arXiv subjects

Ghurumuruhan Ganesan

Publications and source records attributed to Ghurumuruhan Ganesan.

At least 55 records · Page 3Linked to original sources

arXiv2021

Deviation Estimates for Eulerian Edit Numbers of Random Graphs

Consider the random graph~$G(n,p)$ obtained by allowing each edge in the complete graph on~$n$ vertices to be present with probability~$p$ independent of the other edges. In this paper, we study the minimum number of edge edit operations needed to convert~$G(n,p)$ into an Eulerian graph. We obtain deviation estimates for three types Eulerian edit numbers based on whether we perform only edge additions or only edge deletions or a combination of both and show that with high probability, roughly~$\frac{n}{4}$ operations suffice in all three cases.

math.PR↗

arXiv2021

Linear Recurrences over a Finite Field with Exactly Two Periods

In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial~$f$ to have exactly two periods in the sense that the period of any sequence generated by~$f$ is either one or a unique integer greater than one.

math.CO↗

arXiv2020

Recursive methods for some problems in coding and random permutations

In this paper, we study three applications of recursion to problems in coding and random permutations. First, we consider locally recoverable codes with partial locality and use recursion to estimate the minimum distance of such codes. Next we consider weighted lattice representative codes and use recursive subadditive techniques to obtain convergence of the minimum code size. Finally, we obtain a recursive relation involving cycle moments in random permutations and as an illustration, evaluate recursions for the mean and variance.

math.CO↗

arXiv2020

Achieving positive rates with predetermined dictionaries

In the first part of the paper we consider binary input channels that are not necessarily stationary and show how positive rates can be achieved using codes constrained to be within predetermined dictionaries. We use a Gilbert-Varshamov-like argument to obtain the desired rate achieving codes. Next we study the corresponding problem for channels with arbitrary alphabets and use conflict-set decoding to show that if the dictionaries are contained within nice sets, then positive rates are achievable.

cs.IT↗

arXiv2020

Euclidean traveling salesman problem with location dependent and power weighted edges

Consider~$n$ nodes~$\{X_i\}_{1 \leq i \leq n}$ independently distributed in the unit square~$S,$ each according to a distribution~$f$ and let~$K_n$ be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge~$e$ in~$K_n$ we associate a weight~$w(e)$ that may depend on the \emph{individual locations} of the endvertices of~$e$ and is not necessarily a power of the Euclidean length of~$e.$ Denoting~$TSP_n$ to be the minimum weight of a spanning cycle of~$K_n$ corresponding to the travelling salesman problem (TSP) and assuming an equivalence condition on the weight function~$w(.),$ we prove that~$TSP_n$ appropriately scaled and centred converges to zero a.s.\ and in mean as~$n \rightarrow \infty.$ We also obtain upper and lower bound deviation estimates for~$TSP_n.$

math.PR↗

arXiv2018

Correcting an ordered deletion-erasure

In this paper, we show that the single deletion correcting Varshamov-Tenengolts code, with minor modifications, can also correct an ordered deletion-erasure pattern where one deletion and at most one erasure occur and the deletion always occurs before the erasure. For large code lengths, the constructed code has the same logarithmic redundancy as optimal codes.

cs.IT↗

arXiv2018

Records from partial comparisons and discrete approximations

In this paper we study records obtained from partial comparisons within a sequence of independent and identically distributed (i.i.d.) random variables, indexed by positive integers, with a common density~$f.$ Our main result is that if the comparison sets along a subsequence of the indices satisfy a certain compatibility property, then the corresponding record events are independent. Moreover, the record event probabilities do not depend on the density~$f$ and we obtain closed form expressions for the distribution of~$r^{th}$ record value for any integer~$r \geq 1.$ Our proof techniques extend to the discrete case as well and we estimate the difference in record event probabilities associated with a continuous random variable~$X$ and its discrete approximations.

math.PR↗

arXiv2018

Construction and redundancy of codes for correcting deletable errors

Consider a binary word being transmitted through a communication channel that introduces deletable errors where each bit of the word is either retained, flipped, erased or deleted. The simplest code for correcting \emph{all} possible deletable error patterns of a fixed size is the repetition code whose redundancy grows linearly with the code length. In this paper, we relax this condition and construct codes capable of correcting \emph{nearly} all deletable error patterns of a fixed size, with redundancy growing as a logarithm of the word length.

cs.IT↗

arXiv2018

Graph extensions, edit number and regular graphs

A graph G on n vertices is said to be extendable if G can be modified to form a new graph H on more than n vertices, while preserving the degrees of the vertices common to G and H. The added vertices all have the same degree and we define edit numbers to quantify the amount of modification needed to obtain the extended graph. Characterizing graphs with least possible edit numbers, we obtain that graphs with zero edit number can be extended using regular graphs. We also describe an iterative algorithm to construct connected regular graphs on arbitrarily large vertex sets, starting from the complete graph on a fixed set of vertices.

math.CO↗

arXiv2018

Existence of connected regular and nearly regular graphs

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.

math.CO↗

arXiv2018

Traveling salesman problem across dense cities

Consider~$n$ nodes~$\{X_i\}_{1 \leq i \leq n}$ distributed independently across~$N$ cities contained with the unit square~$S$ according to a distribution~$f.$ Each city is modelled as an~$r_n \times r_n$ square contained within~$S$ and let~$TSPC_n$ denote the length of the minimum length cycle containing all the~$n$ nodes, corresponding to the traveling salesman problem (TSP). We obtain variance estimates for~$TSPC_n$ and prove that if the cities are well-connected and densely populated in a certain sense, then~$TSPC_n$ appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for~$TSPC_n.$ Using the proof techniques, we alternately obtain corresponding results for the length~$TSP_n$ of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square~$S.$

math.PR↗

arXiv2018

Minimum spanning trees across dense cities

Consider~$n$ nodes distributed independently across~$N$ cities contained with the unit square~$S$ according to a distribution~$f.$ Each city is modelled as an~$r_n \times r_n$ square contained within~$S$ and~$MSTC_n$ denotes the length of the minimum spanning tree containing all the~$n$ nodes. We use approximation methods to obtain variance estimates for~$MSTC_n$ and prove that if the cities are well-connected and densely populated in a certain sense, then~$MSTC_n$ appropriately centred and scaled converges to zero in probability. Using the proof techniques, we alternately derive corresponding results for the length~$MST_n$ of the minimum spanning tree for the usual case when the nodes are independently distributed throughout the unit square~$S.$ In particular, we obtain that the variance of~$MST_n$ grows at most as a power of the logarithm of~$n$ and use a subsequence argument to get almost sure convergence of~$MST_n$ appropriately centred and scaled.

math.PR↗

arXiv2017

Recurrence region of multiuser Aloha

In this paper, we provide upper and lower bounds for the region of positive recurrence for a general finite user Aloha network.

math.PR↗

arXiv2017

Rates of linear codes with low decoding error probability

Consider binary linear codes obtained from bipartite graphs as follows. There are~$k \geq 1$ left nodes each representing a message bit and there are~$m = m(k)$ right nodes each representing a parity bit, generated from the corresponding set of message node neighbours. Both the message and the parity bits are sent through a memoryless binary input channel that either retains, flips or erases each transmitted bit, independently. Based on the received set of symbols, the decoder at the receiver obtains an estimate of the original message sent. If the decoding error probability~$P_k \longrightarrow 0$ and the average degree per parity node remains bounded as~$k \rightarrow \infty,$ then the rate of the code~$\frac{k}{k+m} \longrightarrow 0$ as~$k \rightarrow \infty.$

math.PR↗

arXiv2017

Duality in percolation via outermost boundaries III: Plus connected components

Tile $\mathbb{R}^2$ into disjoint unit squares $\{S_k\}_{k \geq 0}$ with the origin being the centre of $S_0$ and say that $S_i$ and $S_j$ are star adjacent if they share a corner and plus adjacent if they share an edge. Every square is either vacant or occupied. In this paper, we use the structure of the outermost boundaries derived in Ganesan (2017) to alternately obtain duality between star and plus connected components in the following sense: There is a star connected cycle of vacant squares attached to and surrounding the finite plus connected component containing the origin.

math.PR↗

arXiv2017

Long Paths and Hamiltonian paths in Inhomogenous Random Graphs

In this paper, we study long paths and Hamiltonian paths in inhomogenous random graphs. In the first part of the paper, we consider an inhomogenous Erdős-Rényi random graph $G_E$ with average edge density $p_n.$ We prove that if $np_n^2 \longrightarrow \infty$ as $n \rightarrow \infty,$ then the longest path contains at least $n-ne^{-δ_1 np_n^2}$ nodes with high probability (i.e., with probability converging to one as $n \rightarrow \infty$), for some constant $δ_1> 0 .$ In particular, if $np_n^2 = M\log{n}$ for some constant $M > 0$ large, then $G_E$ is Hamiltonian with high probability; i.e., the longest path contains all the nodes of $G_E.$ In the second part of the paper, we consider a random geometric graph $G_R$ consisting of $n$ nodes, each independently distributed according to a (not necessarily uniform) density $f.$ If $r_n$ is the connectivity radius and $nr_n^2 \longrightarrow \infty,$ then with high probability, the longest cycle contains at least $n-ne^{-δ_2 nr_n^2}$ nodes for some constant $δ_2 > 0.$ As a consequence of our proof, we obtain that if $nr_n^2 = \log{n} + 7\log{\log{n}} + ω_n$ and $ω_n \longrightarrow \infty$ as $n \rightarrow \infty,$ then with high probability $G_R$ contains a Hamiltonian cycle.

math.PR↗

arXiv2017

Cliques and Chromatic Number in Inhomogenous Random Graphs

In this paper, we study cliques and chromatic number of inhomogenous random graphs where the individual edge probabilities could be arbitrarily low. We use a recursive method to obtain estimates on the maximum clique size under a mild positive average edge density assumption. As a Corollary, we also obtain uniform bounds on the maximum clique size and chromatic number for homogenous random graphs for all ranges of the edge probability $p_n$ satisfying $\frac{1}{n^{α_1}} \leq p_n \leq 1-\frac{1}{n^{α_2}}$ for some positive constants $α_1$ and $α_2$.

math.PR↗

arXiv2017

Randomized detection and detection capacity of multidetector networks

In this paper, we study the following detection problem. There are $n$ detectors randomly placed in the unit square $S = \left[-\frac{1}{2},\frac{1}{2}\right]^2$ assigned to detect the presence of a source located at the origin. Time is divided into slots of unit length and $D_i(t) \in \{0,1\}$ represents the (random) decision of the $i^{\rm th}$ detector in time slot $t$. The location of the source is unknown to the detectors and the goal is to design schemes that use the decisions $\{D_i(t)\}_{i,t}$ and detect the presence of the source in as short time as possible. We first determine the minimum achievable detection time $T_{cap}$ and show the existence of \emph{randomized} detection schemes that have detection times arbitrarily close to $T_{cap}$ for almost all configuration of detectors, provided the number of detectors $n$ is sufficiently large. We call such schemes as \emph{capacity achieving} and completely characterize all capacity achieving detection schemes.

math.PR↗