Local Partial Clique Covers for Index Coding

Index coding, or broadcasting with side information, is a network coding problem of most fundamental importance. In this problem, given a directed graph, each vertex represents a user with a need of information, and the neighborhood of each vertex represents the side information availability to that user. The aim is to find an encoding to minimum number of bits (optimal rate) that, when broadcasted, will be sufficient to the need of every user. Not only the optimal rate is intractable, but it is also very hard to characterize with some other well-studied graph parameter or with a simpler formulation, such as a linear program. Recently there have been a series of works that address this question and provide explicit schemes for index coding as the optimal value of a linear program with rate given by well-studied properties such as local chromatic number or partial clique-covering number. There has been a recent attempt to combine these existing notions of local chromatic number and partial clique covering into a unified notion denoted as the local partial clique cover (Arbabjolfaei and Kim, 2014). We present a generalized novel upper-bound (encoding scheme) - in the form of the minimum value of a linear program - for optimal index coding. Our bound also combines the notions of local chromatic number and partial clique covering into a new definition of the local partial clique cover, which outperforms both the previous bounds, as well as beats the previous attempt to combination. Further, we look at the upper bound derived recently by Thapa et al., 2015, and extend their $n$-$\mathsf{GIC}$ (Generalized Interlinked Cycle) construction to $(k,n)$-$\mathsf{GIC}$ graphs, which are a generalization of $k$-partial cliques.Comment: 16 page

Structural Properties of Index Coding Capacity Using Fractional Graph Theory

The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on Information Theory (ISIT

Structural Characteristics of Two-Sender Index Coding

This paper studies index coding with two senders. In this setup, source messages are distributed among the senders possibly with common messages. In addition, there are multiple receivers, with each receiver having some messages a priori, known as side-information, and requesting one unique message such that each message is requested by only one receiver. Index coding in this setup is called two-sender unicast index coding (TSUIC). The main goal is to find the shortest aggregate normalized codelength, which is expressed as the optimal broadcast rate. In this work, firstly, for a given TSUIC problem, we form three independent sub-problems each consisting of the only subset of the messages, based on whether the messages are available only in one of the senders or in both senders. Then we express the optimal broadcast rate of the TSUIC problem as a function of the optimal broadcast rates of those independent sub-problems. In this way, we discover the structural characteristics of TSUIC. For the proofs of our results, we utilize confusion graphs and coding techniques used in single-sender index coding. To adapt the confusion graph technique in TSUIC, we introduce a new graph-coloring approach that is different from the normal graph coloring, which we call two-sender graph coloring, and propose a way of grouping the vertices to analyze the number of colors used. We further determine a class of TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates. Finally, we generalize the results of a class of TSUIC problems to multiple senders.Comment: Submitted for journal publicatio

An Efficient Coded Multicasting Scheme Preserving the Multiplicative Caching Gain

Coded multicasting has been shown to be a promis- ing approach to significantly improve the caching performance of content delivery networks with multiple caches downstream of a common multicast link. However, achievable schemes proposed to date have been shown to achieve the proved order-optimal performance only in the asymptotic regime in which the number of packets per requested item goes to infinity. In this paper, we first extend the asymptotic analysis of the achievable scheme in [1], [2] to the case of heterogeneous cache sizes and demand distributions, providing the best known upper bound on the fundamental limiting performance when the number of packets goes to infinity. We then show that the scheme achieving this upper bound quickly loses its multiplicative caching gain for finite content packetization. To overcome this limitation, we design a novel polynomial-time algorithm based on random greedy graph- coloring that, while keeping the same finite content packetization, recovers a significant part of the multiplicative caching gain. Our results show that the order-optimal coded multicasting schemes proposed to date, while useful in quantifying the fundamental limiting performance, must be properly designed for practical regimes of finite packetization.Comment: 6 pages, 7 figures, Published in Infocom CNTCV 201

Cache-Aided Coded Multicast for Correlated Sources

The combination of edge caching and coded multicasting is a promising approach to improve the efficiency of content delivery over cache-aided networks. The global caching gain resulting from content overlap distributed across the network in current solutions is limited due to the increasingly personalized nature of the content consumed by users. In this paper, the cache-aided coded multicast problem is generalized to account for the correlation among the network content by formulating a source compression problem with distributed side information. A correlation-aware achievable scheme is proposed and an upper bound on its performance is derived. It is shown that considerable load reductions can be achieved, compared to state of the art correlation-unaware schemes, when caching and delivery phases specifically account for the correlation among the content files.Comment: In proceeding of IEEE International Symposium on Turbo Codes and Iterative Information Processing (ISTC), 201

Wireless Bidirectional Relaying, Latin Squares and Graph Vertex Coloring

The problem of obtaining network coding maps for the physical layer network coded two-way relay channel is considered, using the denoise-and-forward forward protocol. It is known that network coding maps used at the relay node which ensure unique decodability at the end nodes form a Latin Square. Also, it is known that minimum distance of the effective constellation at the relay node becomes zero, when the ratio of the fade coefficients from the end node to the relay node, belongs to a finite set of complex numbers determined by the signal set used, called the singular fade states. Furthermore, it has been shown recently that the problem of obtaining network coding maps which remove the harmful effects of singular fade states, reduces to the one of obtaining Latin Squares, which satisfy certain constraints called \textit{singularity removal constraints}. In this paper, it is shown that the singularity removal constraints along with the row and column exclusion conditions of a Latin Square, can be compactly represented by a graph called the \textit{singularity removal graph} determined by the singular fade state and the signal set used. It is shown that a Latin Square which removes a singular fade state can be obtained from a proper vertex coloring of the corresponding singularity removal graph. The minimum number of symbols used to fill in a Latin Square which removes a singular fade state is equal to the chromatic number of the singularity removal graph. It is shown that for any square $M$-QAM signal set, there exists singularity removal graphs whose chromatic numbers exceed $M$ and hence require more than $M$ colors for vertex coloring. Also, it is shown that for any $2^{\lambda}$-PSK signal set, $\lambda \geq 3,$ all the singularity removal graphs can be colored using $2^{\lambda}$ colors.Comment: 18 pages, 19 figure

A New Upperbound on the Broadcast Rate of Index Coding Problems with Symmetric Neighboring Interference

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has equal number of $K$ messages and $K$ receivers, the $k$th receiver $R_{k}$ wanting the $k$th message $x_{k}$ and having the side-information $\mathcal{K}_{k}=(\mathcal{I}_{k} \cup x_{k})^c,$ where ${I}_k= \{x_{k-U},\dots,x_{k-2},x_{k-1}\}\cup\{x_{k+1}, x_{k+2},\dots,x_{k+D}\}$ is the interference with $D$ messages after and $U$ messages before its desired message. The single unicast index coding problem with symmetric neighboring interference (SUICP-SNI) is motivated by topological interference management problems in wireless communication networks. Maleki, Cadambe and Jafar obtained the capacity of this SUICP-SNI with $K$ tending to infinity and Blasiak, Kleinberg and Lubetzky for the special case of $(D=U=1)$ with $K$ being finite. Finding the capacity of the SUICP-SNI for arbitrary $K,D$ and $U$ is a challenging open problem. In our previous work, for an SUICP-SNI with arbitrary $K,D$ and $U$, we defined a set $\mathcal{\mathbf{S}}$ of $2$-tuples such that for every $(a,b)$ in that set $\mathcal{\mathbf{S}}$, the rate $D+1+\frac{a}{b}$ is achieved by using vector linear index codes over every finite field. In this paper, we give an algorithm to find the values of $a$ and $b$ such that $(a,b) \in \mathcal{\mathbf{S}}$ and $\frac{a}{b}$ is minimum. We present a new upperbound on the broadcast rate of SUICP-SNI and prove that this upper bound coincides with the existing results on the exact value of the capacity of SUICP-SNI in the respective settings.Comment: Closely related to our earlier submission arXiv:1705.10614v1 [cs] 28 May 2017. One figure and one tabl

Rate-Distortion-Memory Trade-offs in Heterogeneous Caching Networks

Caching at the wireless edge can be used to keep up with the increasing demand for high-definition wireless video streaming. By prefetching popular content into memory at wireless access points or end-user devices, requests can be served locally, relieving strain on expensive backhaul. In addition, using network coding allows the simultaneous serving of distinct cache misses via common coded multicast transmissions, resulting in significantly larger load reductions compared to those achieved with traditional delivery schemes. Most prior works simply treat video content as fixed-size files that users would like to fully download. This work is motivated by the fact that video can be coded in a scalable fashion and that the decoded video quality depends on the number of layers a user receives in sequence. Using a Gaussian source model, caching and coded delivery methods are designed to minimize the squared error distortion at end-user devices in a rate-limited caching network. The framework is very general and accounts for heterogeneous cache sizes, video popularities and user-file play-back qualities. As part of the solution, a new decentralized scheme for lossy cache-aided delivery subject to preset user distortion targets is proposed, which further generalizes prior literature to a setting with file heterogeneity.Comment: Submitted to Transactions on Wireless Communication

Embedded Index Coding

Motivated by applications in distributed storage and distributed computation, we introduce embedded index coding (EIC). EIC is a type of distributed index coding in which nodes in a distributed system act as both senders and receivers of information. We show how embedded index coding is related to index coding in general, and give characterizations and bounds on the communication costs of optimal embedded index codes. We also define task-based EIC, in which each sending node encodes and sends data blocks independently of the other nodes. Task-based EIC is more computationally tractable and has advantages in applications such as distributed storage, in which senders may complete their broadcasts at different times. Finally, we give heuristic algorithms for approximating optimal embedded index codes, and demonstrate empirically that these algorithms perform well

On weight distributions of perfect colorings and completely regular codes

A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions. Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very simple formula for the weight enumerator of an arbitrary perfect coloring. Codewords: completely regular code; equitable partition; partition design; perfect coloring; perfect structure; regular partition; weight distribution; weight enumerator.Comment: 17pp; partially presented at "Optimal Codes and Related Topics" OC2009, Varna (Bulgaria). V.2: the title was changed (old: "On weight distributions of perfect structures"), Sect.5 "Weight enumerators ..." was adde