Related papers: LLL Algorithm for Lattice Basis Reduction

The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the…

In 2018, the longest vector problem (LVP) and the closest vector problem (CVP) in $p$-adic lattices were introduced. These problems are closely linked to the orthogonalization process. In this paper, we first prove that every $p$-adic…

The Minimum Linear Arrangement problem (MLA) consists of finding a mapping $\pi$ from vertices of a graph to distinct integers that minimizes $\sum_{\{u,v\}\in E}|\pi(u) - \pi(v)|$. In that setting, vertices are often assumed to lie on a…

We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely…

Instead of minimizing the sum of all $n$ squared residuals as the classical least squares (LS) does, Rousseeuw (1984) proposed to minimize the sum of $h$ ($n/2 \leq h < n$) smallest squared residuals, the resulting estimator is called least…

The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound…

A rectangular layout $\mathcal{L}$ is a rectangle partitioned into disjoint smaller rectangles so that no four smaller rectangles meet at the same point. Rectangular layouts were originally used as floorplans in VLSI design to represent…

The parametric lattice-point counting problem is as follows: Given an integer matrix $A \in Z^{m \times n}$, compute an explicit formula parameterized by $b \in R^m$ that determines the number of integer points in the polyhedron $\{x \in…

Starting from the Maxwell-Juettner equilibrium distribution, we develop a relativistic lattice Boltzmann (LB) algorithm capable of handling ultrarelativistic systems with flat, but expanding, spacetimes. The algorithm is validated through…

Computing the LZ factorization (or LZ77 parsing) of a string is a computational bottleneck in many diverse applications, including data compression, text indexing, and pattern discovery. We describe new linear time LZ factorization…

We improve the error terms of some estimates related to counting lattices from recent work of L. Fukshansky, P. Guerzhoy and F. Luca (2017). This improvement is based on some analytic techniques, in particular on bounds of exponential sums…

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent,…

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem…

We present a poly $\log \log n$ time randomized CONGEST algorithm for a natural class of Lovasz Local Lemma (LLL) instances on constant degree graphs. This implies, among other things, that there are no LCL problems with randomized…

Polynomial factorization over $ZZ$ is of great historical and practical importance. Currently, the standard technique is to factor the polynomial over finite fields first and then to lift to integers. Factorization over finite fields can be…

Lattice-based cryptography is one of the leading proposals for post-quantum cryptography. The Shortest Vector Problem (SVP) is arguably the most important problem for the cryptanalysis of lattice-based cryptography, and many lattice-based…

One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with…

Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms…

We revisit Schnorr's lattice-based integer factorization algorithm, now with an effective point of view. We present effective versions of Theorem 2 of Schnorr's "Factoring integers and computing discrete logarithms via diophantine…

Linear temporal logic (LTL) is a specification language for finite sequences (called traces) widely used in program verification, motion planning in robotics, process mining, and many other areas. We consider the problem of learning LTL…