Abstract

Distributed databases often suffer unequal distribution of data among storage nodes, which is known asdata skew'. Data skew arises from a number of causes such as removal of existing storage nodes and addition of new empty nodes to the database. Data skew leads to performance degradations and necessitatesrebalancing'at regular intervals to reduce the amount of skew. We define an -balanced distributed database as a distributed database in which the storage across the nodes has uniform size, and each bit of the data is replicated in distinct storage nodes. We consider the problem of designing such balanced databases along with associated rebalancing schemes which maintain the -balanced property under node removal and addition operations. We present a class of -balanced databases (parameterized by the number of storage nodes) which have the property of structural invariance, ie, the databases designed for different number of storage nodes have the same structure. For this class of -balanced databases, we present rebalancing schemes which use coded transmissions between storage nodes, and characterize their communication loads under node addition and removal. We show that the communication cost incurred to rebalance our distributed database for node addition and removal is optimal, ie, it achieves the minimum possible cost among all possible balanced distributed databases and rebalancing schemes.

Abstract

We consider the distributed computing framework of Map-Reduce, which consists of three phases, the Map phase, the Shuffle phase and the Reduce phase. For this framework, we propose the use of binary matrices (with entries) calledtextit {computing matrices} to describe the map phase and the shuffle phase. Similar binary matrices were recently proposed for the coded caching framework. The structure of ones and zeroes in the binary computing matrix captures the map phase of the Map-reduce framework. We present a new simple coded data shuffling scheme for this binary matrix model, based on atextit {identity submatrix cover} of the computing matrix. This new coded shuffling scheme has in general a larger communication load than existing schemes, but has the advantage of less complexity overhead than the well-known earlier schemes in literature in terms of the file-splitting and associated indexing and coordination required. We also show that there exists a binary matrix based distributed computing scheme with our new data-shuffling scheme which has strictly less than twice than the communication load of the known optimal scheme in literature. The structure of this new scheme enables it to be applied to the framework of Map-reduce with stragglers also, in a straightforward manner, borrowing its advantages and disadvantages from the no-straggler situation. Finally, using binary matrices derived from combinatorial designs, we show specific classes of computing schemes with very lowtextit {file complexity}(number of subfiles in the file), but with higher communication load compared to the optimal scheme for equivalent parameters.

Abstract

Circular perfect graphs are those undirected graphs such that the circular clique number is equal to the circular chromatic number for each induced subgraph. They form a strict superclass of the perfect graphs, whose index coding broadcast rates are well known. We present the broadcast rate of index coding for side-information graphs whose complements are circular perfect, along with an optimal achievable scheme. We thus enlarge the known classes of graphs for which the broadcast rate is exactly characterized. In an attempt to understand the broadcast rate of a graph given that of its complement, we obtain upper and lower bounds for the product and sum of the vector linear broadcast rates of a graph and its complement. We show that these bounds are satisfied with equality even for some perfect graphs. Curating prior results, we show that there are circular perfect but imperfect graphs which satisfy the lower bound on the product of the broadcast rate of the complementary graphs with equality.

Abstract

Coded caching was introduced as a technique of systematically exploiting locally available storage at the clients to increase the channel throughput via coded transmissions. Most known coded caching schemes in literature enable large gains in terms of the rate, however at the cost of subpacketization that is exponential in ( being the number of clients, some positive integer). Building upon recent prior work for coded caching design via line graphs and finite-field projective geometries, we present a new scheme in this work which achieves a subexponential (in ) subpacketization of and rate , for large , and the cached fraction being upper bounded by a constant (for some prime power and constant ). Apart from this asymptotic improvement, we show that through some numerical comparisons that our present scheme has much lower subpacketization than previous comparable …

Abstract

Consider an interference channel consisting of transmitters and receivers with AWGN noise and complex channel gains, and with files in the system. The one-shot for this channel is the maximum number of receivers which can be served simultaneously with vanishing probability of error as the grows large, under a class of schemes known as\textit {one-shot} schemes. Consider that there exists transmitter and receiver side caches which can store fractions and of the library respectively. Recent work for this cache-aided interference channel setup shows that, using a carefully designed prefetching (caching) phase, and a one-shot coded delivery scheme combined with a proper choice of beamforming coefficients at the transmitters, we can achieve a of , where and which was shown to be almost optimal. The existing scheme involves splitting the file into subfiles (the parameter is called the\textit {subpacketization}), where can be extremely large (in fact, with constant cache fractions, it becomes exponential in , for large ). In this work, our first contribution is a scheme which achieves the same of with a smaller subpacketization than prior schemes. Our second contribution is a new coded caching scheme for the interference channel based on projective geometries over finite fields which achieves a one-shot of , with a subpacketization (for some prime power ) that is\textit {subexponential} in , for small constant cache fraction at the receivers. To the best of our knowledge, this is the first coded caching scheme with subpacketization …

Abstract

An index code is said to be locally decodable if each receiver can decode its demand using its side information and by querying only a subset of the transmitted codeword symbols instead of observing the entire codeword. Local decodability can be a beneficial feature in some communication scenarios, such as when the receivers can afford to listen to only a part of the transmissions because of limited availability of power. The locality of an index code is the ratio of the maximum number of codeword symbols queried by a receiver to the message length. In this paper we analyze the optimum locality of linear codes for the family of index coding problems whose min-rank is one less than the number of receivers in the network. We first derive the optimal trade-off between the index coding rate and locality with vector linear coding when the side information graph is a directed cycle. We then provide the optimal trade-off …

Abstract

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. In this work, we propose a binary matrix model to construct the coded caching scheme. The ones in such a caching matrix indicate uncached subfiles at the users. Identity submatrices of the caching matrix represent transmissions in the delivery phase. Using this model, we then propose several novel constructions for coded caching based on the various types of combinatorial designs …

Abstract

Coded Caching is a promising solution to reduce the peak traffic in broadcast networks by prefetching the popular content close to end users and using coded transmissions. One of the chief issues of most coded caching schemes in literature is the issue of large subpacketization, i.e., they require each file to be divided into a large number of subfiles. In this work, we present a coded caching scheme using line graphs of bipartite graphs in conjunction with projective geometries over finite fields. The presented scheme achieves a rate Θ(K/log q K) (K being the number of users, q is some prime power) with subexponential subpacketization q O((logq K)^2) when cached fraction is upper bounded by a constant (M/N ≤ 1/q α , for some positive integer α). Compared to earlier schemes, the presented scheme has a lower subpacketization (albeit possessing a higher rate). We also present a new subpacketization …

Abstract

Coded Caching is a recent technique that optimizes the use of a multi-client broadcast channel by the use of local storage available at the clients and by using coded transmissions to serve multiple clients at once. While large gains in the rate of communication are obtained using coded caching, most existing schemes require that the files at the server be divisible into a large number of parts. In particular, most known coded caching schemes require subpacketization $ F={e^{Oleft ({{K^{frac {1}{r}}}}right)}} $, where K is the number of clients and r is some constant positive integer. While few schemes having subpacketization linear in K are known in literature, unfortunately such schemes require large number of users to exist or offer little gain in rate. In this work, we propose a class of coded caching schemes based on projective geometries over finite fields, generalizing recent results. Our construction achieves subexponential (in K) subpacketization, ie, $ F={q^{O …

Abstract

Coded caching was introduced as a technique of systematically exploiting locally available storage at the clients to increase the channel throughput via coded transmissions. Most known coded caching schemes in literature enable large gains in terms of the rate, however at the cost of subpacketization that is exponential in ( being the number of clients, some positive integer). Building upon recent prior work for coded caching design via line graphs and finite-field projective geometries, we present a new scheme in this work which achieves a subexponential (in ) subpacketization of and rate , for large , and the cached fraction being upper bounded by a constant (for some prime power and constant ) . Apart from this asymptotic improvement, we show that through some numerical comparisons that our present scheme has much lower subpacketization than previous comparable …