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We present stationarity criteria for forest stands, and establish embodiments using a Norwegian empirical stand development model. The natural stationary states only slightly differ from the outcome of long-term simulations previously implemented using the same empirical model. Human interference in terms of diameter-limit cutting is introduced. Consequently, stationary states differing from the natural one appear. Standing volume, growth and monetary value appear low but the financial return rate may be significant. Volume yield and financial return clearly contradict each other, the former arising from harvesting large trees, the latter from frequent removal of small trees. An exponential tree size distribution does not appear to comply with the stationarity criterion.

We introduce the random exponential binary tree (EBT) and study its profile. As customary, the tree is extended by padding each leaf node (considered internal), with the appropriate number of external nodes, so that the outdegree of every internal node is made equal to 2. In a random EBT, at every step, each external node is promoted to an internal node with probability p, stays unchanged with probability 1 - p, and the resulting tree is extended. We study the internal and external profiles of a random EBT and get exact expectations for the numbers of internal and external nodes at each level. Asymptotic analysis shows that the average external profile is richest at level $\frac {2p}{p+1}n$ , and it experiences phase transitions at levels a n, where the a’s are the solutions to an algebraic equation. The rates of convergence themselves go through an infinite number of phase changes in the sublinear range, and then again at the nearly linear levels.

Half a century ago, Chow and Liu proved that the distribution of the first-order dependence tree that optimally approximates an arbitrary joint distribution in terms of Kullback–Leibler cross entropy is the distribution of the maximum-weight dependence tree (MWDT). In a $p$ -dimensional space, this is a tree with $p-1$ branches between pairs of variables with each branch having a weight equal to the mutual information between variables at both ends. Today, MWDTs have applications that are more important in classification than only being used to model first-order dependence structure among variables. Nevertheless, whether the MWDT is the optimal first-order tree in classification has been left unexplored so far. In this letter, we study the optimality of the MWDT structure from the stand-point of classification.

A simple linear-time algorithm for constructing a linear context-free tree grammar of size O(rg + r g log (n/r g))for a given input tree T of size n is presented, where g is the size of a minimal linear context-free tree grammar for T, and r is the maximal rank of symbols in T (which is a constant in many applications). This is the first example of a grammar-based tree compression algorithm with a good, i.e. logarithmic in terms of the size of the input tree, approximation ratio. The analysis of the algorithm uses an extension of the recompression technique from strings to trees., Comment: 45 pages, published in Information and Computation. Approximation ratio improved since the first version, figures improved, some examples added. A small calculation error corrected since the previous version (all claims hold as previously)

A tree $t$-spanner $T$ of a graph $G$ is a spanning tree of $G$ such that the distance in $T$ between every pair of verices is at most $t$ times the distance in $G$ between them. There are efficient algorithms that find a tree $t\cdot O(\log n)$-spanner of a graph $G$, when $G$ admits a tree $t$-spanner. In this paper, the search space is narrowed to $v$-concentrated spanning trees, a simple family that includes all the breadth first search trees starting from vertex $v$. In this case, it is not easy to find approximate tree spanners within factor almost $o(\log n)$. Specifically, let $m$ and $t$ be integers, such that $m>0$ and $t\geq 7$. If there is an efficient algorithm that receives as input a graph $G$ and a vertex $v$ and returns a $v$-concentrated tree $t\cdot o((\log n)^{m/(m+1)})$-spanner of $G$, when $G$ admits a $v$-concentrated tree $t$-spanner, then there is an algorithm that decides 3-SAT in quasi-polynomial time.

A 3-tree is a tree with the maximum degree at most three. Let T be a tree of order n and p(T) . In this paper, we prove that the square of T has a spanning tree F in which every leave of T has degree one or two and F has at most max min n − p(T )+7 2 , n − 1 2 ,2 leaves; This implies that the square graph of a connected graph G has the same conclusion above as a tree. These bounds are all sharp in same sense. We also give a shorter proof of a result in (10).

The Steiner tree problem with revenues, budget and hop-constraints (STPRBH) is a variant of the classical Steiner tree problem. The goal is to find a tree maximizing the collected revenue, which is associated with nodes, subject to a given budget for the edge cost of the tree and a hop-limit for the distance between the given root node and any other node in that tree. In this work, we introduce a novel generic way to model hop-constrained tree problems as integer linear programs and apply it to the STPRBH. Our approach is based on the concept of layered graphs that gained widespread attention in the recent years, due to their computational advantage when compared to previous formulations for modeling hop-constraints. Contrary to previous MIP formulations based on layered graphs (that are arc-based models), our model is node-based. Thus it contains much less variables and allows to tackle large-scale instances and/or instances with large hop-limits, for which the size of arc-based layered graph models may become prohibitive. The aim of our model is to provide a good compromise between quality of root relaxation bounds and the size of the underlying MIP formulation. We implemented a branch-and-cut algorithm for the STPRBH based on our new model. Most of the instances available for the DIMACS challenge, including 78 (out of 86) previously unsolved ones, can be solved to proven optimality within a time limit of 1000 s, most of them being solved within a few seconds only. These instances contain up to 500 nodes and 12,500 edges, with hop-limit up to 25.

For the performance and maintenance cost of the existing index methods, this paper presents a distributed multi access entrance B+ tree index method, which has some features that are important in practice, namely (1) achieve efficient parallel interval queries, and (2) and low maintenance cost for index structure. Our scheme relies on four key techniques to achieve efficient parallel interval queries and low maintenance cost for index structure: (1) It gives a routing table to each leaf node of distributed B+ tree, so as to interval search can be completed from any leaf node of any storage node and break through the Bottleneck caused by the root node in the distributed B+ tree; (2) It construction of balanced binary tree in different levels of a node, and choose select the relevant nodes for its routing table; (3) It uses the layer-by-layer transitivity of B+ tree node splitting information to sense node split position so as to only update routing information within subtree of corresponding particle at the time of node splitting; (4) it uses balanced structure of B+ trees to realize only update the single route information of relevant nodes at the time of node splitting, without adjusting the route tables in a large area. Our scheme has been implemented and evaluated, and the performance results are encouraging.

Dimensionality reduction and manifold learning techniques attempt to recover a lower-dimensional submanifold from the data as encoded in high dimensions. Many techniques, linear or non-linear, have been introduced in the literature. Standard methods, such as Isomap and local linear embedding, map the high-dimensional data points into a low dimension so as to globally minimize a so-called energy function, which measures the mismatch between the precise geometry in high dimensions and the approximate geometry in low dimensions. However, the local effects of such minimizations are often unpredictable, because the energy minimization algorithms are global in nature. In contrast to these methods, the Spanifold algorithm of this paper constructs a tree on the manifold and flattens the manifold in such a way as to approximately preserve pairwise distance relationships within the tree. The vertices of this tree are the data points, and the edges of the tree form a subset of the edges of the nearest-neighbour graph on the data. In addition, the pairwise distances between data points close to the root of the tree undergo minimal distortion as the data are flattened. This allows the user to design the flattening algorithm so as to approximately preserve neighbour relationships in any chosen local region of the data. Copyright © 2015 John Wiley & Sons, Ltd.

This paper addresses the reverse 1-center problem on a weighted tree. Here, a facility has already been located in a predetermined node of the tree network and we want to improve the 1-center objective value at that node as efficiently as possible within a given budget. For solving this problem under uniform linear cost functions, we develop a combinatorial algorithm with running time , where is the number of vertices of the tree.

Sensor nodes observing events, occurring in close proximity, gather data that are highly co-related. This work uses a polynomial regression technique to exploit the spatial correlation of the data in a three dimensional sensor network. The sensing nodes, sense the physical attribute and report their position coordinates (x, y, z) and the sensed value to the nearest tree node. Another set of nodes, categorized as the tree nodes, are responsible for generating a polynomial function of the received data and transmit the coefficients of regression to the parent tree node. The approach proceeds from the bottom to the top. The query from the sink, receives a polynomial function which is generated by the root node of the tree in each cluster to compute the attribute value at any location within the boundary. The proposed approach aims to save a lot of energy in the sensor network. Simulations, performed for different tree heights, indicate that a tree with a depth of four gives the best results.

Some apparently different VLSI circuit design optimization problems can be mapped to the problem of allocating weights (hardware circuits) to the nodes of a tree, such that their total sum (delay) along root-to-leaf paths, or their total product (amplification) along root-to-leaf paths, satisfy given demands (delays or amplifications, respectively) at the tree's leaves. Node's weight is shared by all the leaves of its emanating sub-tree. For both the sum and product constraints cases, $$O(n)$$O(n) weights allocation algorithms are presented, supplying the demands at the leaves, while the total sum of nodes' weights (hardware cost) is minimized. When the assignment of the demands to leaves is not predetermined, it is shown that monotonic order of the demands at leaves is optimal for both cases.

This paper studies the problem of constructing a minimum-cost multicast tree (or Steiner tree) in which each node is associated with a cost that is dependent on its degree in the multicast tree. The cost of a node may depend on its degree in the multicast tree due to a number of reasons. For example, a node may need to perform various processing for sending messages to each of its neighbors in the multicast tree. Thus, the overhead for processing the messages increases as the number of neighbors increases. This paper devises a novel technique to deal with the degree-dependent node costs and applies the technique to develop an approximation algorithm for the degree-dependent node-weighted Steiner tree problem. The bound on the cost of the tree constructed by the proposed approximation algorithm is derived to be 2((ln (k/2))+1)(W_T∗ + B), where k is the size of the set of multicast members, W_T∗ is the cost of a minimum-cost Steiner tree T∗, and B is related to the degree-dependent node costs. Simulations are carried out to study the performance of the proposed algorithm. A distributed implementation of the proposed algorithm is presented. In addition, the proposed algorithm is generalized to solve the degree-dependent node-weighted constrained forest problem.

Employing PTAS to building minimum spanning tree for a large number of equal distribution input terminal nodes can be a effective way in execution time. But applying PTAS to building minimum spanning tree for tremendous unequal distribution node may lead to performance degradation. In this paper, a partial PTAS reflecting the scheme into specific node dense area is presented. In the environment where 90% of 50,000 input terminal nodes stand close together in specific area, approximate minimum spanning tree by our proposed scheme can show about 88.49% execution time less and 0.86%tree length less than by existing PTAS, and about 87.57%execution time less and 1.18% tree length more than by Prim`s naive scheme. Therefore our scheme can go well to many useful applications where a multitude of nodes gathered around specific area should be connected efficiently as soon as possible.

We give two self-stabilizing algorithms for tree networks. The first computes an index, called guide pair, for each process P in O(h) rounds using O(@d"Plogn) space per process, where h is the height of the tree, @d"P the degree of P, and n the number of processes in the network. Guide pairs have numerous applications, including ordered traversal or navigation in the tree. Our second algorithm, which uses the guide pairs computed by the first algorithm, solves in O(n) rounds the ranking problem for an ordered tree, where each process has an input value. This second algorithm has space complexity O(b+@d"Plogn) in each process P, where b is the number of bits needed to store an input value. The first algorithm orders the tree processes according to their topological positions. The second algorithm orders (ranks) the processes according to their input values.

Let T be a tree on a set V of nodes. The p-th power T p of T is the graph on V such that any two nodes u and w of V are adjacent in T p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T, if any, such that G=T p . Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T, if any, such that G=T p . Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n 3) (respectively, O(n 4)) time. In this paper, we give O(n+m)-time algorithms for both problems.

We describe a general technique for obtaining provably correct, non-blocking implementations of a large class of tree data structures where pointers are directed from parents to children. Updates are permitted to modify any contiguous portion of the tree atomically. Our non-blocking algorithms make use of the LLX, SCX and VLX primitives, which are multi-word generalizations of the standard LL, SC and VL primitives and have been implemented from single-word CAS. To illustrate our technique, we describe how it can be used in a fairly straightforward way to obtain a non-blocking implementation of a chromatic tree, which is a relaxed variant of a red-black tree. The height of the tree at any time is O(c + log n), where n is the number of keys and c is the number of updates in progress. We provide an experimental performance analysis which demonstrates that our Java implementation of a chromatic tree rivals, and often significantly outperforms, other leading concurrent dictionaries.

Tree data structure works great in representing computations unambiguously. In this study, the researchers enhanced the existing insert algorithm in B*Tree by introducing an expanded algorithm for inserting key values in B*Tree. This would further delay, if not reduce redistribution and frequency of splitting nodes in B*Tree. The idea of the expanded algorithm is to check for a cousin node within the same level that can accommodate a key value from its nearest sibling. A cascading effect is expected until the nearest sibling of the overflowing node can accommodate a key value with the premise that the nearest siblings of the overflowing node are full. Exploring the potential of this algorithm can really simplify a complex problem. The expanded insert algorithm would ensure that all leaf nodes in the B*Tree would be full and not just 2/3 full before a redistribution process is to be performed. Thus, reducing the nodes used in a B*Tree that would result to a far more favorable priori and posteriori estimates.

HighlightsI proposed an oblique survival tree that is able to cope with right-censored data.The induction algorithm is based on the minimization of the CPL criterion functions.A split-complexity measure with a 10-fold cross-validation was used to prune a tree.The predictive ability of the method was evaluated using synthetic and real data.The outcomes were compared with two existing univariate tree models.The obtained survival trees are small and offer a good predictive ability. ObjectiveRecursive partitioning is a common, assumption-free method of survival data analysis. It focuses mainly on univariate trees, which use splits based on a single variable in each internal node. In this paper, I provide an extension of an oblique survival tree induction technique, in which axis-parallel splits are replaced by hyperplanes, dividing the feature space into areas with a homogeneous survival experience. Method and materialsThe proposed tree induction algorithm consists of two steps. The first covers the induction of a large tree with internal nodes represented by hyperplanes, whose positions are calculated by the minimization of a piecewise-linear criterion function, the dipolar criterion. The other phase uses a split-complexity algorithm to prune unnecessary tree branches and a 10-fold cross-validation technique to choose the best tree. The terminal nodes of the final tree are characterised by KaplanMeier survival functions. A synthetic data set was used to test the performance, while seven real data sets were exploited to validate the proposed method. ResultsThe evaluation of the method was focused on two features: predictive ability and tree size. These were compared with two univariate tree models: the conditional inference tree and recursive partitioning for survival trees, respectively. The comparison of the predictive ability, expressed as an integrated Brier score, showed no statistically significant differences (p=0.486) among the three methods. Similar results were obtained for the tree size (p=0.11), which was calculated as a median value over 20 runs of a 10-fold cross-validation. ConclusionsThe predictive ability of trees generated using piecewise-linear criterion functions is comparable to that of univariate tree-based models. Although a similar conclusion may be drawn from the analysis of the tree size, in the majority of the studied cases, the number of nodes of the dipolar tree is one of the smallest among all the methods.

This paper studies information source detection in networks under the independent cascade (IC) model. Assume the spread of information starts from a single source in a network and a complete snapshot of the network is obtained at some time. The goal is to identify the source based on the observation. We derive the maximum a posterior (MAP) estimator of the source for tree networks and propose a Short-Fat Tree (SFT) algorithm for general networks based on the MAP estimator. The algorithm selects the Jordan infection center [1] and breaks ties according the degree of boundary infected nodes. Loosely speaking, the algorithm selects the node such that the breadth-first search (BFS) tree from it has the minimum depth but the maximum number of leaf nodes. On the Erdos-Renyi (ER) random graph, we establish the following possibility and impossibility results: (i) when the infection duration 0.5, SFT identifies the source with probability 1 (w.p.1) asymptotically (as network size increases to infinity), where n is the network size and μ is the average node degree; (ii) when the infection duration > ⌈log n/ log μ ⌉ + 2, the probability of identifying the source approaches zero asymptotically under any algorithm; and (iii) when infection duration 0, asymptotically, at least 1−δ fraction of the nodes on the BFS tree starting from the source are leaf-nodes, where δ = 3√log n/μ, i.e., the BFS tree starting from the actual source is a fat tree. 1Numerical experiments on tree networks, the ER random graphs and real world networks with different evaluation metrics show that the SFT algorithm outperforms existing algorithms.

The minimizing-deep-coalescence (MDC) approach infers a median (species) tree for a given set of gene trees under the deep coalescence cost. This cost accounts for the minimum number of deep coalescences needed to reconcile a gene tree with a species tree where the leaf-genes are mapped to the leaf-species through a function called leaf labeling. In order to better understand the MDC approach we investigate here the diameter of a gene tree, which is an important property of the deep coalescence cost. This diameter is the maximal deep coalescence costs for a given gene tree under all leaf labelings for each possible species tree topology. While we prove that this diameter is generally infinite, this result relies on the diameter’s unrealistic assumption that species trees can be of infinite size. Providing a more practical definition, we introduce a natural extension of the gene tree diameter that constrains the species tree size by a given constant. For this new diameter, we describe an exact formula, present a complete classification of the trees yielding this diameter, derive formulas for its mean and variance, and demonstrate its ability using comparative studies.

In this paper, PTAS three-dimensional Steiner minimum tree connecting numerous input nodes rapidly in 3D space is proposed. Steiner minimum tree problem belongs to NP problem domain, and when properly devised heuristic introduces, it is generally superior to other algorithms as minimum spanning tree affiliated with P problem domain. But when the number of input nodes is very large, the problem requires excessive execution time. In this paper, a method using PTAS is proposed to solve the difficulty. In experiments for 70,000 input nodes in 3D space, the tree produced by the proposed 8 space partitioned PTAS method reduced 86.88% execution time, compared with the tree by naive 3D steiner minimum tree method, though increased 0.81% tree length. This affirms the proposed method can work well for applications that many nodes of three dimensions are need to connect swifty, enduring slight increase of tree length.

A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T′ if c(T,k) is at least c(T′,k) for all k and strictly greater than c(T′,k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree. In addition, we determine asymptotically the expected values of c(T,k) in special families of trees. C © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 32–43, 2013

Let R and B be sets of red and blue points in the plane in general position. We study the problem of computing a k-level binary space partition (BSP) tree to classify/separate R and B, such that the tree defines a linear decision at each internal node and each leaf of the tree corresponds to a (convex) cell of the partition that contains only red or only blue points. Specifically, we show that a 2-level tree can be computed, if one exists, in time O(n2). We show that a minimum-level (3 ≤ k ≤ log n) tree can be computed in time nO( log n). In the special case of axis-parallel partitions, we show that 2-level and 3-level trees can be computed in time O(n), while a minimum-level tree can be computed in time O(n5).

In general, Euclidean minimum spanning tree is a tree connecting input nodes with minimum connecting cost. But the tree may not be optimal when applied to real world problems of three dimension. In this paper, three dimension Euclidean minimum spanning tree is proposed, connecting all input nodes of 3-dimensional space with minimum cost. In experiments for 30,000 input nodes with 100% space ratio, the tree produced by the proposed method can reduce 90.0% connection cost tree, compared with the tree by two dimension Prim`s minimum spanning tree. In two dimension plane, the proposed tree increases 251.2% connecting cost, which is pointless in 3-dimensional real world. Therefore, the proposed method can work well for many connecting problems in real world space of three dimensions.

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive lengths is tree spectral if it is the path spectrum of a tree. We show that for each even integer s ⩾ 2 at least 34.57% of all subsets of the set { 2 , 3 , … , s } are tree spectral, and for each odd integer s ⩾ 2 at least 27.44% of all subsets of the set { 2 , 3 , … , s } are tree spectral.

For concurrent database access, the B-link tree is a well-suited data structure; this paper describes a modification of the B-link tree to reduce the expense associated with splits and merges while moving down the tree. The modification is keyed to the concept of stability in each B-link tree node. Being stable amounts to a node having somewhere between a small percentage more than the minimum number and a small percentage fewer than the maximum number of child links such a node may have. When additions or deletions lead to instability in a node, either an overflow or underflow is considered to be imminent. Upon visiting a node, an update process will determine if the node's content is at the B-link maximum or minimum; if this is the case, the process performs whatever split or merge and redistribution operations necessary to make this and any related node stable. As a result of this action, there will always be room for an overflowed child to insert an entry in its parent and likewise room for an underflowed child to remove an entry from its parent; backward propagation of insertions or deletions further up the Blink tree are thus eliminated, improving the possible degree of concurrency. Further, no node will be left without a parent, obviating horizontal traversal within the tree in all but highly infrequent process scheduling scenarios.

The d-dimensional k-ary hypercube tree Tk(d) is a generalization of the hypercube tree, also known in the literature as the spanning binomial tree. We present some of its structural properties and investigate in detail its average distance. For instance, it is shown that the binary hypercube tree has the anomaly of having two nodes in its centre as opposed to having one in hypercube trees of arity k>2. However, in all dimensions, the centre, centroid and median coincide. We show that its total distance is [image omitted] , which is minimum. Consequently, for d≥2, its average distance is μ(Tk(d))=(2d(k-1)kd-1)/(kd-1)-2/k, whose limiting value is 2. This answers a generalization of a conjecture of Dobrynin et al. [Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), pp. 211-249] for the binary hypercube by the affirmative.

We give a (ln n+1)-approximation for the decision tree (DT) problem. An instance of DT is a set of m binary tests T=(T 1,…,T m ) and a set of n items X=(X 1,…,X n ). The goal is to output a binary tree where each internal node is a test, each leaf is an item and the total external path length of the tree is minimized. Total external path length is the sum of the depths of all the leaves in the tree. DT has a long history in computer science with applications ranging from medical diagnosis to experiment design. It also generalizes the problem of finding optimal average-case search strategies in partially ordered sets which includes several alphabetic tree problems. Our work decreases the previous best upper bound on the approximation ratio by a constant factor. We provide a new analysis of the greedy algorithm that uses a simple accounting scheme to spread the cost of a tree among pairs of items split at a particular node. We conclude by showing that our upper bound also holds for the DT problem with weighted tests.

The L(2, 1)-labeling problem for a graph G is a variation of the standard graph coloring problem. Here, we seek to assign a label (color) to each node of G such that nodes a distance of two apart are assigned unique labels and adjacent nodes receive labels which are at least two apart. In a previous paper—presented at the 23rd IASTED International Multi-Conference: Parallel and Distributed Computing and Networks, Innsbruck, Austria—we presented, to the best of our knowledge, the first self-stabilizing algorithm which {Δ + 2}-L(2, 1)-labels rooted trees. That algorithm was shown to require an exponential number of moves to stabilize on a global solution (which is not uncommon in self-stabilizing systems). In this paper, we present two self-stabilizing algorithms which {Δ + 2}-L(2, 1)-label a given rooted tree T in only O(nh) moves (where h is the height and n is the number of nodes in the tree T) under a central scheduler. We also show how the algorithms may be adapted to unrooted trees, dynamic topology changes, and consider the correctness of the protocols under the distributed scheduler model.

We study two enumeration problems for up–down alternating trees, i.e., rooted labelled trees T, where the labels v1,v2,v3,… on every path starting at the root of T satisfy v1v3⋯. First we consider various tree families of interest in combinatorics (such as unordered, ordered, d-ary and Motzkin trees) and study the number Tn of different up–down alternating labelled trees of size n. We obtain for all tree families considered an implicit characterization of the exponential generating function T(z) leading to asymptotic results of the coefficients Tn for various tree families. Second we consider the particular family of up–down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters label of the root node, degree of the root node and depth of a random node in a random tree of size n. This leads to exact enumeration results and limiting distribution results.

Tree-structured computations are among the most popular and important and powerful computing paradigms in computer science. In many such applications, the size and shape of a tree that represents a parallel computation is dynamic and unpredictable at compile time. The tree evolves gradually during the course of the computation. When such an application is executed on a multicomputer system with a static network, the dynamic tree evolution problem is to distribute the tree nodes to the processors of the network such that all the processors receive roughly the same amount of load and that communicating nodes are assigned to neighboring processors. The main contribution of the paper is to conduct a unified study of dynamic tree evolution on regular networks. Our regular networks include d-dimensional torus networks, d-dimensional lattice networks, complete d-partite networks, and random regular networks. These networks include several fundamental networks such as rings, k-ary d-cubes, complete networks, and hypercubes as special cases. We describe a simple random-walk-based asymptotically optimal dynamic tree evolution algorithm on regular networks and analyze the exponential speed at which the performance ratio converges to the optimal. Our strategy is to prove that the Markov chain of a random walk on a regular network is rapidly mixing in the sense that, starting from any initial distribution, the Markov chain converges to its stationary distribution in a small number of steps. We also show that larger dilation results in better tree node distribution.

This article proposes an approximation of the tree edit distance through the string edit distance for binary tree codes, instead of for Euler strings introduced by Akutsu (2006). Here, a binary tree code is a string obtained by traversing a binary tree representation with two kinds of dummy nodes of a tree in preorder. Then, we show that σ/2 ≤ τ ≤ (h + 1)σ + h, where τ is the tree edit distance between trees, and σ is the string edit distance between their binary tree codes and h is the minimum height of the trees.

A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on n leaves, sample sequences of length Ω(log n) are needed. It was conjectured by Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than $${p^{\ast} = (\sqrt{2}-1)/2^{3/2}}$$ , then the tree can be recovered from sequences of length O(log n). The value p* is given by the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel’s conjecture was proven by the second author in the special case where the tree is “balanced.” The second author also proved that if all edges have mutation probability larger than p* then the length needed is n Ω(1). Here we show that Steel’s conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from O(log n)-length sequences when the mutation probabilities are discretized and less than p*. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.

We expand on the work of Hosoya to describe a generalization of continued fractions called “tree expressions.” Each rooted tree will be shown to correspond to a unique tree expression which can be evaluated as a rational number (not necessarily in lowest terms) whose numerator is equal to the Hosoya index of the entire tree and whose denominator is equal to the tree with the root deleted. In the development, we use Z(G) to define a natural candidate ζ(G, v) for a “vertex topological index” which is a value applied to each vertex of a graph, rather than a value assigned to the graph overall. Finally, we generalize the notion of tree expression to “labeled tree expressions” that correspond to labeled trees and show that such expressions can be evaluated as quotients of determinants of matrices that resemble adjacency matrices.

Given a tree with n nodes, we consider the problem of finding the most profitable subtree of that tree with at most K nodes which is known as the Cardinality Subtree of a Tree Problem. We present a new exact linear extended formulation with O(nK) two-indexed variables and O(nK) constraints.

It is a known fact that the Wiener index (i.e. the sum of all distances between pairs of vertices in a graph) of a tree with an odd number of vertices is always even. In this paper, we consider the distribution of the Wiener index and the related tree parameter “internal path length” modulo 2 by means of a generating functions approach as well as by constructing bijections for plane trees.

We give an intuitive method — using local, cyclic replica symmetry — to isolate exponential tree decay in truncated (connected) correlations. We give an expansion and use the symmetry to show that all terms vanish, except those displaying replica condensation. The condensation property ensures exponential tree decay. We illustrate our method in a low-temperature Ising system, but expect that one can use a similar method in other random field and quantum field problems. While considering the illustration, we prove an elementary upper bound on the entropy of random lattice surfaces.

Determining the optimal location of a switching center in a tree network of users is accurately modeled by the median problem. A real-time approach is used in this paper to investigate the dynamics of such a communication network in two cases: (1) a growing tree of nodes associated with equal demand rates, and (2) a stream of corrections that arbitrarily change the demand rates at the nodes. The worst-case analysis performed in both situations clearly demonstrates the importance of parallelism in such real-time paradigms. It is shown that the error generated by the best sequential algorithm in the first case can be arbitrarily large. A synergistic behavior is revealed when the quality-up is investigated in the second case.

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of: the generalized Bethe tree B, a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B, a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices, a graph obtained from the cycle Cr by attaching B, by its root, to each vertex of the cycle, and a tree obtained from the path Pr by attaching B, by its root, to each vertex of the path, is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

We study for various tree families the distribution of the number of edge-disjoint paths required to cover the edges of a random tree of size n. For all tree families considered we can show a central limit theorem with expectation ~@mn and variance ~@nn with constants @m, @n depending on the specific tree family.

We consider three random variables X n , Y n and Z n , which represent the numbers of the nodes with 0, 1, and 2 children, in the binary search trees of size n. The expectation and variance of the three above random variables are got, and it is also shown that X n , Y n and Z n are all asymptotically normal as n → ∞ by applying the contraction method.

Our context of interest is tree-structured exact search in metric spaces. We make the simple observation that, the deeper a data item is within the tree, the higher the probability of that item being excluded from a search. Assuming a fixed and independent probability p of any subtree being excluded at query time, the probability of an individual data item being accessed is \( (1-p)^d\) for a node at depth d. In a balanced binary tree half of the data will be at the maximum depth of the tree so this effect should be significant and observable. We test this hypothesis with two experiments on partition trees. First, we force a balance by adjusting the partition/exclusion criteria, and compare this with unbalanced trees where the mean data depth is greater. Second, we compare a generic hyperplane tree with a monotone hyperplane tree, where also the mean depth is greater. In both cases the tree with the greater mean data depth performs better in high-dimensional spaces. We then experiment with increasing the mean depth of nodes by using a small, fixed set of reference points to make exclusion decisions over the whole tree, so that almost all of the data resides at the maximum depth. Again this can be seen to reduce the overall cost of indexing. Furthermore, we observe that having already calculated reference point distances for all data, a final filtering can be applied if the distance table is retained. This reduces further the number of distance calculations required, whilst retaining scalability. The final structure can in fact be viewed as a hybrid between a generic hyperplane tree and a LAESA search structure.

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i.e., labellings of the nodes by distinct integers of the set {1,…,n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity degree of node j in a random tree of size n and give closed formulae for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via a tree evolution process. Furthermore limiting distribution results of this parameter are given, which completely characterize the phase change behavior depending on the growth of j compared to n.

We study the problem of uniformly partitioning the edge set of a tree with n edges into k connected components, where k=

International audience; Semantic Backpropagation (SB) was introduced in GP so as to take into account the semantics of a GP tree at all intermediate states of the program execution, i.e., at each node of the tree. The idea is to compute the optimal " should-be " values each subtree should return, whilst assuming that the rest of the tree is unchanged, and to choose a subtree that matches as well as possible these target values. A single tree is evolved by iteratively replacing one of its nodes with the best subtree from a static library according to this local fitness, with tree size as a secondary criterion. Previous results for standard Boolean GP benchmarks that have been obtained by the authors with another variant of SB are improved in term of tree size. SB is then applied for the first time to categorical GP benchmarks, and outperforms all known results to date for three variable finite algebras.

In a tree with each edge associated with a length and a weight (positive, negative, or zero are possible) we develop an O(nlognloglogn) time algorithm for finding a path such that its sum of weights is maximized and its sum of lengths does not exceed a given value. The previously best-known result is O(), where n is the number of nodes in the tree.

The Quality of Service Multicast Tree Problem is a generalization of the Steiner tree problem which appears in the context of multimedia multicast and network design. In this generalization, each node possesses a rate and the cost of an edge with length l in a Steiner tree T connecting the source to non-zero rate nodes is l · re, where re is the maximum node rate in the component of T-{e} that does not contain the source. The best previously known approximation ratios for this problem (based on the best known approximation factor of 1.549 for the Steiner tree problem in networks) are 2.066 for the case of two non-zero rates and 4.212 for the case of an unbounded number of rates. In this paper we give improved approximation algorithms with ratios of 1.960 and 3.802, respectively. When the minimum spanning tree heuristic is used for finding approximate Steiner trees, then the previously best known approximation ratios of 2.667 for two non-zero rates and 5.542 for an unbounded number of rates are reduced to 2.414 and 4.311, respectively.

A multiple bus network (MBN) connects a set of processors via set of buses. Two important parameters of an MBN are its loading (largest number of connections on a bus) and its degree (largest number of connections to a processor). These parameters determine the cost, speed, and implementability of the MBN. The smallest degree that any useful MBN can have is 2. In this paper, we study the relationship between running time, degree, and loading of degree-2 MBNs running a fundamental class of algorithms called binary tree algorithms. (A binary tree algorithm reduces 2/sup n/ inputs at the leaves of a balanced-binary tree to a single result at the root of the tree.) Specifically, we establish a nontrivial /spl Omega/(n/logn) loading lower bound for any degree-2 MBN running a 2/sup n/ input binary tree algorithm optimally in n steps. We show that this bound does not hold if the restriction on the degree or the running time is relaxed. That is, optimal-time, degree-3, constant loading MBNs and suboptimal-time, degree-2, constant loading MBNs exist for binary tree algorithms. We also derive a lower bound on the additional time (beyond the optimal) needed to run binary tree algorithms on a degree-2, loading-L MBN, for any L>3.

In many tree-structured parallel computations, the size and structure of a tree that represents a parallel computation is unpredictable at compile-time; the tree evolves gradually during the course of the computation. When such a computation is performed on a static network, the dynamic tree embedding problem is to distribute the tree nodes to the processors of the network such that all the processors receive roughly the same amount of load and that communicating nodes are assigned to neighboring processors. Furthermore, when a new tree node is generated, it should be immediately assigned to a processor for execution without any information on the further evolving of the tree; and load distribution is performed by all processors in a totally distributed fashion. We study the problem of embedding dynamically evolving trees in hypercubic networks, including shuffle-exchange, de Bruijn, cube-connected cycles, wrapped butterfly, and hypercube networks. The performance of a random-walk-based randomized tree embedding algorithm is evaluated. Several random tree models are considered. We develop recurrence relations for analyzing the performance of embedding of complete-tree-based random trees and randomized complete trees, and linear systems of equations for reproduction trees. We present more efficient recurrence relations and linear systems of equations for symmetric networks. We also demonstrate extensive numerical data of the performance ratio and make a number of interesting observations of randomized tree embedding in the five hypercubic networks. Copyright © 2004 John Wiley & Sons, Ltd.