Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. We give an efficient, linear-time approximation algorithm, which estimates the rotation distance, within a provable factor of 2, between ordered rooted binary trees. .
Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a single rotation. There are no known efficient algorithms to compute rotation distance between trees and thus distances in rotation graphs. Limiting the allowed locations of where rotations are permitted gives rise to a number of notions of restricted rotation distances. Allowing rotations at a minimal such set of locations gives restricted rotation distance…
Rotation distance between rooted binary trees is the minimum number of simple rotations needed to transform one tree into the other. Computing the rotation distance between a pair of rooted trees can be quickly reduced in cases where there is a common edge between the trees, or where a single rotation introduces a common edge. Tree pairs which do not have such a reduction are difficult tree pairs, where there is no generally known first step. Here, we describe efforts to count and estimate the number of such difficult tree pairs, and find that the fraction decreases exponentially fast toward z…
Tree rotations (left and right) are basic local deformations allowing to transform between two unlabeled binary trees of the same size. Hence, there is a natural problem of practically finding such transformation path with low number of rotations, the optimal minimal number is called the rotation distance. Such distance could be used for instance to quantify similarity between two trees for various machine learning problems, for example to compare hierarchical clusterings or arbitrarily chosen spanning trees of two graphs, like in SMILES notation popular for describing chemical molecules. Ther…
We show that the edit distance between two strings of length $n$ can be computed within a factor of $f(ε)$ in $n^{1+ε}$ time as long as the edit distance is at least $n^{1-δ}$ for some $δ(ε) > 0$.
Due to hybridization events in evolution, studying two different genes of a set of species may yield two related but different phylogenetic trees for the set of species. In this case, we want to measure the dissimilarity of the two trees. The rooted subtree prune and regraft (rSPR) distance of the two trees has been used for this purpose. The problem of computing the rSPR distance of two given trees has many applications but is unfortunately NP-hard. The previously best approximation algorithm for rSPR distance achieves a ratio of 2.5 and it was open whether a better approximation algorithm fo…
The min-distance between two nodes $u, v$ is defined as the minimum of the distance from $v$ to $u$ or from $u$ to $v$, and is a natural distance metric in DAGs. As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in $\tilde{O}(mn)$ time. So it is natural to resort to approximation algorithms in $\tilde{O}(mn^{1-ε})$ time for some positive $ε$. Abboud, Vassilevska W., and Wang [SODA …
It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees. The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step which is guaranteed to be the beginning of a minimal length path. Of interest, therefore, is how to sample such difficult pairs of trees of a fixed size. We show that it is possible to do so effic…
Given a pair of strings, the problems of computing their Longest Common Subsequence and Edit Distance have been extensively studied for decades. For exact algorithms, LCS and Edit Distance (with character insertions and deletions) are equivalent; the state of the art running time is (almost) quadratic and this is tight under plausible fine-grained complexity assumptions. But for approximation algorithms the picture is different: there is a long line of works with improved approximation factors for Edit Distance, but for LCS (with binary strings) only a trivial $1/2$-approximation was known. In…
The edit distance is a basic string similarity measure used in many applications such as text mining, signal processing, bioinformatics, and so on. However, the computational cost can be a problem when we repeat many distance calculations as seen in real-life searching situations. A promising solution to cope with the problem is to approximate the edit distance by another distance with a lower computational cost. There are, indeed, many distances have been proposed for approximating the edit distance. However, their approximation accuracies are evaluated only theoretically: many of them are ev…
Maximum parsimony distance is a measure used to quantify the dissimilarity of two unrooted phylogenetic trees. It is NP-hard to compute, and very few positive algorithmic results are known due to its complex combinatorial structure. Here we address this shortcoming by showing that the problem is fixed parameter tractable. We do this by establishing a linear kernel i.e., that after applying certain reduction rules the resulting instance has size that is bounded by a linear function of the distance. As powerful corollaries to this result we prove that the problem permits a polynomial-time consta…
A distance-approximation algorithm for a graph property $\mathcal{P}$ in the adjacency-matrix model is given an approximation parameter $ε\in (0,1)$ and query access to the adjacency matrix of a graph $G=(V,E)$. It is required to output an estimate of the \emph{distance} between $G$ and the closest graph $G'=(V,E')$ that satisfies $\mathcal{P}$, where the distance between graphs is the size of the symmetric difference between their edge sets, normalized by $|V|^2$. In this work we introduce property covers, as a framework for using distance-approximation algorithms for "simple" properties to d…
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally treewidth-fragile graph classes, a property that is true for many natural graph classes with sublinear separators. We also provide quasipolynomial-time approximation schemes for these problems in all classes with sublinear separators.
Evolution of large scale networks demand for efficient way of communication in the networks. One way to propagate information in the network is to find vertex cover. In this paper we describe a variant of vertex cover problem naming it N-distance Vertex Minimal Cover(N-MVC) Problem to optimize information propagation throughout the network. A minimum subset of vertices of a unweighted and undirected graph G = (V, E) is called N-MVC if for all v in V , v is at distance less than or equal to N from at least one of the the vertices in N-MVC. In the following paper, this problem is defined, formul…
