Arbitrary Tree Research Articles

The computational complexity of calculating partition functions of optimal medians with Hamming distance

We study the complexity of computing the partition function of medians for binary strings with Hamming distance using various weight functions. When the weight function is the factorial function, this partition function has application in bioinformatics, counting the most parsimonious scenarios on a star tree under the Single Cut-or-Join model for genome rearrangement. Although this model is computationally simple, we show that it is #P-complete to compute the partition function. Our results are also extended to binary trees as we show that it is #P-complete to calculate the most parsimonious scenarios on an arbitrary binary tree under the Single Cut-or-Join model. These results also apply to substitution models for many biological sequences.

Statistical shape analysis of simplified neuronal trees

Neuron morphology plays a central role in characterizing cognitive health and functionality of brain structures. The problem of quantifying neuron shapes and capturing statistical variability of shapes is difficult because neurons differ both in geometry and in topology. This paper develops a mathematical representation of neuronal trees, restricting to the trees that consist of: (1) a main branch viewed as a parameterized curve in $\mathbb{R}^{3}$, and (2) some number of secondary branches—also parameterized curves in $\mathbb{R}^{3}$—which emanate from the main branch at arbitrary points. It imposes a metric on the representation space, in order to compare neuronal shapes, and to obtain optimal deformations (geodesics) across arbitrary trees. The key idea is to impose certain equivalence relations that allow trees with different geometries and topologies to be compared efficiently. The combinatorial problem of matching side branches across trees is reduced to a linear assignment with well-known efficient solutions. This framework is then applied to comparing, clustering, and classifying neurons using fully automated algorithms. The framework is illustrated on three datasets of neuron reconstructions, specifically showing geodesics paths and cross-validated classification between experimental groups.

Combinatorial Scoring of Phylogenetic Trees and Networks Based on Homoplasy-Free Characters.

Construction of phylogenetic trees and networks for extant species from their characters represents one of the key problems in phylogenomics. While solution to this problem is not always uniquely defined and there exist multiple methods for tree/network construction, it becomes important to measure how well the constructed networks capture the given character relationship across the species. We propose a novel method for measuring the specificity of a given phylogenetic network in terms of the total number of distributions of homoplasy-free character states at the leaves that the network may impose. While for binary phylogenetic trees, this number has an exact formula and depends only on the number of leaves and character states but not on the tree topology, the situation is much more complicated for nonbinary trees or networks. Nevertheless, we develop an algorithm for combinatorial enumeration of such distributions, which is applicable for arbitrary trees and networks under some reasonable assumptions. We further extend our algorithm to a special class of characters that follow Dollo's law of irreversibility.

Random self-similar trees and a hierarchical branching process

We study self-similarity in random binary rooted trees. In a well-understood case of Galton–Watson trees, a distribution on a space of trees is said to be self-similar if it is invariant with respect to the operation of pruning, which cuts the tree leaves. This only happens for the critical Galton–Watson tree (a constant process progeny), which also exhibits other special symmetries. We extend the prune-invariance setup to arbitrary binary trees with edge lengths. In this general case the class of self-similar processes becomes much richer and covers a variety of practically important situations. The main result is construction of the hierarchical branching processes that satisfy various self-similarity definitions (including mean self-similarity and self-similarity in edge-lengths) depending on the process parameters. Taking the limit of averaged stochastic dynamics, as the number of trajectories increases, we obtain a deterministic system of differential equations that describes the process evolution. This system is used to establish a phase transition that separates fading and explosive behavior of the average process progeny. We describe a class of critical Tokunaga processes that happen at the phase transition boundary. They enjoy multiple additional symmetries and include the celebrated critical binary Galton–Watson tree with independent exponential edge length as a special case. Finally, we discuss a duality between trees and continuous functions, and introduce a class of extreme-invariant processes, constructed as the Harris paths of a self-similar hierarchical branching process, whose local minima has the same (linearly scaled) distribution as the original process.

Algorithm and complexity of the two disjoint connected dominating sets problem on trees

In this paper, we consider a variation of the classic dominating set problem - The Two Disjoint Connected Dominating Sets (DCDS) problem, which finds applications in many real domains including wireless sensor networks. In the DCDS problem, we are given a graph G=(V,E) and required to find a new edge set E′ with minimum cardinality such that the resulting new graph after the adding of E′ has a pair of disjoint connected dominating sets. This problem is very hard in general graphs, and we show that it is NP-hard even restricted to trees. We also present a polynomial time approximation algorithm for the DCDS problem for arbitrary trees with performance ratio 32 asymptotically.

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.

Trees are all around us: Farmers' management of wood pastures in the light of a controversial policy

Wood pastures are some of the most species-rich environments found in Europe and therefore essential habitats for biodiversity conservation. Society also puts faith in multiple values of trees, ranging from climate change mitigation to socio-cultural traditions. Therefore, the seemingly arbitrary tree density limit for pasture environments imposed by the EU through its Common Agricultural Policy (CAP) threatened both biological and societal values. In this study on farmers' perspectives, we target the effects of the CAP tree density limit on management of wood pastures in a low-intensively managed agricultural landscape of southern Sweden. The case of simplifying nature by using simple number limitations was used as an entry point in semi-structured, open-ended, interviews with farmers and officials about their view on trees and pasture management in relation to policy directives. The interviews showed that the policy incentive shifted the management focus from grazing quality to the number of trees and that farmers are willing to cut in order to get subsidies and timber revenues, however not unreflectingly. Farmers had high knowledge about the wide ranging social, cultural and natural values of trees, and are often themselves as good regulators of tree management as policies intend to be. Our study reveals many difficulties in managing the complex relations within landscapes with simplified legal measures, opening up for further discussion about improving policy instruments to preserve both social and biological values of wood pastures. However, although the tree density limit has been criticised on many points related to biodiversity conservation, this study shows that other values linked to pasture trees, e.g. the aesthetic values and their importance as shelter for grazing animals, could be an argument to actually keep the focus on trees as indicators of pasture management quality. We suggest that trees in general and wood-pastures in particular therefore are good starting points, or boundary-objects, for collaboration between production and conservation interests of farming and environmental management.

Characterization of the induced matching extendable graphs with [formula omitted] vertices and [formula omitted] edges

A graph G is induced matching extendable or IM-extendable if every induced matching of G is contained in a perfect matching of G. In 1998, Yuan proved that a connected IM-extendable graph on 2n vertices has at least 3n−2 edges, and that the only IM-extendable graph with 2n vertices and 3n−2 edges is T×K2 , where T is an arbitrary tree on n vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with 2n≥6 vertices and 3n−1 edges is T×K2+e, where T is an arbitrary tree on n vertices and e is an edge connecting two vertices that lie in different copies of T and have distance 3 between them in T×K2. In this paper, we introduced the definition of Q-joint graph and characterized the connected IM-extendable graphs with 2n≥4 vertices and 3n edges.

Submodular unsplittable flow on trees

We study the Unsplittable Flow problem (mathsf {UFP}) on trees with a submodular objective function. The input to this problem is a tree with edge capacities and a collection of tasks, each characterized by a source node, a sink node, and a demand. A subset of the tasks is feasible if the tasks can simultaneously send their demands from the source to the sink without violating the edge capacities. The goal is to select a feasible subset of the tasks that maximizes a submodular objective function. Our main result is an O(klog n)-approximation algorithm for Submodular UFP on trees where k denotes the pathwidth of the given tree. Since every tree has pathwidth O(log n), we obtain an O(log ^2 n) approximation for arbitrary trees. This is the first non-trivial approximation guarantee for the problem, matching the best known approximation ratio for UFP on trees with a linear objective function. Our main technical contribution is a new geometric relaxation for UFP on trees that builds on the recent work of Bonsma et al. (2014, FOCS), Anagnostopoulos et al. (Amazing 2+epsilon approximation for unsplittable flow on a path, SIAM, pp 26–41, 2014) for UFP on paths with a linear objective. Our relaxation is very structured, so it can be combined with the contention resolution framework of Chekuri et al. (2009, STOC). Our approach is robust and extends to several related problems, such as UFP with bag constraints and the Storage Allocation Problem. Additionally, we study the special case of UFP on trees with a linear objective and upward instances where, for each task, the source node is a descendant of the sink node. Such instances generalize UFP on paths. We build on the work of Bansal et al. (A quasi-PTAS for unsplittable flow on line graphs, ACM, pp 721–729, 2006) for UFP on paths and obtain a QPTAS for upward instances when the input data is quasi-polynomially bounded. We complement this result by showing that, unlike the path setting, upward instances are mathsf {APX}-hard if the input data is arbitrary.

Hierarchical Approximate Proper Orthogonal Decomposition

Proper Orthogonal Decomposition (POD) is a widely used technique for the construction of low-dimensional approximation spaces from high-dimensional input data. For large-scale applications and an increasing amount of input data vectors, however, computing the POD often becomes prohibitively expensive. This work presents a generic, easy-to-implement approach to compute an approximate POD based on arbitrary tree hierarchies of worker nodes, where each worker computes a POD of only a small amount of input vectors. The tree hierarchy can be freely adapted to optimally suit the available computational resources. In particular, this hierarchical approximate POD (HAPOD) allows for both simple parallelization with low communication overhead, as well as live sequential POD computation under restricted memory capacities. Rigorous error estimates ensure the reliability of our approach, and extensive numerical examples underline its performance.

Reducibility of Computable Metrics on the Real Line

We study computable reducibility of computable metrics on R induced by reducibility of their respective Cauchy representations. It is proved that this ordering has a subordering isomorphic to an arbitrary countable tree. Also we introduce a weak version of computable reducibility and construct a countable antichain of computable metrics that are incomparable with respect to it. Informally, copies of the real line equipped with these metrics are pairwise homeomorphic but not computably homeomorphic.

Securing Outsourced Data in the Multi-Authority Cloud with Fine-Grained Access Control and Efficient Attribute Revocation

Data outsourcing is a promising service for data owners, where their data are stored on a cloud storage provider. Since the cloud is not fully trusted, data access control has become a challenging issue in the Cloud Storage System (CSS). Ciphertext-Policy Attribute-Based Encryption (CP-ABE) is a feasible technique for ensuring access control in the CSS, where an attribute authority is responsible to manage attributes and distribute keys. In this paper, we propose a novel revocable Multi-Authority CP-ABE scheme, in which the access policy can be constructed as an arbitrary tree rather than a matrix used by existing schemes. The tree-like policy makes our scheme more flexible. Consequently, the encryption, decryption and attribute revocation operations are also more efficient. Our scheme is also proved to be secure under the standard assumption. It can resist user collusion attack, while the attribute revocation operation also achieves both forward security and backward security. Simulation results show that our scheme is highly efficient

The pos/neg-weighted 2-medians in balanced trees with subtree-shaped customers

This paper deals with the pos/neg-weighted p-median problem on tree graphs where all customers are modeled as subtrees. We present a polynomial algorithm for the 2-median problem on an arbitrary tree. Then we improve the time complexity to O(n log n) for the problem on a balanced tree, where n is the number of the vertices in the tree.

Bounded Repairability for Regular Tree Languages

We study the problem of bounded repairability of a given restriction tree language R into a target tree language T . More precisely, we say that R is bounded repairable with respect to T if there exists a bound on the number of standard tree editing operations necessary to apply to any tree in R to obtain a tree in T . We consider a number of possible specifications for tree languages: bottom-up tree automata (on curry encoding of unranked trees) that capture the class of XML schemas and document type definitions (DTDs). We also consider a special case when the restriction language R is universal (i.e., contains all trees over a given alphabet). We give an effective characterization of bounded repairability between pairs of tree languages represented with automata. This characterization introduces two tools—synopsis trees and a coverage relation between them—allowing one to reason about tree languages that undergo a bounded number of editing operations. We then employ this characterization to provide upper bounds to the complexity of deciding bounded repairability and show that these bounds are tight. In particular, when the input tree languages are specified with arbitrary bottom-up automata, the problem is coNExp-complete. The problem remains coNExp-complete even if we use deterministic nonrecursive DTDs to specify the input languages. The complexity of the problem can be reduced if we assume that the alphabet, the set of node labels, is fixed: the problem becomes PS pace -complete for nonrecursive DTDs and coNP-complete for deterministic nonrecursive DTDs. Finally, when the restriction tree language R is universal, we show that the bounded repairability problem becomes E xp -complete if the target language is specified by an arbitrary bottom-up tree automaton and becomes tractable (P-complete, in fact) when a deterministic bottom-up automaton is used.

Constructing Maximum-Lifetime Data-Gathering Tree in WSNs Based on Compressed Sensing

Data gathering is one of the most important operations in many wireless sensor networks (WSNs) applications. In order to implement data gathering, a tree structure rooted at the sink is usually defined. In most wireless sensor networks, nodes are powered by batteries with limited energy. Prolonging network lifetime is a critical issue for WSNs. As a technique for signal processing, compressed sensing (CS) is being increasingly applied to wireless sensor networks for saving energy. Compressive sensing can reduce the number of data transmissions and balance the traffic load throughout networks. In this paper, we investigate data gathering in wireless sensor networks using CS and aim at constructing a maximum-lifetime data-gathering tree. The lifetime of the network is defined as the number of data-gathering rounds until the first node depletes its energy. Based on the hybrid-CS data-gathering model, we first construct an arbitrary data-gathering tree and then use the random switching decision and optimal parent node selecting strategy to adjust the load of the bottleneck node and prolong the network lifetime. Simulation results show that the proposed algorithm outperforms several existing approaches in terms of network lifetime.

A characterization of single-peaked preferences via random social choice functions

The paper proves the following result: every path-connected domain of preferences that admits a strategy-proof, unanimous, tops-only random social choice function satisfying a compromise property, is single-peaked. Conversely, every single-peaked domain admits a random social choice function satisfying these properties. Single-peakedness is dened with respect to arbitrary trees. The paper provides a justication of the salience of single-peaked preferences and evidence in favour of the Gul conjecture (Barber a (2010)).

Arbitrary Decision Tree for Weather Prediction

Arbitrary Decision Tree for Weather Prediction

Global defensive sets in graphs

In the paper we study a new problem of finding a minimum global defensive set in a graph which is a generalization of the global alliance problem. For a given graph G and a subset S of a vertex set of G, we define for every subset X of S the predicate SEC(X)=true if and only if |N[X]∩S|≥|N[X]∖S| holds, where N[X] is a closed neighbourhood of X in graph G. A set S is a defensive alliance if and only if for each vertex v∈S we have SEC({v})=true. If S is also a dominating set of G (i.e., N[S]=V(G)), we say that S is a global defensive alliance.We introduce the concept of defensive sets in graph G as follows: set S is a defensive set in G if and only if for each vertex v∈S we have SEC({v})=true or there exists a neighbour u of v such that u∈S and SEC({v,u})=true. Similarly, if S is also a dominating set of G, we say that S is a global defensive set. We also study the problems of total dominating alliances (total alliances) and total dominating defensive sets (total defensive sets), i.e., S is a dominating set and the induced graph G[S] has no isolated vertices.In the paper we proved the NP-completeness for planar bipartite subcubic graphs of the decision versions of the following minimalization problems: a global and total alliance, a global and total defensive set. We proposed polynomial time algorithms solving in trees the problem of finding the minimum total and global defensive set and the total alliance. We obtained the lower bound on the minimum size of a global defensive set in arbitrary graphs and trees.

On Construction of Quantum Markov Chains on Cayley trees

The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.

Potentially nilpotent tridiagonal sign patterns of order 4

An $n\times n$ sign pattern ${\cal A}$ is said to be potentially nilpotent (PN) if there exists some nilpotent real matrix $B$ with sign pattern ${\cal A}$. In [M.~Arav, F.~Hall, Z.~Li, K.~Kaphle, and N.~Manzagol.Spectrally arbitrary tree sign patterns of order $4$, {\em Electronic Journal of Linear Algebra}, 20:180--197, 2010.], the authors gave some open questions, and one of them is the following: {\em For the class of $4 \times 4$ tridiagonal sign patterns, is PN (together with positive and negative diagonal entries) equivalent to being SAP?}\ In this paper, a positive answer for this question is given.