Related papers: On Deterministically Finding an Element of High Or…

A new algorithm, Guidesort, for sorting in the uniprocessor variant of the parallel disk model (PDM) of Vitter and Shriver is presented. The algorithm is deterministic and executes a number of (parallel) I/O operations that comes within a…

Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most \[ O\left( \frac{N^{1/5}…

The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…

Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an…

Higher-order unification has been shown to be undecidable. Miller discovered the pattern fragment and subsequently showed that higher-order pattern unification is decidable and has most general unifiers. We extend the algorithm to…

Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em…

Grover's algorithm is one of the pioneering demonstrations of the advantages of quantum computing over its classical counterpart, providing - at most - a quadratic speed-up over the classical solution for unstructured database search. The…

Grover's algorithm solves the unstructured search problem. Grover's algorithm can find the target state with certainty only if searching one out of four. Designing the deterministic search algorithm can avoid any repetition of the…

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the…

Withdrawn by the author due to irreparable errors. We present a quantum algorithm that in the black-box model performs a search in an ordered list of N elements. Using 3/4 log N + O(1) queries, it achieves a success probability of at least…

We introduce a new deterministic factoring algorithm, which could be described in the cryptographically fashionable term of "factoring with hints": we show that, given the knowledge of the factorisations of $O(N^{1/3+\epsilon})$ terms…

We want to find a marked element out of a black box containing N elements. When the number of marked elements is known this can be done elegantly with Grover's algorithm, a variant of which even gives a correct result with certainty. On the…

We examine sorting algorithms for $n$ elements whose basic operation is comparing $t$ elements simultaneously (a $t$-comparator). We focus on algorithms that use only a single round or two rounds -- comparisons performed in the second round…

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its…

We introduce an algorithm which, in the context of nonlinear regression on vector-valued explanatory variables, chooses those combinations of vector components that provide best prediction. The algorithm devotes particular attention to…

Grover's algorithm provides a quadratic speedup over classical algorithms to search for marked elements in an unstructured database. The original algorithm is probabilistic, returning a marked element with bounded error. There are several…

When facing a very large stream of data, it is often desirable to extract most important statistics online in a short time and using small memory. For example, one may want to quickly find the most influential users generating posts online…

Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio…

Let $G$ be a finite abelian group $G$ with $N$ elements. In this paper we give a O(N) time algorithm for computing a basis of $G$. Furthermore, we obtain an algorithm for computing a basis from a generating system of $G$ with $M$ elements…

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…