Capacity of Some 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. Maleki, Cadambe and Jafar obtained the capacity of this symmetric neighboring interference single unicast index coding problem (SNI-SUICP) with $(K)$ tending to infinity and Blasiak, Kleinberg and Lubetzky for the special case of $(D=U=1)$ with $K$ being finite. In this work, for any finite $K$ and arbitrary $D$ we obtain the capacity for the case $U=gcd(K,D+1)-1.$ Our proof is constructive, i.e., we give an explicit construction of a linear index code achieving the capacity.Comment: Considerable overlap with that of arXiv:1705.03192v1 [cs.IT] 9 may 2017 (especially the introductory parts). Few typos in version1 have been fixed. Figures=9; Table=

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

Reduced Complexity Index Codes and Improved Upperbound on Broadcast Rate for Neighboring Interference Problems

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has $K$ messages and $K$ receivers, the $k$th receiver $R_{k}$ wanting the $k$th message $x_{k}$ and having the interference with $D$ messages after and $U$ messages before its desired message. Maleki, Cadambe and Jafar studied SUICP(SNI) because of its importance in topological interference management problems. Maleki \textit{et. al.} derived the lowerbound on the broadcast rate of this setting to be $D+1$. In our earlier work, for SUICP(SNI) with arbitrary $K,D$ and $U$, we defined set $\mathbf{S}$ of 2-tuples and for every $(a,b) \in \mathbf{S}$, we constructed $b$-dimensional vector linear index code with rate $D+1+\frac{a}{b}$ by using an encoding matrix of dimension $Kb \times (b(D+1)+a)$. In this paper, we use the symmetric structure of the SUICP(SNI) to reduce the size of encoding matrix by partitioning the message symbols. The rate achieved in this paper is same as that of the existing constructions of vector linear index codes. More specifically, we construct $b$-dimensional vector linear index codes for SUICP(SNI) by partitioning the $Kb$ messages into $b(U+1)+c$ sets for some non-negative integer $c$. We use an encoding matrix of size $\frac{Kb}{b(U+1)+c} \times \frac{b(D+1)+a}{b(U+1)+c}$ to encode each partition separately. The advantage of this method is that the receivers need to store atmost $\frac{b(D+1)+a}{b(U+1)+c}$ number of broadcast symbols (index code symbols) to decode a given wanted message symbol. We also give a construction of scalar linear index codes for SUICP(SNI) with arbitrary $K,D$ and $U$. We give an improved upperbound on the braodcast rate of SUICP(SNI).Comment: 9 pages and 4 figures. Extension of our previous arXiv submission: arXiv:1707.00455 and hence some overla

An Approximation Algorithm for Optimal Clique Cover Delivery in Coded Caching

Coded caching can significantly reduce the communication bandwidth requirement for satisfying users' demands by utilizing the multicasting gain among multiple users. Most existing works assume that the users follow the prescriptions for content placement made by the system. However, users may prefer to decide what files to cache. To address this issue, we consider a network consisting of a file server connected through a shared link to $K$ users, each equipped with a cache which has been already filled arbitrarily. Given an arbitrary content placement, the goal is to find a delivery strategy for the server that minimizes the load of the shared link. In this paper, we focus on a specific class of coded multicasting delivery schemes known as the "clique cover delivery scheme". We first formulate the optimal clique cover delivery problem as a combinatorial optimization problem. Using a connection with the weighted set cover problem, we propose an approximation algorithm and show that it provides an approximation ratio of $(1 + \log K)$, while the approximation ratio for the existing coded delivery schemes is linear in $K$. Numerical simulations show that our proposed algorithm provides a considerable bandwidth reduction over the existing coded delivery schemes for almost all content placement schemes.Comment: Accepted for publication in IEEE Transactions on Communication