In the index coding problem a sender holds a message @math and wishes to broadcast information to @math receivers in a way that enables the @math th receiver to retrieve the @math th bit @math . Every receiver has prior side information comprising a subset of the bits of @math , and the goal is to minimize the length of the information sent via the broadcast channel. Porter and Wootters have recently introduced the model of embedded index coding, where the receivers also play the role of the sender and the goal is to minimize the total length of their broadcast information. An embedded index code is said to be task-based if every receiver retrieves its bit based only on the information provided by one of the receivers. This short paper studies the effect of the task-based restriction on linear embedded index coding. It is shown that for certain side information maps there exists a linear embedded index code of length quadratically smaller than that of any task-based embedded index code. The result attains, up to a multiplicative constant, the largest possible gap between the two quantities. The proof is by an explicit construction and the analysis involves spectral techniques.
The index coding problem, introduced in @cite_0 and further developed in @cite_7 , has been studied in various variations and extensions. This research is motivated by applications such as distributed storage @cite_2 , wireless communication @cite_16 , and the more general problem of network coding @cite_8 . The variant called embedded index coding, introduced in @cite_17 , can be viewed as a special case of the multi-sender index coding model studied in @cite_4 which allows multiple senders and multiple receivers but as disjoint sets of vertices (see also @cite_14 ). The framework of index coding studied in @cite_17 is more general than the one considered in the current work and allows the receivers to request multiple messages.
@cite_14 @cite_4 @cite_7 @cite_8 @cite_0 @cite_2 @cite_16 @cite_17
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We consider a Multi-Sender Unicast Index-Coding (MSUIC) problem, where in a broadcast network, multiple senders collaboratively send distinct messages to multiple receivers, each having some subset of the messages a priori . The aim is to find the shortest index code that minimizes the total number of coded bits sent by the senders. In this paper, built on the classic single-sender minrank concept, we develop a new rank-minimization framework for MSUIC that explicitly takes into account the sender message constraints and minimizes the sum of the ranks of encoding matrices subject to the receiver decoding requirements. This framework provides a systematic way to construct multi-sender linear index codes and to study their capability in achieving the shortest index codelength per message length (i.e., the optimal broadcast rate). In particular, we establish the optimal broadcast rate for all critical MSUIC instances with up to four receivers and show that a binary linear index code is optimal for all, except 15 instances with four receivers. We also propose a heuristic algorithm (in lieu of exhaustive search) to solve the rank-minimization problem. The effectiveness of the algorithm is validated by numerical studies of MSUIC instances with four or more receivers. Index coding studies multiterminal source-coding problems where a set of receivers are required to decode multiple (possibly different) messages from a common broadcast, and they each know some messages a priori . In this paper, at the receiver end, we consider a special setting where each receiver knows only one message a priori , and each message is known to only one receiver. At the broadcasting end, we consider a generalized setting where there could be multiple senders, and each sender knows a subset of the messages. The senders collaborate to transmit an index code. This paper looks at minimizing the number of total coded bits the senders are required to transmit. When there is only one sender, we propose a pruning algorithm to find a lower bound on the optimal (i.e., the shortest) index codelength, and show that it is achievable by linear index codes. When there are two or more senders, we propose an appending technique to be used in conjunction with the pruning technique to give a lower bound on the optimal index codelength; we also derive an upper bound based on cyclic codes. While the two bounds do not match in general, for the special case where no two distinct senders know any message in common, the bounds match, giving the optimal index codelength. The results are expressed in terms of strongly connected components in directed graphs that represent the index-coding problems. Motivated by a problem of transmitting data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R_1, . . . , R_n. He holds an input x 0, 1 ^n and wishes to broadcast a single message so that each receiver R_i can recover the bit x_i. Each R_i has prior side information about x, induced by a directed graph G on n nodes; R_i knows the bits of x in the positions j | (i, j) is an edge of G . We call encoding schemes that achieve this goal INDEX codes for 0, 1 ^n with side information graph G. In this paper we identify a measure on graphs, the minrank, which we conjecture to exactly characterize the minimum length of INDEX codes. We resolve the conjecture for certain natural classes of graphs. For arbitrary graphs, we show that the minrank bound is tight for both linear codes and certain classes of non-linear codes. For the general problem, we obtain a (weaker) lower bound that the length of an INDEX code for any graph G is at least the size of the maximum acyclic induced subgraph of G. We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems. The Informed-Source Coding On Demand (ISCOD) approach for efficiently supplying nonidentical data from a central server to multiple caching clients over a broadcast channel is presented. The key idea underlying ISCOD is the joint exploitation of the data blocks already cached by each client, the server's full knowledge of client-cache contents and client requests, and the fact that each client only needs to be able to derive the blocks requested by it rather than all the blocks ever transmitted or even the union of the blocks requested by the different clients. We present two-phase ISCOD algorithms: the server first creates ad-hoc error-correction sets based on its knowledge of client states; next, it uses erasure-correction codes to construct the data for transmission. Each client uses its cached data and the received supplemental data to derive its requested blocks. The result is up to a several-fold reduction in the amount of transmitted supplemental data. Also, we define k-partial cliques in a directed graph and cast ISCOD in terms of partial-clique covers. This paper studies linear interference networks, both wired and wireless, with no channel state information at the transmitters except a coarse knowledge of the end-to-end one-hop topology of the network that only allows a distinction between weak (zero) and significant (nonzero) channels and no further knowledge of the channel coefficients' realizations. The network capacity (wired) and degrees of freedom (DoF) (wireless) are found to be bounded above by the capacity of an index coding problem for which the antidote graph is the complement of the given interference graph. The problems are shown to be equivalent under linear solutions. An interference alignment perspective is then used to translate the existing index coding solutions into the wired network capacity and wireless network DoF solutions, as well as to find new and unified solutions to different classes of all three problems. 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.