Related papers: Distinguishing virtual braids in polynomial time

The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive…

We consider the following problem: given an unsorted array of $n$ elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which uses O(n) space…

In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.

Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem of deciding if two phylogenetic networks embed the same sets of phylogenetic trees is computationally hard, \blue{in…

A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of $O(\log n)$ bits, where $n$ is the size of the input. We assume that a simple $n$-gon is given by the ordered sequence of its…

This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple…

For $n$-vertex graphs with treewidth $k = O(n^{1/2-\epsilon})$ and an arbitrary $\epsilon>0$, we present a word-RAM algorithm to compute vertex separators using only $O(n)$ bits of working memory. As an application of our algorithm, we give…

For an arbitrary word $w$ on an alphabet, we can define the alternating symbol graph, $G(w)$, as the graph in which the edge $(a, b)$ is in $E$ iff the letters $a$ and $b$ alternate in the word $w$. A graph $G = (V, E)$ is said to be…

A graph separator is a subset of vertices of a graph whose removal divides the graph into small components. Computing small graph separators for various classes of graphs is an important computational task. In this paper, we present a…

Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure…

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an $n\times n$ matrix over a finite field that requires $O(n^3)$ field operations and O(n) random vectors, and is well suited for successful practical…

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations…

Let $V$ be the set of real common solutions to $F = (f_1, \ldots, f_s)$ in $\mathbb{R}[x_1, \ldots, x_n]$ and $D$ be the maximum total degree of the $f_i$'s. We design an algorithm which on input $F$ computes the dimension of $V$. Letting…

We investigate analytically the problem of enumeration of nonequivalent primitive words in the braid group B_n for n >> 1 by analysing the random word statistics and the target space on the basis of the locally free group approximation. We…

Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $\pi\in S_n$ such that $\chi_\lambda(\pi)$ is provably different…

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of…

We construct an algorithm to decide whether two given Legendrian or transverse links are equivalent. In general, the complexity of the algorithm is too high for practical implementation. However, in many cases, when the symmetry group of…

We use Kauffman's bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the…

Classically, the edit distance of two length-$n$ strings can be computed in $O(n^2)$ time, whereas an $O(n^{2-\epsilon})$-time procedure would falsify the Orthogonal Vectors Hypothesis. If the edit distance does not exceed $k$, the running…

Graph isomorphism is an important computer science problem. The problem for the general case is unknown to be in polynomial time. The base algorithm for the general case works in quasi-polynomial time. The solutions in polynomial time for…