Related papers: Walks on Primes in Imaginary Quadratic Fields
The Gaussian Moat problem asks whether one can walk to infinity in the Gaussian integers using the Gaussian primes as stepping stones and taking bounded length steps or not. In this paper, we have analyzed the Gaussian primes and also…
An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring…
The Gaussian Moat Problem asks whether it is possible to walk from the origin to infinity in the complex plane using only Gaussian primes as stepstones and steps of bounded length. We prove that this is not possible.
The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the…
In this paper, we have developed an algorithm for the prime searching in $\mathbb{R}^3$. This problem was proposed by M. Das [Arxiv,2019]. This paper is an extension of her work. As we know the distribution of primes will get more irregular…
Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very…
We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set…
We provide a spectrum of new theoretical insights and practical results for finding a Minimum Dilation Triangulation (MDT), a natural geometric optimization problem of considerable previous attention: Given a set $P$ of $n$ points in the…
We present a novel quantum walk-based approach to solve the Minimum Spanning Tree (MST) problem under a maximal degree constraint (MDC). By recasting the classical MST problem as a quantum walk on a graph, where vertices are encoded as…
We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such…
Insight into a number of interesting questions in cosmology can be obtained from the first crossing distributions of physically motivated barriers by random walks with correlated steps. We write the first crossing distribution as a formal…
The list of norm-Euclidean imaginary quadratic fields is known and finite. For each known case, we give a division algorithm that finds a remainder at distance less than the Euclidean minimum of the field.
The distributional analysis of Euclidean algorithms was carried out by Baladi and Vall\'{e}e. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the…
$\newcommand{\eps}{\varepsilon}$ In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path…
We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences…
A trajectory-following primal--dual interior-point method solves nonlinear optimization problems with inequality and equality constraints by approximately finding points satisfying perturbed Karush--Kuhn--Tucker optimality conditions for a…
New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the…
We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTASs) to several well-known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor.…
In root finding and optimization, there are many cases where there is a closed set $A$ one likes that the sequence constructed by one's favourite method will not converge to A (here, we do not assume extra properties on $A$ such as being…
This paper presents a new multi-query motion planning algorithm for linear Gaussian systems with the goal of reaching a Euclidean ball with high probability. We develop a new formulation for ball-shaped ambiguity sets of Gaussian…