Related papers: The Accelerated Euclidean Algorithm

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…

We consider the problem of inserting a new item into an ordered list of N-1 items. The length of an algorithm is measured by the number of comparisons it makes between the new item and items already on the list. Classically, determining the…

One of the great triumphs in the history of numerical methods was the discovery of the Conjugate Gradient (CG) algorithm. It could solve a symmetric positive-definite system of linear equations of dimension N in exactly N steps. As many…

This paper presents a quantum algorithm that computes the product of two $n\times n$ Boolean matrices in $\tilde O(n\sqrt{\ell}+\ell\sqrt{n})$ time, where $\ell$ is the number of non-zero entries in the product. This improves the previous…

For vertices $u$ and $v$ of an $n$-vertex graph $G$, a $uv$-trail of $G$ is an induced $uv$-path of $G$ that is not a shortest $uv$-path of $G$. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known…

Let $V$ be a set of $n$ points in the plane. The unit-disk graph $G = (V, E)$ has vertex set $V$ and an edge $e_{uv} \in E$ between vertices $u, v \in V$ if the Euclidean distance between $u$ and $v$ is at most 1. The weight of each edge…

This paper deals with circulant matrices. It is shown that a circulant matrix can be multiplied by a vector in time O(n log(n)) in a ring with roots of unity without making use of an FFT algorithm. With our algorithm we achieve a speedup of…

We give an algorithm for counting self-avoiding walks or self-avoiding polygons that runs in time $\exp(C\sqrt{n\log n})$ on 2-dimensional lattices and time $\exp(C_dn^{(d-1)/d}\log n)$ on $d$-dimensional lattices for $d>2$.

We propose k^2-means, a new clustering method which efficiently copes with large numbers of clusters and achieves low energy solutions. k^2-means builds upon the standard k-means (Lloyd's algorithm) and combines a new strategy to accelerate…

This paper proposes a new algorithm for reducing Approximate Nearest Neighbor problem to Approximate Near Neighbor problem. The advantage of this algorithm is that it achieves O(log n) query time. As a reduction problem, the uery time…

A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…

The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their…

In this paper, we suggest a new efficient algorithm in order to compute S-polynomial reduction rapidly in the known algorithm for computing Grobner bases, and compare the complexity with others.

We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree $n$ over a finite field $\F_q$, the average-case complexity of our algorithm is an expected $O(n^{1 + o(1)} \log^{2 +…

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…

A new algorithms for computing discrete logarithms on elliptic curves defined over finite fields is suggested. It is based on a new method to find zeroes of summation polynomials. In binary elliptic curves one is to solve a cubic system of…

We describe a new algorithm that computes the n-th Bernoulli number in n^(4/3 + o(1)) bit operations. This improves on previous algorithms that had complexity n^(2 + o(1)).

It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…

An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…

This note introduces a new class of integer factoring algorithms. Two versions of this method will be described, deterministic and probabilistic. These algorithms are practical, and can factor large classes of balanced integers N = pq, p <…