Related papers: Divisors in Residue Classes Revisited
Divide-and-conquer is a central paradigm for the design of algorithms, through which some fundamental computational problems, such as sorting arrays and computing convex hulls, are solved in optimal time within $\Theta(n\log{n})$ in the…
The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…
The main purpose of this paper is to study the NP-complete subset-sum problem, not in the usual context of time-complexity-based classification of the algorithms (exponential/polynomial), but through a new kind of algorithmic classification…
A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking…
Differential (Ore) type polynomials with "approximate" polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest…
In this paper we give a polynomial time algorithm to compute $\varphi(N)$ for an RSA module $N$ using as input the order modulo $N$ of a randomly chosen integer. This provides a new insight in the very important problem of factoring an RSA…
Given a set of N points, we have discovered an algorithm that can separate these points from one another by n-dimensional planes. Each point is chosen at random and put into a set S and planes which separate them are determined and put into…
We revisit the problem of rigorously and deterministically finding elements of large order in the multiplicative group of integers modulo a natural number $N$. Solving this problem is an essential step in several recent deterministic…
The fastest known algorithm for factoring a degree $n$ univariate polynomial over a finite field $\mathbb{F}_q$ runs in time $O(n^{3/2 + o(1)}\text{polylog } q)$, and there is a reason to believe that the $3/2$ exponent represents a…
We give the first polynomial-time algorithm for performing linear or polynomial regression resilient to adversarial corruptions in both examples and labels. Given a sufficiently large (polynomial-size) training set drawn i.i.d. from…
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work.…
We present a Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the…
Given a rectangle $R$ with area $A$ and a set of areas $L=\{A_1,...,A_n\}$ with $\sum_{i=1}^n A_i = A$, we consider the problem of partitioning $R$ into $n$ sub-regions $R_1,...,R_n$ with areas $A_1,...,A_n$ in a way that the total…
Suppose we are given an $n$-dimensional order-3 symmetric tensor $T \in (\mathbb{R}^n)^{\otimes 3}$ that is the sum of $r$ random rank-1 terms. The problem of recovering the rank-1 components is possible in principle when $r \lesssim n^2$…
Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…
We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in…
Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of…
The problem of when a given digraph contains a subdivision of a fixed digraph $F$ is considered. Bang-Jensen et al. laid out foundations for approaching this problem from the algorithmic point of view. In this paper we give further support…
We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $\tilde{O}(n^{1/2})$ rounds. In contrast, the previous…