Online Steiner Tree with Deletions

In the online Steiner tree problem, the input is a set of vertices that appear one-by-one, and we have to maintain a Steiner tree on the current set of vertices. The cost of the tree is the total length of edges in the tree, and we want this cost to be close to the cost of the optimal Steiner tree at all points in time. If we are allowed to only add edges, a tight bound of $\Theta(\log n)$ on the competitiveness is known. Recently it was shown that if we can add one new edge and make one edge swap upon every vertex arrival, we can maintain a constant-competitive tree online. But what if the set of vertices sees both additions and deletions? Again, we would like to obtain a low-cost Steiner tree with as few edge changes as possible. The original paper of Imase and Waxman had also considered this model, and it gave a greedy algorithm that maintained a constant-competitive tree online, and made at most $O(n^{3/2})$ edge changes for the first $n$ requests. In this paper give the following two results. Our first result is an online algorithm that maintains a Steiner tree only under deletions: we start off with a set of vertices, and at each time one of the vertices is removed from this set: our Steiner tree no longer has to span this vertex. We give an algorithm that changes only a constant number of edges upon each request, and maintains a constant-competitive tree at all times. Our algorithm uses the primal-dual framework and a global charging argument to carefully make these constant number of changes. We then study the natural greedy algorithm proposed by Imase and Waxman that maintains a constant-competitive Steiner tree in the fully-dynamic model (where each request either adds or deletes a vertex). Our second result shows that this algorithm makes only a constant number of changes per request in an amortized sense.Comment: An extended abstract appears in the SODA 2014 conferenc

Some Efficient Solutions to Yao's Millionaire Problem

We present three simple and efficient protocol constructions to solve Yao's Millionaire Problem when the parties involved are non-colluding and semi-honest. The first construction uses a partially homomorphic Encryption Scheme and is a 4-round scheme using 2 encryptions, 2 homomorphic circuit evaluations (subtraction and XOR) and a single decryption. The second construction uses an untrusted third party and achieves a communication overhead linear in input bit-size with the help of an order preserving function.Moreover, the second construction does not require an apriori input bound and can work on inputs of different bit-sizes. The third construction does not use a third party and, even though, it has a quadratic communication overhead, it is a fairly simple construction.Comment: 17 page

Simulation of optical properties of quantum dots

Nanotechnology devices based on quantum dots have very interesting applications because of the unique characteristics that quantum dots exhibit. In addition to the very small size and operation almost at the speed of light, many important characteristics of quantum dots such as absorption and extinction can be exploited to design devices with a wide range of applications. The study of metallic quantum dots is also a very exciting and interesting field in engineering because of their immense application and relevance to Surface Enhanced Raman Spectroscopy (SERS); In this thesis, the variation in absorption spectra over a changing array of parameters has been studied. The objective is to look for tunability for various materials and sizes of quantum dots. Extinction spectra and electric field intensities over a range of substrates and for various sizes of spherical gold (Au) and silver (Ag) nanoparticles have been simulated. Important parameters such as suitable size of metal nanoparticle, surrounding semiconductor substrate, and wavelength of incident light for achieving highest electric field intensity have been proposed on the basis of simulations

Perturbations in higher derivative gravity beyond maximally symmetric spacetimes

We study (covariant) scalar-vector-tensor (SVT) perturbations of infinite derivative gravity (IDG), at the quadratic level of the action, around conformally-flat, covariantly constant curvature backgrounds which are not maximally symmetric spacetimes (MSS). This extends a previous analysis of perturbations done around MSS, which were shown to be ghost-free. We motivate our choice of backgrounds which arise as solutions of IDG in the UV, avoiding big bang and black hole singularities. Contrary to MSS, in this paper we show that, generically, all SVT modes are coupled to each other at the quadratic level of the action. We consider simple examples of the full IDG action, and illustrate this mixing and also a case where the action can be diagonalized and ghost-free solutions constructed. Our study is widely applicable for both non-singular cosmology and black hole physics where backgrounds depart from MSS. In appendices, we provide SVT perturbations around conformally-flat and arbitrary backgrounds which can serve as a compendium of useful results when studying SVT perturbations of various higher derivative gravity models.Comment: 36 pages, 1 figur

Study of Sericulture & Cocoon Production in Janjgir- Champa Diistrict of Chhattisgarh (India)

Sericulture is a growing business for rural development since it integrates well with farming practices and has the ability to produce lucrative income all year round. It boasts affordable startup costs and offers jobs all year round. Sericulture, a cottage and small-scale industry, is a labor-intensive, economically appealing, and environmentally friendly form of agriculture. Per square meter of land, sericulture produces a lot of work and cash. Sericulture offers many opportunities for improving human resource employability and can successfully slow down population migration to cities. When compared to other crop operations in terms of generating money, sericulture is the most lucrative. The cultivation of mulberries, the generation of silkworm seeds, the rearing of silkworms, the reeling and weaving of silk, the collecting of byproducts, and their processing are all aspects of the sericulture industry that generate a significant amount of work and, consequently, a source of income for rural and tribal people. Sericulture is recognized as a thriving rural sector primarily because it offers families and labor year-round, remunerative employment, and also guarantees periodic income even with tiny land holdings