9 research outputs found
Linear Fractional Network Coding and Representable Discrete Polymatroids
A linear Fractional Network Coding (FNC) solution over is a
linear network coding solution over in which the message
dimensions need not necessarily be the same and need not be the same as the
edge vector dimension. Scalar linear network coding, vector linear network
coding are special cases of linear FNC. In this paper, we establish the
connection between the existence of a linear FNC solution for a network over
and the representability over of discrete
polymatroids, which are the multi-set analogue of matroids. All previously
known results on the connection between the scalar and vector linear
solvability of networks and representations of matroids and discrete
polymatroids follow as special cases. An algorithm is provided to construct
networks which admit FNC solution over from discrete
polymatroids representable over Example networks constructed
from discrete polymatroids using the algorithm are provided, which do not admit
any scalar and vector solution, and for which FNC solutions with the message
dimensions being different provide a larger throughput than FNC solutions with
the message dimensions being equal.Comment: 8 pages, 5 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1301.300
Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids
Discrete polymatroids are the multi-set analogue of matroids. In this paper,
we explore the connections among linear network coding, linear index coding and
representable discrete polymatroids. We consider vector linear solutions of
networks over a field with possibly different message and edge
vector dimensions, which are referred to as linear fractional solutions. We
define a \textit{discrete polymatroidal} network and show that a linear
fractional solution over a field exists for a network if and
only if the network is discrete polymatroidal with respect to a discrete
polymatroid representable over An algorithm to construct
networks starting from certain class of discrete polymatroids is provided.
Every representation over for the discrete polymatroid, results
in a linear fractional solution over for the constructed
network. Next, we consider the index coding problem and show that a linear
solution to an index coding problem exists if and only if there exists a
representable discrete polymatroid satisfying certain conditions which are
determined by the index coding problem considered. El Rouayheb et. al. showed
that the problem of finding a multi-linear representation for a matroid can be
reduced to finding a \textit{perfect linear index coding solution} for an index
coding problem obtained from that matroid. We generalize the result of El
Rouayheb et. al. by showing that the problem of finding a representation for a
discrete polymatroid can be reduced to finding a perfect linear index coding
solution for an index coding problem obtained from that discrete polymatroid.Comment: 24 pages, 6 figures, 4 tables, some sections reorganized, Section VI
newly added, accepted for publication in IEEE Transactions on Information
Theor
Guessing Games on Undirected Graphs
PhDGuessing games for directed graphs were introduced by Riis for studying multiple unicast
network coding problems. In a guessing game, the players toss generalised die
and can see some of the other outcomes depending on the structure of an underlying
digraph. They later simultaneously guess the outcome of their own die. Their objective
is to find a strategy that maximises the probability that they all guess correctly.
The performance of the optimal strategy for a digraph is measured by the guessing
number.
In general, the existence of an algorithm for computing guessing numbers of a graph
is unknown. In the case of undirected graphs, Christofides and Markstr om defined
a strategy that they conjectured to be optimal. One of the main results of this
thesis is a disproof of this conjecture. In particular, we illustrate an undirected
graph on 10 vertices having guessing number which is strictly larger than the lowerbound
provided by Christofides and Markstr om's method. Moreover, even in case
the undirected graph is triangle-free, we establish counter examples to this conjecture
based on combinatorial objects known as Steiner systems.
The main tool thus far for computing guessing numbers of graphs has been information
theoretic inequalities. Using this method, we are able to derive the guessing
numbers of new families of undirected graphs, which in general cannot be computed
directly using a computer. A new result of the thesis is that Shannon's information
inequalities, which work particularly well for a wide range of graph classes, are not
sufficient for computing the guessing number.
Another contribution of this thesis is a firm answer to the question concerning irreversible
guessing games. In particular, we construct a directed graph G with Shannon
upper-bound that is larger than the same bound obtained when we reverse all edges
of G.
Finally, we initialize a study on noisy guessing game, which is a generalization of
noiseless guessing game defined by Riis.
We pose a few more interesting questions, some of which we can answer and some
which we leave as open problems.
5School of Electronic Engineering and Computer
Science
Network coding for robust wireless networks
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 157-167).Wireless networks and communications promise to allow improved access to services and information, ubiquitous connectivity, and mobility. However, current wireless networks are not well-equipped to meet the high bandwidth and strict delay requirements of future applications. Wireless networks suffer from frequent losses and low throughput. We aim to provide designs for robust wireless networks. This dissertation presents protocols and algorithms that significantly improve wireless network performance and effectively overcome interference, erasures, and attacks. The key idea behind this dissertation is in understanding that wireless networks are fundamentally different from wired networks, and recognizing that directly applying techniques from wired networks to wireless networks limits performance. The key ingredient underlying our algorithms and protocols is network coding. By recognizing the algebraic nature of information, network coding breaks the convention of routing networks, and allows mixing of information in the intermediate nodes and routers. This mixing has been shown to have numerous performance benefits, e.g. increase in throughput and robustness against losses and failures. We present three protocols and algorithms, each using network coding to harness a different characteristic of the wireless medium. We address the problem of interference, erasures, and attacks in wireless networks with the following network coded designs. -- Algebraic NC exploits strategic interference to provide a distributed, randomized code construction for multi-user wireless networks. Network coding framework simplifies the multi-user wireless network model, and allows us to describe the multi-user wireless networks in an algebraic framework. This algebraic framework provides a randomized, distributed code construction, which we show achieves capacity for multicast connections as well as a certain set of non-multicast connections. -- TCP/NC efficiently and reliably delivers data over unreliable lossy wireless networks. TCP, which was designed for reliable transmission over wired networks, often experiences severe performance degradation in wireless networks. TCP/NC combines network coding's erasure correction capabilities with TCP's congestion control mechanism and reliability. We show that TCP/NC achieves significantly higher throughput than TCP in lossy networks; therefore, TCP/NC is well suited for reliable communication in lossy wireless networks. -- Algebraic Watchdog takes advantage of the broadcast nature of wireless networks to provide a secure global self-checking network. Algebraic Watchdog allows nodes to detect malicious behaviors probabilistically, and police their neighbors locally using overheard messages. Unlike traditional detection protocols which are receiver-based, this protocol gives the senders an active role in checking the nodes downstream. We provide a trellis-based inference algorithm and protocol for detection, and analyze its performance. The main contribution of this dissertation is in providing algorithms and designs for robust wireless networks using network coding. We present how network coding can be applied to overcome the challenges of operating in wireless networks. We present both analytical and simulation results to support that network coded designs, if designed with care, can bring forth significant gains, not only in terms of throughput but also in terms of reliability, security, and robustness.by MinJi Kim.Ph.D