding problem for the error-free c…
We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, $s_1-t_1$ and $s_2-t_2$. Current characterizations of the multiple unicast capacity region in this setting have a large number of inequalities, which makes them hard to explicitly evaluate. In this work we consider a slightly different problem. We assume that we only know certain minimum cut values for the network, e.g., mincut$(S_i, T_j)$, where $S_i \subseteq \{s_1, s_2\}$ and $T_j \subseteq \{t_1, t_2\}$ for different subsets $S_i$ and $T_j$. Based on these …
A new precoding-based intersession network coding (NC) scheme has recently been proposed, which applies the interference alignment technique, originally devised for wireless interference channels, to the 3-unicast problem of directed acyclic networks. The main result of this work is a graph-theoretic characterization of the feasibility of the 3-unicast interference alignment scheme. To that end, we first investigate several key relationships between the point-to-point network channel gains and the underlying graph structure. Such relationships turn out to be critical when characterizing graph-…
We consider a two-unicast-$Z$ network over a directed acyclic graph of unit capacitated edges; the two-unicast-$Z$ network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-$Z$ networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to derivin…
We establish a duality result between linear index coding and Locally Repairable Codes (LRCs). Specifically, we show that a natural extension of LRCs we call Generalized Locally Repairable Codes (GLCRs) are exactly dual to linear index codes. In a GLRC, every node is decodable from a specific set of other nodes and these sets induce a recoverability directed graph. We show that the dual linear subspace of a GLRC is a solution to an index coding instance where the side information graph is this GLRC recoverability graph. We show that the GLRC rate is equivalent to the complementary index coding…
We study joint unicast and multigroup multicast transmission in single-cell massive multiple-input-multiple-output (MIMO) systems, under maximum ratio transmission. For the unicast transmission, the objective is to maximize the weighted sum spectral efficiency (SE) of the unicast user terminals (UTs) and for the multicast transmission the objective is to maximize the minimum SE of the multicast UTs. These two problems are coupled to each other in a conflicting manner, due to their shared power resource and interference. To address this, we formulate a multiobjective optimization problem (MOOP)…
We consider the problem of network coding across three unicast sessions over a directed acyclic graph, when each session has min-cut one. Previous work by Das et al. adapted a precoding-based interference alignment technique, originally developed for the wireless interference channel, specifically to this problem. We refer to this approach as precoding-based network alignment (PBNA). Similar to the wireless setting, PBNA asymptotically achieves half the minimum cut; different from the wireless setting, its feasibility depends on the graph structure. Das et al. provided a set of feasibility con…
A sum-network is a directed acyclic network in which all terminal nodes demand the `sum' of the independent information observed at the source nodes. Many characteristics of the well-studied multiple-unicast network communication problem also hold for sum-networks due to a known reduction between instances of these two problems. Our main result is that unlike a multiple unicast network, the coding capacity of a sum-network is dependent on the message alphabet. We demonstrate this using a construction procedure and show that the choice of a message alphabet can reduce the coding capacity of a s…
We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where each sender and the receiver is connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the network only performs random linear network coding, and restrict our approaches to precoding-based linear schemes, where the senders use precoding matrices to encode source symbols. We adapt a precoding-based interference alignment technique, originally developed for the wireless interference channel, to construct a precoding-based linear sc…
The study of feedback has been mostly limited to single-hop communication settings. In this paper, we consider Gaussian networks where sources and destinations can communicate with the help of intermediate relays over multiple hops. We assume that links in the network can be bidirected providing opportunities for feedback. We ask the following question: can the information transfer in both directions of a link be critical to maximizing the end-to-end communication rates in the network? Equivalently, could one of the directions in each bidirected link (and more generally at least one of the lin…
A sum-network is an instance of a network coding problem over a directed acyclic network in which each terminal node wants to compute the sum over a finite field of the information observed at all the source nodes. Many characteristics of the well-studied multiple unicast network communication problem also hold for sum-networks due to a known reduction between instances of these two problems. In this work, we describe an algorithm to construct families of sum-network instances using incidence structures. The computation capacity of several of these sum-network families is characterized. We dem…
The algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes, at any given time instant, is a \mathbb{F}_{q}$-linear combination of the input symbols across different generations, where $\mathbb{F}_{q}$ denotes the field over which the network operates. We use finite-field discrete fourier transform (DFT) to convert the output symbols at the sink nodes, at any give…
We study the rate region of variable-length source-network codes that are used to compute a function of messages observed over a network. The particular network considered here is the simplest instance of a directed acyclic graph (DAG) that is not a tree. Existing work on zero-error function computation in DAG networks provides bounds on the \textit{computation capacity}, which is a measure of the amount of communication required per edge in the worst case. This work focuses on the average case: an achievable rate tuple describes the expected amount of communication required on each edge, wher…
We consider the network communication scenario, over directed acyclic networks with unit capacity edges in which a number of sources $s_i$ each holding independent unit-entropy information $X_i$ wish to communicate the sum $\sum{X_i}$ to a set of terminals $t_j$. We show that in the case in which there are only two sources or only two terminals, communication is possible if and only if each source terminal pair $s_i/t_j$ is connected by at least a single path. For the more general communication problem in which there are three sources and three terminals, we prove that a single path connecting…
The algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes, at any given time instant, is a $\mathbb{F}_{p^m}$-linear combination of the input symbols across different generations where, $\mathbb{F}_{p^m}$ denotes the field over which the network operates ($p$ is prime and $m$ is a positive integer). We use finite-field discrete fourier transform (DFT) to convert…
Partial clique covering is one of the most basic coding schemes for index coding problems, generalizing clique and cycle covering on the side information digraph and further reducing the achievable broadcast rate. In this paper, we start with partition multicast, a special case of partial clique covering with cover number 1, and show that partition multicast achieves the optimal broadcast rate of the multiple-unicast index coding if and only if the side information digraph is partially acyclic. A digraph is said to be partially acyclic if its sub-digraph induced by the vertex with maximum in-d…
When there exists a malicious attacker in the network, we need to consider the possibilities of eavesdropping and the contamination simultaneously. Under an acyclic broadcast network, the optimality of linear codes was shown when Eve is allowed to attack any $r$ edges. The optimality of linear codes is not shown under a different assumption for Eve. As a typical example of an acyclic unicast network, we focus on the one-hop relay network under the single transmission scheme by assuming that Eve attacks only one edge in each level. Surprisingly, as a result, we find that a non-linear code signi…
We consider a network (that is capable of network coding) with a set of sources and terminals, where each terminal is interested in recovering the sum of the sources. Considering directed acyclic graphs with unit capacity edges and independent, unit-entropy sources, we show the rate region when (a) there are two sources and $n$ terminals, and (b) $n$ sources and two terminals. In these cases as long as there exists at least one path from each source to each terminal we demonstrate that there exists a valid assignment of coding vectors to the edges such that the terminals can recover the sum of…
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 {\it sum-networks}, and present some bounds in terms of min-cut, and the numbers of sources and terminals.
We consider the problem of communicating the sum of $m$ sources to $n$ terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either $m=2$ or $n=2$, the sum of the sources can be communicated to the terminals if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can…
We consider a directed acyclic network with three sources and three terminals such that each source independently generates one symbol from a given field $F$ and each terminal wants to receive the sum (over $F$) of the source symbols. Each link in the network is considered to be error-free and delay-free and can carry one symbol from the field in each use. We call such a network a 3-source 3-terminal {\it $(3s/3t)$ sum-network}. In this paper, we give a necessary and sufficient condition for a $3s/3t$ sum-network to allow all the terminals to receive the sum of the source symbols over \textit{…
A sum-network is a directed acyclic network where each source independently generates one symbol from a given field $\mathbb F$ and each terminal wants to receive the sum $($over $\mathbb F)$ of the source symbols. For sum-networks with two sources or two terminals, the solvability is characterized by the connection condition of each source-terminal pair [3]. A necessary and sufficient condition for the solvability of the $3$-source $3$-terminal $(3$s$/3$t$)$ sum-networks was given by Shenvi and Dey [6]. However, the general case of arbitrary sources/sinks is still open. In this paper, we inve…
We consider directed acyclic {\em sum-networks} with $m$ sources and $n$ terminals where the sources generate symbols from an arbitrary alphabet field $F$, and the terminals need to recover the sum of the sources over $F$. We show that for any co-finite set of primes, there is a sum-network which is solvable only over fields of characteristics belonging to that set. We further construct a sum-network where a scalar solution exists over all fields other than the binary field $F_2$. We also show that a sum-network is solvable over a field if and only if its reverse network is solvable over the s…
We consider a directed acyclic network with multiple sources and multiple terminals where each terminal is interested in decoding the sum of independent sources generated at the source nodes. We describe a procedure whereby a simple undirected graph can be used to construct such a sum-network and demonstrate an upper bound on its computation rate. Furthermore, we show sufficient conditions for the construction of a linear network code that achieves this upper bound. Our procedure allows us to construct sum-networks that have any arbitrary computation rate $\frac{p}{q}$ (where $p,q$ are non-neg…
For zero-error function computation over directed acyclic networks, existing upper and lower bounds on the computation capacity are known to be loose. In this work we consider the problem of computing the arithmetic sum over a specific directed acyclic network that is not a tree. We assume the sources to be i.i.d. Bernoulli with parameter $1/2$. Even in this simple setting, we demonstrate that upper bounding the computation rate is quite nontrivial. In particular, it requires us to consider variable length network codes and relate the upper bound to equivalently lower bounding the entropy of d…
In Index Coding, the goal is to use a broadcast channel as efficiently as possible to communicate information from a source to multiple receivers which can possess some of the information symbols at the source as side-information. In this work, we present a duality relationship between index coding (IC) and multiple-unicast network coding (NC). It is known that the IC problem can be represented using a side-information graph $G$ (with number of vertices $n$ equal to the number of source symbols). The size of the maximum acyclic induced subgraph, denoted by $MAIS$ is a lower bound on the \texti…
A model, called the linear transform network (LTN), is proposed to analyze the compression and estimation of correlated signals transmitted over directed acyclic graphs (DAGs). An LTN is a DAG network with multiple source and receiver nodes. Source nodes transmit subspace projections of random correlated signals by applying reduced-dimension linear transforms. The subspace projections are linearly processed by multiple relays and routed to intended receivers. Each receiver applies a linear estimator to approximate a subset of the sources with minimum mean squared error (MSE) distortion. The mo…
Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a group of $c_i$ edge-disjoint paths from $S_i$ to $R_i$, which will be referred to as a group of Menger's paths from $S_i$ to $R_i$ in this paper. Although within the same group they are edge-disjoint, the Menger's paths from different groups may have to merge with each other. It is known that by choosing Menger's paths appropriately, the number of mergings among diff…
We take a unified view of network coding and decentralized control. Precisely speaking, we consider both as linear time-invariant systems by appropriately restricting channels and coding schemes of network coding to be linear time-invariant, and the plant and controllers of decentralized control to be linear time-invariant as well. First, we apply linear system theory to network coding. This gives a novel way of converting an arbitrary relay network to an equivalent acyclic single-hop relay network, which we call Network Linearization. Based on network linearization, we prove that the fundamen…
This work studies the problem of distributed compression of correlated sources with an action-dependent joint distribution. This class of problems is, in fact, an extension of the Slepian-Wolf model, but where cost-constrained actions taken by the encoder or the decoder affect the generation of one of the sources. The purpose of this work is to study the implications of actions on the achievable rates. In particular, two cases where transmission occurs over a rate-limited link are studied; case A for actions taken at the decoder and case B where actions are taken at the encoder. A complete sin…
