Vertices Of Tree Research Articles

Improved quantum backtracking algorithms using effective resistance estimates

We investigate quantum backtracking algorithms of a type previously introduced by Montanaro (arXiv:1509.02374). These algorithms explore trees of unknown structure, and in certain cases exponentially outperform classical procedures (such as DPLL). Some of the previous work focused on obtaining a quantum advantage for trees in which a unique marked vertex is promised to exist. We remove this restriction and re-characterise the problem in terms of the effective resistance of the search space. To this end, we present a generalisation of one of Montanaro's algorithms to trees containing $k \geq 1$ marked vertices, where $k$ is not necessarily known \textit{a priori}. Our approach involves using amplitude estimation to determine a near-optimal weighting of a diffusion operator, which can then be applied to prepare a superposition state that has support only on marked vertices and ancestors thereof. By repeatedly sampling this state and updating the input vertex, a marked vertex is reached in a logarithmic number of steps. The algorithm thereby achieves the conjectured bound of $\widetilde{\mathcal{O}}(\sqrt{TR_{\mathrm{max}}})$ for finding a single marked vertex and $\widetilde{\mathcal{O}}\left(k\sqrt{T R_{\mathrm{max}}}\right)$ for finding all $k$ marked vertices, where $T$ is an upper bound on the tree size and $R_{\mathrm{max}}$ is the maximum effective resistance encountered by the algorithm. This constitutes a speedup over Montanaro's original procedure in both the case of finding one and finding multiple marked vertices in an arbitrary tree. If there are no marked vertices, the effective resistance becomes infinite, and we recover the scaling of Montanaro's existence algorithm.

Recovering normal networks from shortest inter-taxa distance information.

Phylogenetic networks are a type of leaf-labelled, acyclic, directed graph used by biologists to represent the evolutionary history of species whose past includes reticulation events. A phylogenetic network is tree-child if each non-leaf vertex is the parent of a tree vertex or a leaf. Up to a certain equivalence, it has been recently shown that, under two different types of weightings, edge-weighted tree-child networks are determined by their collection of distances between each pair of taxa. However, the size of these collections can be exponential in the size of the taxa set. In this paper, we show that, if we have no "shortcuts", that is, the networks are normal, the same results are obtained with only a quadratic number of inter-taxa distances by using the shortest distance between each pair of taxa. The proofs are constructive and give cubic-time algorithms in the size of the taxa sets for building such weighted networks.

On the Trees With Maximal Augmented Zagreb Index

The augmented Zagreb index (AZI), a variant of the well-known atom-bond connectivity (ABC) index, was shown to have the best predicting ability for a variety of physicochemical properties among several tested vertex-degree-based topological indices. However, contrasting to the extensive research on Problem A: characterizing $n$ -vertex tree(s) with minimal ABC index, few works have been done on Problem B: characterizing $n$ -vertex tree(s) with maximal AZI. Ali and Bhatti conjectured that a tree has maximal AZI iff it has minimal ABC index, with the implication that Problem B is as difficult as Problem A. In this paper, we first prove that among connected graphs with given degree sequence, there exists a breadth-first searching graph maximizing the AZI. Using this, an efficient algorithm based on tree degree sequences is designed to search the $n$ -vertex tree(s) with maximal AZI up to $n=200$ . We find that the balanced double star uniquely maximizes the AZI for $19\le n\le 200$ , and consequently, we disprove the aforementioned conjecture posed by Ali and Bhatti. Naturally, the balanced double star is conjectured to be the unique tree with maximal AZI among $n$ -vertex trees for $n\ge 19$ . Toward our conjecture, we prove that all the pendent paths are of length 1 in an $n$ -vertex tree with maximal AZI if $n\ge 19$ .

Co-Periodicity Isomorphisms between Forests of Finite <I>p</I>-Groups

Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.

On the reduced second Zagreb index of trees

The current note is devoted to investigate the trees, which maximize or minimize the reduced second Zagreb index among all [Formula: see text]-vertex trees with fixed number of segments. This note also involves development of some results, which may be used to characterize the extremal trees with respect to the aforementioned index among all [Formula: see text]-vertex trees having fixed number of branching vertices.

Column planarity and partially-simultaneous geometric embedding

We introduce the notion of column planarity of a subset $R$ of the vertices of a graph $G$. Informally, we say that $R$ is column planar in $G$ if we can assign $x$-coordinates to the vertices in $R$ such that any assignment of $y$-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of $G$. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for the maximum size of column planar subsets of trees: every tree on $n$ vertices contains a column planar set of size at least $14n/17$ and for any $\epsilon > 0$ and any sufficiently large $n$, there exists an $n$-vertex tree in which every column planar subset has size at most $(5/6 + \epsilon)n$. In addition, we show that every outerplanar graph has a column planar set of size at least $n/2$. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (PSGE). A PSGE of two graphs $G_1$ and $G_2$ allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct $k$-PSGEs, which are PSGEs in which at least $k$ vertices are mapped to the same point for both graphs. In particular, we show that every two trees on $n$ vertices admit an $11n/17$-PSGE and every two outerplanar graphs admit an $n/4$-PSGE.

Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees.

SummaryEvolutionary relationships are represented by phylogenetic trees, and a phylogenetic analysis of gene sequences typically produces a collection of these trees, one for each gene in the analysis. Analysis of samples of trees is difficult due to the multi-dimensionality of the space of possible trees. In Euclidean spaces, principal component analysis is a popular method of reducing high-dimensional data to a low-dimensional representation that preserves much of the sample’s structure. However, the space of all phylogenetic trees on a fixed set of species does not form a Euclidean vector space, and methods adapted to tree space are needed. Previous work introduced the notion of a principal geodesic in this space, analogous to the first principal component. Here we propose a geometric object for tree space similar to the n}{}kth principal component in Euclidean space: the locus of the weighted Fréchet mean of n}{}k+1 vertex trees when the weights vary over the n}{}k-simplex. We establish some basic properties of these objects, in particular showing that they have dimension n}{}k, and propose algorithms for projection onto these surfaces and for finding the principal locus associated with a sample of trees. Simulation studies demonstrate that these algorithms perform well, and analyses of two datasets, containing Apicomplexa and African coelacanth genomes respectively, reveal important structure from the second principal components.

Constructing Tree-Child Networks from Distance Matrices

A tree-child network is a phylogenetic network with the property that each non-leaf vertex is the parent of a tree vertex or a leaf. In this paper, we show that a tree-child network on taxa (leaf) set X with an outgroup and a positive real-valued weighting of its edges is essentially determined by the multi-set of all path-length distances between elements in X provided, for each reticulation, the edges directed into it have equal weight. Furthermore, we give a polynomial-time algorithm for reconstructing such a network from this inter-taxa distance information. Such constructions are of central importance in evolutionary biology where phylogenetic networks represent the ancestral history of a collection of present-day taxa.

Bounded-Degree Spanning Trees in Randomly Perturbed Graphs

We show that for any fixed dense graph $G$ and bounded-degree tree $T$ on the same number of vertices, a modest random perturbation of $G$ will typically contain a copy of $T$. This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c=c(\alpha,\Delta)$ such that if $G$ is an $n$-vertex graph with minimum degree at least $\alpha n$, and $T$ is an $n$-vertex tree with maximum degree at most $\Delta$, then if we add $cn$ uniformly random edges to $G$, the resulting graph will contain $T$ asymptotically almost surely (as $n\to\infty$). Our proof uses a lemma concerning the decomposition of a dense graph into superregular pairs of comparable sizes, which may be of independent interest.

Spotting Trees with Few Leaves

We show two results related to finding trees and paths in graphs. First, we show that in $O^*(1.657^k2^{l/2})$ time one can either find a $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph or conclude that such a tree does not exist. Our solution can be applied as a subroutine to solve the $k$-Internal Spanning Tree problem in $O^*(min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the running time can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for Hamiltonicity and $k$-Path in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8. Our results extend the technique by Bjorklund [SIAM J. Comput., 43 (2014), pp. 280--299] and Bjorklund et al...

Planning Metering for Power Distribution Systems Monitoring with Topological Reconfiguration

This paper presents a methodology for the optimal meter placement in real-time power distribution systems monitoring with state estimation (SE), taking into account aspects such as observability; absence of critical data; possible changes in the network configuration. It is shown that this problem is considered as a tree vertex cover problem and can be solved in O(n) polynomial time. To accomplish this task, a simple, practical value and constructive Heuristic for RTUs and Measurements Planning (HRMP) algorithm that consider inherent characteristics of the meter placement problem is proposed. Simulation results show a reduction of up to 5 units of remote transmission in the 37-B system with topology reconfiguration, when compared with the solution obtained by metaheuristic Genetic algorithm, resulting in a reduction cost (up to 20.4 %) of the measurement system. As a problem solved in O(n) polynomial time, we achieve a reduction of up to 207,900 % in computational time, when compared with the resolution via metaheuristic with \(O(n^{3})\).

HORTON LAW IN SELF-SIMILAR TREES

Self-similarity of random trees is related to the operation of pruning. Pruning [Formula: see text] cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton–Strahler order of a vertex [Formula: see text] and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at [Formula: see text]. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers [Formula: see text] and [Formula: see text] are defined as the expected number of branches of order [Formula: see text], and the expected number of order-[Formula: see text] branches that merged order-[Formula: see text] branches, [Formula: see text], respectively, in a finite tree of order [Formula: see text]. The Tokunaga coefficients are defined as [Formula: see text]. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: [Formula: see text]. We show that for self-similar trees, the condition [Formula: see text] is necessary and sufficient for the existence of the strong Horton law: [Formula: see text], as [Formula: see text] for some [Formula: see text] and every [Formula: see text]. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.

A statistic related to trees and words on a finite alphabet

Let [Formula: see text] denote the set of trees on [Formula: see text] labeled vertices. In this paper, we consider the statistic on [Formula: see text] which records the number of vertices of odd degree and count trees having a prescribed number of such vertices. In doing so, we make use of a well-known bijection between members of [Formula: see text] and the set of [Formula: see text]-ary words of length [Formula: see text]. In fact, we are able to study a generalization mod [Formula: see text] of the aforementioned statistic for all [Formula: see text]. In the cases when [Formula: see text] and [Formula: see text], further enumerative results for both words and trees may be given.

Artin Transfer Patterns on Descendant Trees of Finite p-Groups

Based on a thorough theory of the Artin transfer homomorphism from a group G to the abelianization of a subgroup of finite index , and its connection with the permutation representation and the monomial representation of G, the Artin pattern , which consists of families , resp. , of transfer targets, resp. transfer kernels, is defined for the vertices of any descendant tree T of finite p-groups. It is endowed with partial order relations and , which are compatible with the parent-descendant relation of the edges of the tree T. The partial order enables termination criteria for the p-group generation algorithm which can be used for searching and identifying a finite p-group G, whose Artin pattern is known completely or at least partially, by constructing the descendant tree with the abelianization of G as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns and explaining the stabilization, resp. polarization, of their components in descendant trees T of finite p-groups.

A GRASP for the Steiner Tree Problem in Graphs to Support Multicast Routing

In computer networks, multicast routing is that in which a single message is transmitted to multiple recipients from a subset of devices, called multicast group. To address the multicast routing problem, it is usual to model the computer network as a graph. In line with this, some strategies aim at building a routing tree that minimizes the total costs of transmission between the vertices of the routing tree that contains the multicast group, the Steiner tree. However, in large scale applications, local-based strategies may be a better alternative, since they can also be easily adapted to tackle distributed systems. In the literature, to the extent of our knowledge, to construct approximate Steiner tree using this type of approach remains a challenge. In this paper, we present a neighborhood-based metaheuristic to find an approximate solution for the Steiner tree in graphs. In the computational experiments, we used benchmark instances of small, medium and large scales and compared the results achieved by the proposed strategy with the best found so far. In addition, we analyze a real word instance for which the proposed algorithm has obtained good performance.

An estimate of the objective function optimum for the network Steiner problem

A complete weighted graph, \(G(X,\varGamma ,W)\), is considered. Let \(\tilde{X}\subset X\) be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes set \(\tilde{X}\). The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of vertices \(\tilde{X}\) The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one.

The rainbow vertex-index of complementary graphs

A vertex-colored graph $G$ is \emph{rainbow vertex-connected} if two vertices are connected by a path whose internal vertices have distinct colors. The \emph{rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u,v$ of distinct vertices, $G$ contains a vertex-rainbow $u-v$ geodesic, then $G$ is \emph{strongly rainbow vertex-connected}. The minimum $k$ for which there exists a $k$-coloring of $G$ that results in a strongly rainbow-vertex-connected graph is called the \emph{strong rainbow vertex number} $srvc(G)$ of $G$. Thus $rvc(G)\leq srvc(G)$ for every nontrivial connected graph $G$. A tree $T$ in $G$ is called a \emph{rainbow vertex tree} if the internal vertices of $T$ receive different colors. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{rainbow vertex $S$-tree} if the internal vertices of $T$ receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of $G$ such that there is a rainbow vertex $S$-tree for every $k$-set $S$ of $V(G)$ is called the {\it $k$-rainbow vertex-index} of $G$, denoted by $rvx_k(G)$. In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The $k$-rainbow vertex-index of complementary graphs are also studied.

Dynamic Maintenance of a Shortest-Path Tree on Homogeneous Batches of Updates

A dynamic graph algorithm is called batch if it is able to update efficiently the solution of a given graph problem after multiple updates at a time (i.e., a batch) take place on the input graph. In this article, we study batch algorithms for maintaining a single-source shortest-path tree in graphs with positive real edge weights. In particular, we focus our attention on homogeneous batches, that is, either incremental (containing only edge insertion and weight decrease operations) or decremental (containing only edge deletion and weight increase operations) batches, which model realistic dynamic scenarios like transient vertex failures in communication networks and traffic congestion/decongestion phenomena in road networks. We propose two new algorithms to process either incremental or decremental batches, respectively, and a combination of these two algorithms that is able to process arbitrary sequences of incremental and decremental batches. All these algorithms are update sensitive ; namely, they are efficient with respect to the number of vertices in the shortest-path tree that change their parents and/or their distances from the source as a consequence of a batch. This makes unfeasible an effective comparison on a theoretical basis of our new algorithms with the solutions known in the literature, which in turn are analyzed with respect to others and different parameters. For this reason, in order to evaluate the quality of our approach, we provide also an extensive experimental study including our new algorithms and the most efficient previous batch algorithms. Our experimental results complement previous studies and show that the various solutions can be consistently ranked on the basis of the type of homogeneous batch and of the underlying network. As a result, our work can be helpful in selecting a proper solution depending on the specific application scenario.

Balance Layout Problem for 3D-Objects: Mathematical Model and Solution Methods

The paper introduces a general mathematical model of the optimal layout of 3D-objects (full-spheres, right circular cylinders, right regular prisms, and right rectangular parallelepipeds) in a container (straight circular cylinder, paraboloid of revolution, truncated circular cone) with circular racks. The model takes into account the minimum and maximum admissible distances between objects as well as the behavior constraints of the mechanical system (equilibrium, moments of inertia, and stability constraints). We propose solution methods based on Shor's r-algorithm, multistart algorithm, and accelerated search of terminal vertices of the decision tree.

Randomized fault-tolerant virtual backbone tree to improve the lifetime of wireless sensor networks

Backbone nodes are effective for routing in wireless networks because they reduce the energy consumption in sensor nodes. Packet delivery only occurs through the backbone nodes, which depletes the energy in the backbone drastically. Several backbone construction algorithms, including energy-aware virtual backbone tree, virtual backbone tree algorithm for minimal energy consumption and multihop cluster-based stable backbone tree, fail to form a complete backbone when converting important nodes, such as a cut vertex tree node to a non-backbone node. Thus we propose a fault-tolerant virtual backbone tree (FTVBT) algorithm that addresses all of these conflicts and we give theoretical derivations of the bounds for the probability that a sensor node can connect with the backbone. Furthermore, randomized FTVBT improves FTVBT by redistributing non-tree nodes randomly among all the eligible tree nodes based on their fitness values, thereby decreasing the rapid depletion of energy in a particular node and increasing the network lifetime. We performed simulations in NS2 and analyzed the experimental results.