Refined algorithms to compute syzygies

Algorithms to Compute the Lyndon Array

In the Lyndon array λ = λx[1..n] of a string x = x[1..n], λ[i] is the length of the longest Lyndon word starting at position i of x. The computation of λ has recently become of great interest, since it was shown (Bannai et al., The “Runs” Theorem [2]) that the runs in x are computable in linear time from λx. Here we first describe three algorithms for computing λx that have been suggested in th...

Eleven Simple Algorithms to Compute Fibonacci Numbers

The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used in introductory algorithms courses. In this paper, we present eleven of these well-known algorithms and some of their properties. These algorithms, though ver...

Signature-based algorithms to compute standard bases

Signature-based algorithms to compute Groebner bases

This paper describes a Buchberger-style algorithm to compute a Gröbner basis of a polynomial ideal, allowing for a selection strategy based on " signatures ". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and...

How to Compute Times of Random Walks Based Distributed Algorithms

Random walk based distributed algorithms make use of a token that circulates in the system according to a random walk scheme to achieve their goal. To study their efficiency and compare it to one of the deterministic solutions, one is led to compute certain quantities, namely the hitting times and the cover time. Until now, only bounds on these quantities were defined. First, this paper present...