An O(M(n) log n) algorithm for the Jacobi symbol

The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n) log n), using Sch\"onhage's fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n) log n) algorithm based on the binary recursive gcd algorithm of Stehl\'e and Zimmermann. Our implementation - which to our knowledge is the first to run in time O(M(n) log n) - is faster than GMP's quadratic implementation for inputs larger than about 10000 decimal digits.Comment: Submitted to ANTS IX (Nancy, July 2010

How Fast Can We Multiply Large Integers on an Actual Computer?

We provide two complexity measures that can be used to measure the running time of algorithms to compute multiplications of long integers. The random access machine with unit or logarithmic cost is not adequate for measuring the complexity of a task like multiplication of long integers. The Turing machine is more useful here, but fails to take into account the multiplication instruction for short integers, which is available on physical computing devices. An interesting outcome is that the proposed refined complexity measures do not rank the well known multiplication algorithms the same way as the Turing machine model.Comment: To appear in the proceedings of Latin 2014. Springer LNCS 839

Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems

We introduce a new problem that combines the well known All Pairs Shortest Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to compute the shortest paths for all pairs of vertices for all possible flow amounts. We call this new problem the All Pairs Shortest Paths for All Flows (APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with unit edge costs and real edge capacities in $\tilde{O}(\sqrt{t}n^{(\omega+9)/4}) = \tilde{O}(\sqrt{t}n^{2.843})$ time, where $n$ is the number of vertices, $t$ is the number of distinct edge capacities (flow amounts) and $O(n^{\omega}) < O(n^{2.373})$ is the time taken to multiply two $n$-by-$n$ matrices over a ring. Secondly we extend the problem to graphs with positive integer edge costs and present an algorithm with $\tilde{O}(\sqrt{t}c^{(\omega+5)/4}n^{(\omega+9)/4}) = \tilde{O}(\sqrt{t}c^{1.843}n^{2.843})$ worst case time complexity, where $c$ is the upper bound on edge costs

Chains of large gaps between primes

Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that $$G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that $$G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.Comment: 16 pages, no figure

A one-dimensional Vlasov-Maxwell equilibrium for the force-free Harris sheet

In this paper the first non-linear force-free Vlasov-Maxwell equilibrium is presented. One component of the equilibrium magnetic field has the same spatial structure as the Harris sheet, but whereas the Harris sheet is kept in force balance by pressure gradients, in the force-free solution presented here force balance is maintained by magnetic shear. Magnetic pressure, plasma pressure and plasma density are constant. The method used to find the equilibrium is based on the analogy of the one-dimensional Vlasov-Maxwell equilibrium problem to the motion of a pseudo-particle in a two-dimensional conservative potential. This potential is equivalent to one of the diagonal components of the plasma pressure tensor. After finding the appropriate functional form for this pressure tensor component, the corresponding distribution functions can be found using a Fourier transform method. The force-free solution can be generalized to a complete family of equilibria that describe the transition between the purely pressure-balanced Harris sheet to the force-free Harris sheet.Comment: 10 pages, 2 figures, submitted to PRL, revised versio

Gradual sub-lattice reduction and a new complexity for factoring polynomials

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well

Computing with and without arbitrary large numbers

In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some problems. However, this has only been shown so far for specific problems. We provide a characterization of the power of such extra inputs for general problems. To do so, we first correct a classical result by Simon and Szegedy (1992) as well as one by Simon (1981). In the former we show mistakes in the proof and correct these by an entirely new construction, with no great change to the results. In the latter, the original proof direction stands with only minor modifications, but the new results are far stronger than those of Simon (1981). In both cases, the new constructions provide the theoretical tools required to characterize the power of arbitrary large numbers.Comment: 12 pages (main text) + 30 pages (appendices), 1 figure. Extended abstract. The full paper was presented at TAMC 2013. (Reference given is for the paper version, as it appears in the proceedings.

Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant $C$ such that $\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on $n$ vertices with independence number $\alpha$.Comment: 14 page