Unicast Index Research Articles

On the Optimal Broadcast Rate of the Two-Sender Unicast Index Coding Problem With Fully-Participated Interactions

The problem of two-sender unicast index coding consists of two senders and a set of receivers. Each receiver demands a unique message not demanded by any other receiver and has a subset of messages as its side information. Every demanded message is available with at least one of the senders. The senders avail the knowledge of the side information at all the receivers to reduce the total number of transmissions required to satisfy the demands of all the receivers. The objective is to find the minimum total of number of transmissions per message length (known as the optimal broadcast rate with finite length messages) and its limiting value as the message length tends to infinity (called the optimal broadcast rate). Achievable broadcast rates are provided for a class of the two-sender unicast index coding problem based on a special graph coloring technique called two-sender graph coloring. This result illustrates the utility of graph products in the two-sender unicast index coding problem for the first time in the literature. For another class, achievable broadcast rates are provided based on the optimal broadcast rates of three single-sender sub-problems with finite message length. This employs a code construction for the two-sender unicast index coding problem using optimal codes (including non-linear codes) of the sub-problems. Optimal broadcast rates are provided for a special class of the TUICP for which only an upper bound was known prior to this work. The optimal broadcast rates presented in this work also consider non-linear coding schemes at the two senders.

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.

Optimal Linear Broadcast Rates of Some Two-Sender Unicast Index Coding Problems

The two-sender unicast index coding problem consists of two senders, each having a different set of messages. Some messages may be common to both the senders. Each receiver demands a unique message and has a subset of messages known as its side-information. The senders transmit coded messages by availing the knowledge of the side-information of all the receivers, such that all the receivers are able to decode their demands. The aim is to find the optimal aggregate number of coded transmissions per message length (also called the optimal broadcast rate with finite length messages), and its limiting value as the message length tends to infinity (also called the optimal broadcast rate). In this paper, only linear coding schemes are considered. Optimal linear broadcast rate for any finite message length and optimal linear broadcast rate for a basic class of the two-sender unicast index coding problem are established. Optimal code-constructions are also provided. These results are given in terms of the corresponding results of three independent single-sender sub-problems of the two-sender unicast index coding problem. Proof techniques used to obtain the results for the two-sender problem are shown to be useful in obtaining the results for some classes of the multi-sender unicast index coding problem.

Optimal Error Correcting Index Codes for Some Generalized Index Coding Problems

The index coding with side-information problem introduced by Birk and Kol has been generalized to the case, where the transmissions are subjected to errors by Dau et al.. Lower and upper bounds on the optimal number of transmissions required to correct a specified number δ of errors were established. In another generalization of index coding problem, namely, generalized index coding (GIC) problem introduced by Dai et al., linear combination of the messages can be demanded and held as side information by the receivers. Error correction for GIC problems was studied and the bounds for optimal number of transmissions required for δ-error correcting generalized index codes were established by Byrne and Calderini. In this paper, for a particular class of GIC problems construction of optimal scalar linear index codes are discussed. For this class of GIC problems, the optimal linear δ-error correcting index codes are also constructed. As special cases, optimal linear error correcting codes are obtained for two classes of original index coding problems, namely, single-prior index coding problems and single unicast index coding problems with symmetric neighboring consecutive side information.

Capacity of Symmetric Index Coding Problems With X-Network Setting With <inline-formula> <tex-math notation="LaTeX">$2\times2$ </tex-math> </inline-formula> Local Connectivity

In $X$ -network setting with ${L \times L}$ local connectivity, we have a locally connected network where each receiver is connected to $L$ consecutive base stations and each base station has a distinct message for each connected receiver. Maleki et al. modeled the ${X}$ -network setting with ${L \times L}$ local connectivity as a multiple unicast index coding problem and proved that the capacity of symmetric index coding problems with locally connected ${X}$ -network with ${K}$ number of receivers and number of messages ${M=KL}$ and ${K}$ tending to infinity is ${({{2}}/[{{L(L+1)}}]})$ and for finite number of receivers this was shown to be an upper bound on the capacity. In this letter: 1) we show that when ${L=2}$ the upper bound is exact, i.e., the capacity is (1/3) and 2) for this case we give an explicit construction of optimal linear index codes to achieve this capacity by using interference alignment.

Characterizing the Number of Optimal Index Codes and Their Relation to Min–Max Probability of Error Performance

An index coding problem arises when there is a single source with a number of messages and multiple receivers each wanting a subset of messages knowing a different set of messages a priori . The noiseless index coding problem is to identify the minimum number of transmissions (optimal length) to be made by the source through a noiseless channel so that all receivers can decode their wanted messages using the transmitted symbols and their respective prior information. Recently (A. Thomas, R. Kavitha, A. Chandramouli, and B. Sundar Rajan, “Optimal index coding with min-max probability of error over fading channels,” in Proc. IEEE 26th Annu. Int. Symp. Pers. Indoor Mobile Radio Commun. , Hong Kong, 2015, pp. 889–894), it is shown that different optimal length codes perform differently in a noisy channel for single-uniprior index coding problems. Towards identifying the best optimal length index code, one needs to know the number of optimal length index codes. In this paper, we consider unicast index coding problems and present results on the number of optimal length index codes making use of the representation of an index coding problem by an equivalent network code. Our formulation results in matrices of smaller sizes compared to the approach of Koetter and Medard (R. Koetter and M. Medard, “An algebraic approach to network coding,” IEEE/ACM Trans. Netw. , vol. 11, no. 5, pp. 782–795, Oct. 2003.). Our formulation leads to a lower bound on the minimum number of optimal length codes possible for all unicast index coding problems (L. Ong and C. K. Ho, “Optimal index codes for a class of multicast networks with receiver side information,” in Proc. IEEE Int. Conf. Commun. , 2012, pp. 2213–2218), which is met with equality for several special cases of the unicast index coding problem. Furthermore, for noisy broadcast channels, a method to identify the best among the optimal length codes for unicast index coding problems in terms of minimum–maximum error probability is presented.

Optimal Finite-Length and Asymptotic Index Codes for Five or Fewer Receivers

Index coding models broadcast networks in which a sender sends different messages to different receivers simultaneously, where each receiver may know some of the messages a priori. The aim is to find the minimum (normalised) index codelength that the sender sends. This paper considers unicast index coding, where each receiver requests exactly one message, and each message is requested by exactly one receiver. Each unicast index-coding instances can be fully described by a directed graph and vice versa, where each vertex corresponds to one receiver. For any directed graph representing a unicast index-coding instance, we show that if a maximum acyclic induced subgraph (MAIS) is obtained by removing two or fewer vertices from the graph, then the minimum index codelength equals the number of vertices in the MAIS, and linear codes are optimal for the corresponding index-coding instance. Using this result, we solved all unicast index-coding instances with up to five receivers, which correspond to all graphs with up to five vertices. For 9819 non-isomorphic graphs among all graphs up to five vertices, we obtained the minimum index codelength for all message alphabet sizes; for the remaining 28 graphs, we obtained the minimum index codelength if the message alphabet size is $k^2$ for any positive integer $k$. This work complements the result by Arbabjolfaei et al. (ISIT 2013), who solved all unicast index-coding instances with up to five receivers in the asymptotic regime, where the message alphabet size tends to infinity.

Index Coding Capacity: How Far Can One Go With Only Shannon Inequalities?

An interference alignment perspective is used to identify the simplest instances (minimum possible number of edges in the alignment graph, no more than 2 interfering messages at any destination) of index coding problems where non-Shannon information inequalities are necessary for capacity characterization. In particular, this includes the first known example of a multiple unicast (one destination per message) index coding problem where non-Shannon information inequalities are shown to be necessary. The simplest multiple unicast example has 7 edges in the alignment graph and 11 messages. The simplest multiple groupcast (multiple destinations per message) example has 6 edges in the alignment graph, 6 messages, and 10 receivers. For both the simplest multiple unicast and multiple groupcast instances, the best outer bound based on only Shannon inequalities is $\frac{2}{5}$, which is tightened to $\frac{11}{28}$ by the use of the Zhang-Yeung non-Shannon type information inequality, and the linear capacity is shown to be $\frac{5}{13}$ using the Ingleton inequality. Conversely, identifying the minimal challenging aspects of the index coding problem allows an expansion of the class of solved index coding problems up to (but not including) these instances.

Index Coding—An Interference Alignment Perspective

Author(s): Maleki, H; Cadambe, VR; Jafar, SA | Abstract: The index coding problem is studied from an interference alignment perspective providing new results as well as new insights into, and generalizations of, previously known results. An equivalence is established between the capacity of multiple unicast index coding (where each message is desired by exactly one receiver), and groupcast index coding (where a message can be desired by multiple receivers), which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings, where this question has previously been answered. Necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints, and generalizations to feasibility of rate 1/(L+1)per message when each destination desires at least L messages, are similarly obtained. Finally, capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks. The solutions are based on vector linear coding. © 1963-2012 IEEE.