arXiv subjects

Ashwin Ganesan

Publications and source records attributed to Ashwin Ganesan.

27 records · Page 2Linked to original sources

arXiv2012

On some upper bounds on the fractional chromatic number of weighted graphs

Given a weighted graph $G_\bx$, where $(x(v): v \in V)$ is a non-negative, real-valued weight assigned to the vertices of G, let $B(G_\bx)$ be an upper bound on the fractional chromatic number of the weighted graph $G_\bx$; so $χ_f(G_\bx) \le B(G_\bx)$. To investigate the worst-case performance of the upper bound $B$, we study the graph invariant $$β(G) = \sup_{\bx \ne 0} \frac{B(G_\bx)}{χ_f(G_\bx)}.$$ \noindent This invariant is examined for various upper bounds $B$ on the fractional chromatic number. In some important cases, this graph invariant is shown to be related to the size of the largest star subgraph in the graph. This problem arises in the area of resource estimation in distributed systems and wireless networks; the results presented here have implications on the design and performance of decentralized communication networks.

cs.IT↗

arXiv2012

Minimal resolving sets for the hypercube

For a given undirected graph $G$, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in $S$. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining resolving sets that are minimal or that are minimum (of minimal cardinality). Let $Q^n$ denote the $n$-dimensional hypercube with vertex set ${0,1}^n$. In Erdös and Renyi (Erdos & Renyi, 1963) it was shown that a particular set of $n$ vertices forms a resolving set for the hypercube. The main purpose of this note is to prove that a proper subset of that set of size $n-1$ is also a resolving set for the hypercube for all $n \ge 5$ and that this proper subset is a minimal resolving set.

cs.DM↗

arXiv2011

Diameter of Cayley graphs of permutation groups generated by transposition trees

Let $Γ$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $Γ$ is bounded as $$\diam(Γ) \le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))},$$ where the maximization is over all permutations $π$, $c(π)$ denotes the number of cycles in $π$, and $\dist_T$ is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $β$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. $$\diam(Γ) \le β\le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))}.$$

cs.DM↗

arXiv2011

The regular number of a graph

Let $G$ be a simple undirected graph. The regular number of $G$ is defined to be the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is regular. In this work, we obtain the regular number of some families of graphs and discuss some general bounds on this parameter. Also, some of the lower or upper bounds proved in \cite{Kulli:Janakiram:Iyer:2001} are shown here to hold with equality.

cs.DM↗

arXiv2011

On a diameter bound for Cayley graphs generated by transposition trees

Let $Γ$ be a Cayley graph generated by a transposition tree. A natural problem is to understand how the properties of the Cayley graph depend on those of the underlying transposition tree. We focus here on diameter and distance related questions. We examine some related bounds and prove some properties about them. This paper has been withdrawn and a new version with these results and further extensions is at abs/1111.3114

math.CO↗

arXiv2009

On some sufficient conditions for distributed Quality-of-Service support in wireless networks

Given a wireless network where some pairs of communication links interfere with each other, we study sufficient conditions for determining whether a given set of minimum bandwidth Quality of Service (QoS) requirements can be satisfied. We are especially interested in algorithms which have low communication overhead and low processing complexity. The interference in the network is modeled using a conflict graph whose vertices are the communication links in the network. Two links are adjacent in this graph if and only if they interfere with each other due to being in the same vicinity and hence cannot be simultaneously active. The problem of scheduling the transmission of the various links is then essentially a fractional, weighted vertex coloring problem, for which upper bounds on the fractional chromatic number are sought using only localized information. We present some distributed algorithms for this problem, and discuss their worst-case performance. These algorithms are seen to be within a bounded factor away from optimal for some well known classes of networks and interference models.

cs.IT↗

arXiv2006

On Universally Decodable Matrices for Space-Time Coding

The notion of universally decodable matrices (UDMs) was recently introduced by Tavildar and Viswanath while studying slow fading channels. It turns out that the problem of constructing UDMs is tightly connected to the problem of constructing maximum distance separable (MDS) codes. In this paper, we first study the properties of UDMs in general and then we discuss an explicit construction of a class of UDMs, a construction which can be seen as an extension of Reed-Solomon codes. In fact, we show that this extension is, in a sense to be made more precise later on, unique. Moreover, the structure of this class of UDMs allows us to answer some open conjectures by Tavildar, Viswanath, and Doshi in the positive, and it also allows us to formulate an efficient decoding algorithm for this class of UDMs. It turns out that our construction yields a coding scheme that is essentially equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Moreover, we point out connections to so-called repeated-root cyclic codes.

cs.IT↗

arXiv2006

On the Existence of Universally Decodable Matrices

Universally decodable matrices (UDMs) can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $N$ and a prime power $q$. The main result of this paper is that the simple condition $L \leq q+1$ is both necessary and sufficient for $(L,N,q)$-UDMs to exist. The existence proof is constructive and yields a coding scheme that is equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Our work resolves an open problem posed recently in the literature.

cs.IT↗

arXiv2005

An Explicit Construction of Universally Decodable Matrices

Universally decodable matrices can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $n$ and a prime power $q$. Based on Pascal's triangle we give an explicit construction of universally decodable matrices for any non-zero integers $L$ and $n$ and any prime power $q$ where $L \leq q+1$. This is the largest set of possible parameter values since for any list of universally decodable matrices the value $L$ is upper bounded by $q+1$, except for the trivial case $n = 1$. For the proof of our construction we use properties of Hasse derivatives, and it turns out that our construction has connections to Reed-Solomon codes, Reed-Muller codes, and so-called repeated-root cyclic codes. Additionally, we show how universally decodable matrices can be modified so that they remain universally decodable matrices.

cs.IT↗