Brijesh Kumar Rai

On the capacity of sum-networks

In this paper, we consider sum-networks, which is a class of directed acyclic networks where each terminal in the network requires sum of all the sources. In the literature, all reported sum-networks are shown to have only certain rational valued coding capacities. For example, it has been shown that a sum-network, having 3 sources and 3 terminals, has coding capacity either 0, 2/3 or ≥ 1. It is an open problem whether every positive rational number is the coding capacity of some sum-network. In this paper, for every positive rational number k/n, we show the existence of a sum-network which has coding capacity equal to k/n. The constructed sum-network also demonstrate that the gap between min-cut bound and coding capacity can be unbounded for sum-networks.

A low delay cross-layer contention based synchronous MAC protocol for a multi-hop WSN

Recently designed cross-layer contention based synchronous MAC protocols like the PRMAC protocol, for wireless sensor networks (WSNs) enable a node to schedule multi-hop transmission of multiple data packets in a cycle. However, these systems accommodate both the request-to-send data process and the confirmation-to-send data process in the same data transmission scheduling window (i.e. data window). This reduces the length of the multi-hop flow setup in the data window. In a multi-hop scenario, this degrades both the packet delivery ratio (PDR) and the end-to-end transmission delay (E2ETD). In this paper, we propose a cross-layer contention based synchronous MAC protocol, which accommodates the request-to-send data process in the data window and the confirmation-to-send data process in the sleep window for increased efficiency. We evaluate our proposed protocol through ns-2.35 simulations and compare its performance with the PRMAC protocol. Results suggest that in multi-hop scenario, proposed protocol outperforms PRMAC both in terms of the E2ETD and the packet delivery ratio (PDR).

Sum-Networks: Min-Cut = 2 Does Not Guarantee Solvability

In this paper, we consider sum-networks where each terminal requires sum of all the sources. For a sum-network having 3 sources and 3 terminals, it is known that the network is solvable if the min-cut between each source and each terminal is at least 2. However, it is not known whether this condition is sufficient for sum-networks having arbitrary number of sources and terminals. In the present work, we present a sum-network which is not solvable even though the min-cut between any source and any terminal is at least 2.

A Novel Framework to Enhance the Performance of Contention-Based Synchronous MAC Protocols

In this paper, we propose a novel framework to improve the end-to-end transmission delay (E2ETD) and packet delivery ratio (PDR) of existing contention-based synchronous MAC protocols designed for wireless sensor networks, without increasing the duty cycle. This is achieved by partitioning the n deployed sensor nodes into k disjoint sets, which are of almost equal size. It then suitably modifies the cycle structure followed by the existing contention-based synchronous MAC protocols by mapping the data transmission process of k existing cycles into one restructured cycle. To evaluate the performance of this approach, we implement RMAC, PRMAC, and CLMAC protocols in the proposed framework using ns2.35 simulator. Results indicate that our scalable framework reduces the E2ETD and increases the PDR significantly at the cost of a very small increase in average energy consumption.

On the Message Dimensions of Vector Linearly Solvable Networks

It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is 2. It is not known whether this result can be generalized for an arbitrary message dimension. In this letter, we show that there exists a network that admits an m dimensional vector linear solution, where m is a positive integer greater than or equal to 2, but does not have a vector linear solution over any finite field when the message dimension is less than m.

On the capacity of ms/3t and 3s/nt sum-networks

We consider directed acyclic networks where each terminal requires sum of all the sources. Such a class of networks has been termed as sum-networks in the literature. A sum-network having m sources and n terminals has been termed as a ms/nt sum-network. There has been previous works on the capacity of sum-networks, specifically, it has been shown that the capacity of a 3s/3t sum-network is either 0,2/3 or ≥ 1. In this paper, we consider some generalizations of 3s/3t sum-networks, namely, ms/3t and 3s/nt sum-networks, where m, n ≥ 3. For ms/3t and 3s/nt sum-networks, where m, n ≥ 3, if the mincut between each source and each terminal is at least 1, the capacity is known to be at least 2/3. In this paper, we show that there exist ms/3t and 3s/nt sum-networks whose capacities lie between 2/3 and 1. Specifically, we show that for any positive integer k ≥ 2, there exists a ms/3t sum-network (and also a 3s/nt sum-network) whose capacity is k/k+1. We conjecture that the capacity of a ms/3t sum-network, where m > 3 (and also of a 3s/nt sum-network, where n > 3) is either 0, ≥ 1 or of the form k/k+1, where k is a positive integer greater than or equal to 2.

On the number of sources and terminals of sum-networks with capacity p/q

Rai et al. in [1] showed that any rational number p/q is the capacity of some sum-network. The authors proved this result by constructing such sum-networks. The authors first constructed a sum-network with capacity 1/q, and then they connected p of such networks in parallel to create a sum-network of capacity p/q. Such construction requires 2q2 - q number of sources and 2q2-q number of terminals. In this paper, we ask the following question: is it possible to construct such sum-networks with lesser number of sources or terminals? The result in [2], where it is shown that there exists a k sources and 3 terminal sum-network with capacity k-1 over k; confirms that at least for some values of p and q, the answer is affirmative. For various values of p and q, we have made an attempt to construct sum-networks having lesser number of sources and terminals than used in [1], and have been able to construct various such sum-networks. In this paper, we present three such example sum-networks, with capacities 3 over 5; 4 over 7 and 5 over 12 respectively. However, we have not yet been able to come up with a systematic method to construct such sum-networks. We believe that studying these examples is a step towards constructing a sum-network, with capacity p over q, which requires minimum number of sources and terminals.

Generalized Fano and non-Fano networks

It is known that the Fano network has a vector linear solution if and only if the characteristic of the finite field is 2; and the non-Fano network has a vector linear solution if and only if the characteristic of the finite field is not 2. Using these properties of Fano and non-Fano networks it has been shown that linear network coding is insufficient. In this paper we generalize the properties of Fano and non-Fano networks. Specifically, by adding more nodes and edges to the Fano network, we construct a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field belongs to an arbitrary given set of primes {p<inf>1</inf>, p<inf>2</inf>, …, p<inf>l</inf>}. Similarly, by adding more nodes and edges to the non-Fano network, we construct a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field does not belong to an arbitrary given set of primes {p<inf>1</inf>, p<inf>2</inf>, …, p<inf>l</inf>}.

Adaptive erasure code based distributed storage systems

Consider the following scenario: A data storage service provider provides an erasure code based distributed storage system (DSS). For the same data, the service provider gives several options: an (n_i, k_i) erasure code based DSS for i = 1,2, ..., m. The service provider charges differently for different options (say dollar Pi for an (n_i, k_i) erasure code based DSS for the data B of size |B|). A client had initially chosen for an (n_i, k_i) erasure code based DSS. At some point of time, the client wants to change for another option, say for an (n_j, k_j) erasure code based DSS for the same data, where 1≤ i, j ≤ m, i ≠ j. Thus, service provider would require to convert the (n_i, k_i) erasure code based DSS into an (n_j, k_j) erasure code based DSS. The service provider has the following problem: How to design an erasure code based DSS so that the conversion of an (n_i, k_i) erasure code based DSS into an (n_j, k_j) erasure code based DSS, for 1 ≤ i, j ≤ m, i ≠ j, can be done by downloading the minimum amount of data? In this paper, we present an adaptive coding scheme which requires to download the minimum amount of data while converting an (n_i, k_i) erasure code based DSS to an (n_j, k_j) erasure code based DSS, where 1 ≤ i, j ≤ m, i ≠ j.

Sum-networks: Dependency on characteristic of the finite field under linear network coding

Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a rate 1 linear network coding solution if and only if the characteristic of the finite field belongs to the given set. It has also been shown that for any positive rational number k/n, there exists a sum-network which has capacity equal to k/n. It is a natural question whether, for any positive rational number k/n, and for any finite set/co-finite set of primes {p1, p2, …, pl}, there exists a sum-network which has a capacity achieving rate k/n fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. We show that indeed there exists such a sum-network by giving an explicit construction.