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Register a new userInset Edges Effect and Average Distance of Trees
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Added by
M.H. Khalifeh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. When k = 1 the complexity of the problem is polynomial. In this paper, we further find the single inset edge(s) of a tree with the closest change on the average distance to a given input. To do that we may require the effect of each inset edge for the set of inset edges. For this, we propose an algorithm with the time complexity between O(m) and O(m/m) and an average of less than O( m.log(m)), where m stands for the number of possible inset edges. Then it takes up to O(log(m)) to find the target inset edges for a custom change on the average distance. Using theoretical tools, the algorithm strictly avoids recalculating the distances with no changes, after adding a new edge to a tree. Then reduces the time complexity of calculating remaining distances using some matrix tools which first introduced in [8] with one additional technique. This gives us a dynamic time complexity and absolutely depends on the input tree which is proportion to the Wiener index of the input tree.
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An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. However, when k = 1 the complexity of the problem is polynomial. In this paper, some tools for a precise analysis of the problem for the trees are established. Using the tools, we can avoid using the distance matrix. This leads to more efficient algorithms and a better analysis of the problem. Several applications of the tools as well as a tight bound for the change of average distance when an inset edge is added to a tree are presented.
PQ-trees and PC-trees are data structures that represent sets of linear and circular orders, respectively, subject to constraints that specific subsets of elements have to be consecutive. While equivalent to each other, PC-trees are conceptually much simpler than PQ-trees; updating a PC-trees so that a set of elements becomes consecutive requires only a single operation, whereas PQ-trees use an update procedure that is described in terms of nine transformation templates that have to be recursively matched and applied. Despite these theoretical advantages, to date no practical PC-tree implementation is available. This might be due to the original description by Hsu and McConnell in some places only sketching the details of the implementation. In this paper, we describe two alternative implementations of PC-trees. For the first one, we follow the approach by Hsu and McConnell, filling in the necessary details and also proposing improvements on the original algorithm. For the second one, we use a different technique for efficiently representing the tree using a Union-Find data structure. In an extensive experimental evaluation we compare our implementations to a variety of other implementations of PQ-trees that are available on the web as part of academic and other software libraries. Our results show that both PC-tree implementations beat their closest fully correct competitor, the PQ-tree implementation from the OGDF library, by a factor of 2 to 4, showing that PC-trees are not only conceptually simpler but also fast in practice. Moreover, we find the Union-Find-based implementation, while having a slightly worse asymptotic runtime, to be twice as fast as the one based on the description by Hsu and McConnell.
Tree comparison metrics have proven to be an invaluable aide in the reconstruction and analysis of phylogenetic (evolutionary) trees. The path-length distance between trees is a particularly attractive measure as it reflects differences in tree shape as well as differences between branch lengths. The distance equals the sum, over all pairs of taxa, of the squared differences between the lengths of the unique path connecting them in each tree. We describe an $O(n log n)$ time for computing this distance, making extensive use of tree decomposition techniques introduced by Brodal et al. (2004).
We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(log log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.
We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise $epsilon$-close to the uniform distribution, in an emph{amortized-efficient} fashion. We consider the adjacency list query model, where access to a graph $G$ is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let $n$ and $m$ denote the number of vertices and edges of $G$, respectively. Eden and Rosenbaum provided upper and lower bounds of $Theta^*(n/sqrt m)$ for sampling a single edge in general graphs (where $O^*(cdot)$ suppresses $textrm{poly}(1/epsilon)$ and $textrm{poly}(log n)$ dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples $q$ in advance, has an overall cost that is sublinear in $q$, namely, $O^*(sqrt q cdot(n/sqrt m))$, which is strictly preferable to $O^*(qcdot (n/sqrt m))$ cost resulting from $q$ invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tv{e}tek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.
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