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Local Time Sharing for Index Coding
"... Abstract—A series of extensions of the index coding schemes based on time sharing by Birk and Kol, by Blasiak, Kleinberg, and Lubetzky, and by Shanmugam, Dimakis, and Langberg are presented. Each extension strictly improves upon the previous extensions as well as the existing schemes. The main idea ..."
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Abstract—A series of extensions of the index coding schemes based on time sharing by Birk and Kol, by Blasiak, Kleinberg, and Lubetzky, and by Shanmugam, Dimakis, and Langberg are presented. Each extension strictly improves upon the previous extensions as well as the existing schemes. The main idea behind these extensions is local time sharing over subproblems introduced by Shanmugam et al., in which the local side information available at each receiver is exploited to send the subproblm indices with a fewer number of transmissions. The final extension, despite being the best in this class of coding schemes, is shown to be still suboptimal, characterizing the fundamental limit of local time sharing. I.
Fundamental Limits of Caching in Wireless D2D Networks
, 2014
"... We consider a wireless DevicetoDevice (D2D) network where communication is restricted to be singlehop. Users make arbitrary requests from a finite library of files and have precached information on their devices, subject to a pernode storage capacity constraint. A similar problem has already be ..."
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We consider a wireless DevicetoDevice (D2D) network where communication is restricted to be singlehop. Users make arbitrary requests from a finite library of files and have precached information on their devices, subject to a pernode storage capacity constraint. A similar problem has already been considered in an “infrastructure ” setting, where all users receive a common multicast (coded) message from a single omniscient server (e.g., a base station having all the files in the library) through a shared bottleneck link. In this work, we consider a D2D “infrastructureless ” version of the problem. We propose a caching strategy based on deterministic assignment of subpackets of the library files, and a coded delivery strategy where the users send linearly coded messages to each other in order to collectively satisfy their demands. We also consider a random caching strategy, which is more suitable to a fully decentralized implementation. Under certain conditions, both approaches can achieve the information theoretic outer bound within a constant multiplicative factor. In our previous work, we showed that a caching D2D wireless network with onehop communication, random caching, and uncoded delivery (direct file transmissions), achieves the same throughput scaling law of the infrastructurebased coded multicasting scheme, in the regime of large number of users and
Structural properties of index coding capacity using fractional graph theory
 IEEE INT. SYMP. INF. THEORY
, 2015
"... 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, ..."
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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 graphtheoretic 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, oneway, or complete.
Linear codes are optimal for indexcoding instances with five or fewer receivers
, 2014
"... Abstract—We study zeroerror unicast indexcoding instances, where each receiver must perfectly decode its requested message set, and the message sets requested by any two receivers do not overlap. We show that for all these instances with up to five receivers, linear index codes are optimal. Althou ..."
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Abstract—We study zeroerror unicast indexcoding instances, where each receiver must perfectly decode its requested message set, and the message sets requested by any two receivers do not overlap. We show that for all these instances with up to five receivers, linear index codes are optimal. Although this class contains 9847 nonisomorphic instances, by using our recent results and by properly categorizing the instances based on their graphical representations, we need to consider only 13 nontrivial instances to solve the entire class. This work complements the result by Arbabjolfaei et al. (ISIT 2013), who derived the capacity region of all unicast indexcoding problems with up to five receivers in the diminishingerror setup. They employed randomcoding arguments, which require infinitelylong messages. We consider the zeroerror setup; our approach uses graph theory and combinatorics, and does not require long messages. I.
Index Coding via Random Coding
, 2014
"... The index coding problem is a simple distributed source coding problem in which a sender broadcasts multiple messages to their respective receivers with side information about other messages. This problem arises in many applications such as content broadcasting, distributed caching, and wireless i ..."
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The index coding problem is a simple distributed source coding problem in which a sender broadcasts multiple messages to their respective receivers with side information about other messages. This problem arises in many applications such as content broadcasting, distributed caching, and wireless interference management. At the same time, it is a canonical instance of the multipleunicast network coding problem that captures the essence of broadcasting multiple interfering streams. Reflecting the importance as well as the difficulty of the index coding problem, several coding schemes have been proposed that are built on tools from graph theory, linear network coding, combinatorial optimization, and interference alignment. This paper studies the composite coding scheme based on random coding in information theory. Despite its conceptual simplicity that allows for rather straightforward analysis, the scheme uniformly outperforms the existing coding schemes by Birk and Kol (1998), Blasiak, Kleinberg, and Lubetzky (2013), and Shanmugam, Dimakis, and Langberg (2013), and is optimal for all index coding problems with up to five messages.
Topological Interference Management for Hexagonal Cellular Networks
"... Abstract—We consider the topological interference management problem for a downlink hexagonal cellular network, where the channel state information at the transmitters is limited to just the network topology. Recent work by Jafar showed that if interference is limited to only near the cell boundary ..."
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Abstract—We consider the topological interference management problem for a downlink hexagonal cellular network, where the channel state information at the transmitters is limited to just the network topology. Recent work by Jafar showed that if interference is limited to only near the cell boundary, then, an aligned frequency reuse pattern achieves the optimal value of 6/7 degrees of freedom (DoF) per cell, as opposed to the conventional frequency reuse baseline of 1/3 DoF per cell. We generalize the setting to include interference from multiple layers of adjacent cells and characterize how the gains of the optimal solution over basic frequency reuse diminish with increasing number of interference layers. Next, we focus on singlelayer interference and explore the sensitivity of the idealized assumptions behind the connectivity model of Jafar, which achieves higher DoF but only at the cost of a higher effective noise floor than the baseline, and under idealized placements of users. A modified connectivity model that operates at a comparable noisefloor to the baseline is then studied, and its DoF are shown to be bounded above by 6/11 and below by 1/2. Through numerical simulations, we compare the solutions that achieve 6/7, 1/2, and 1/3 DoF per cell and find that, while both the 6/7 and the 1/2 DoF solutions beat the baseline 1/3 figure, between them, the 1/2 DoF aligned frequency reuse pattern is more robust for small cell networks particularly for random users ’ distribution on the cell boundaries. Index Terms—Hexagonal cellular network, degree of freedom, aligned frequency reuse, topological interference management. I.
Some New Results on Index Coding When The Number of Data Is Less Than The Number of Receivers
"... Abstract—In this paper, index coding problems in which the number (m) of receivers is larger than that (n) of data are considered. Unlike the case that the two numbers are same (n = m), index coding problems with n ≤ m are more general and hard to handle. To circumvent this difficulty, problems with ..."
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Abstract—In this paper, index coding problems in which the number (m) of receivers is larger than that (n) of data are considered. Unlike the case that the two numbers are same (n = m), index coding problems with n ≤ m are more general and hard to handle. To circumvent this difficulty, problems with n < m are approached via corresponding problems with n = m. It is shown that in certain cases, the symmetric capacity and code construction for index coding problems with n < m can be obtained from the existing symmetric capacity result and codes for index coding problems with n = m. Such cases include cases with n < m ≤ 5. I.
On Critical Index Coding Problems
"... Abstract—The question of under what condition some side information for index coding can be removed without affecting the capacity region is studied, which was originally posed by Tahmasbi, Shahrasbi, and Gohari. To answer this question, the notion of unicycle for the side information graph is intr ..."
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Abstract—The question of under what condition some side information for index coding can be removed without affecting the capacity region is studied, which was originally posed by Tahmasbi, Shahrasbi, and Gohari. To answer this question, the notion of unicycle for the side information graph is introduced and it is shown that any edge that belongs to a unicycle is critical, namely, it cannot be removed without reducing the capacity region. Although this sufficient condition for criticality is not necessary in general, a partial converse is established, which elucidates the connection between the notion of unicycle and the maximal acylic induced subgraph outer bound on the capacity region by BarYossef, Birk, Jayram, and Kol. I.
Index Coding and Network Coding via Rank Minimization
"... Abstract—Index codes reduce the number of bits broadcast by a wireless transmitter to a number of receivers with different demands and with side information. It is known that the problem of finding optimal linear index codes is NPhard. We investigate the performance of different heuristics based on ..."
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Abstract—Index codes reduce the number of bits broadcast by a wireless transmitter to a number of receivers with different demands and with side information. It is known that the problem of finding optimal linear index codes is NPhard. We investigate the performance of different heuristics based on rank minimization and matrix completion methods, such as alternating projections and alternating minimization, for constructing linear index codes over the reals. As a summary of our results, the alternating projections method gives the best results in terms of minimizing the number of broadcast bits and convergence rate and leads to up to 13 % savings in average communication cost compared to graph coloring algorithms studied in the literature. Moreover, we describe how the proposed methods can be used to construct linear network codes for nonmulticast networks. Our computer code is available online. I.
Graph Theory versus Minimum Rank for Index Coding
"... Abstract—We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is ..."
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Abstract—We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number. I.