字典序地生成有序根树的一个算法

§1.引言由于树的生成在计算机科学中有着重要应用,近年来许多文章研究了树的生成,其中大多数文章是讨论2分树及 k 分树的生成.研究一般有序根树的文章尚少.文献[1]给出了有序根树的一个序列表示法,并描述了一个生成有序根树的算法.文献[2]及[3]讨论了生成2分树及 k 分树的算法.本文用0,1序列表示有序根树,并给出了一个字典序地生成具有 n 个顶点的所有有序根树的算法.本文的表示法及算法与文献[1]中所提方法不同.本算法亦可用来生成具有 n 个叶子的所有2分树.它比[2]中的算法更简单.本文中未加说明的术语皆见[1].     

Balanced ordered trees

A 0-balanced ordered three is an ordered tree all of whose root-to-leaf paths have the same length. In this article, we persent exact and asymptotic enumeration and distribution ersults for classes of 0-balanced ordered trees with n nodes. These results enable us to compute various expected values of interesting parameters defined on these trees. © 1994 John Wiley & Sons, Inc.     

On the average number of nodes in a subtree of a tree

For any tree T (labelled, not rooted) of order n, it will be shown that the average number of nodes in a subtree of T is at least $\text{(n + 2)}\text{3}$, with this minimum achieved iff T is a path. For T rooted, the average number of nodes in a subtree containing the root is at least $\text{(n + 1)}\text{2}$ and always exceeds the average over all unrooted subtrees. For the maximum mean, examples show that there are arbitrarily large trees in which the average subtree contains essentially all of the nodes. The mean subtree order as a function on trees is also shown to be monotone with respect to inclusion.     

On Weighted Depths in Random Binary Search Trees

Following the model introduced by Aguech et al. (Probab Eng Inf Sci 21:133–141, 2007), the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyse weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search process, the weighted path length and the weighted Wiener index in a random binary search tree. We establish three regimes of nodes depending on whether the second-order behaviour of their weighted depths follows from fluctuations of the keys on the path, the depth of the nodes or both. Finally, we investigate a random distribution function on the unit interval arising as scaling limit for weighted depths of nodes with at most one child.     

递归树的若干枚举特征

递归树由Meir和Moon定义作非平面增长树的一种，且所有节点出度都是允许的．本文首先在n个节点的递归树集合和n-1个元素的排列之间建立一个新的──对应，这个对应能同时给出树叶子和排列中的路段之间的对应和树叶子数和排列中的路段数之间的密切关系．同时还研究递归树的各种枚举特征，诸如节点的分类枚举（内节点和叶子节点、偶节点和奇节点，具不同出度的节点）和通路长度枚举（接各种节点分类）．     

On optimal trees

The merits of different shapes of trees as data storage structures are compared. Given a tree data structure, the worst-case cost of searching the tree is studied, under weak assumptions about the cost of searches. In particular, it is assumed that the cost of a search path is the sum of the costs of the nodes on it, and that the cost of a node depends only on the outdegree of that node. The main results of the paper are that there are regular trees (as defined in the paper) which are nearly optimal among trees with a given number of nodes, and that minimal-cost trees are often nearly regular.     

On the average shape of monotonically labelled tree structures

Let $\text{B}$_k denote one of the families of binary trees, t-aray trees (t> 2) or ordered trees with nodes labelled monotonically by elements of {1 < 2 < ? < k}. The average height of the j-th leaf of the trees of $\text{B}$_k with exactly n nodes is shown to converge to a finite limit (depending on k and j) for n → ∞. The limit is determined explicitly for small values of k and its asymptotic behaviour in j and k is investigated. Some recent results on the average shape of rooted tree structures appear as special cases.     

A deep-submicron steiner tree

An A-Tree is a rectilinear Steiner tree in which every sink is connected to a driver by a shortest length path, while simultaneously minimizing total wire length. This paper presents a polynomial approximation algorithm for the generalized version of an A-Tree problem with upper-bounded delays along each path from the driver to the sinks and with restrictions on the number of Steiner nodes. We refer to it as “Deep-submicron Steiner tree”, as minimizing the number of Steiner nodes is crucial for signal integrity issues in deep-submicron Very-Large-Scaled-Integrated-circuit (VLSI) designs. The idea behind the algorithm is to control two parameters in order to construct a feasible (with respect to the paths delays and the number of Steiner nodes) tree of small cost.The simulation results show the high efficiency of our approach.     

Optimized network structure for full-synchronization

A network of Kuramoto oscillators with different natural frequencies is optimized for enhanced synchronizability. All node inputs are normalized by the node connectivity and some important properties of the network structure are determined in this case: (i) optimized networks present a strong anti-correlation between natural frequencies of adjacent nodes; (ii) this anti-correlation should be as high as possible since the average path length between nodes is maintained as small as in random networks; and (iii) high anti-correlation is obtained without any relation between nodes natural frequencies and the degree of connectivity. We also propose a network construction model with which it is shown that high anti-correlation and small average paths may be achieved by randomly rewiring a fraction of the links of a totally anti-correlated network, and that these networks present optimal synchronization properties.     

Partitioning multi-edge graphs

We introduce a new kind of graph called multi-edge graph which arises in many applications such as finite state Markov chains and broadcasting communication networks. A cluster in such a graph is a set of nodes such that for any ordered pair of nodes, there is a path of multi-edges from one to the other such that these edges remain within the same set. We give two algorithms to partition a multi-edge graph into maximal clusters. Both these algorithms are based on the depth-first search algorithm to find strongly connected components of the directed graph. We also discuss some applications of clustering in multi-edge graphs.     

Optimal layout of edge-weighted forests

The layout problem for trees with weighted edges is motivated by the design of very-large-scale integrated circuits. Some of the nodes are fixed and the object is to position the remainder so that the total weighted edge cost is minimized. The cost of each edge is the product of its weight and its length under some appropriate norm. Optimization for planar layouts is shown to be NP-hard. If crossings are permitted, then optimal layouts under the L₁ norm can be efficiently computed. Suitable algorithms and data structures are presented, and explicit exact cost functions are given for two classes of weighted complete binary trees.     

Orderly Algorithm to Enumerate Central Groupoids and Their Graphs

A graph has the unique path property UPPn if there is a unique path of length n between any ordered pair of nodes. This paper reiterates Royle and MacKay＇s technique for constructing orderly algorithms. We wish to use this technique to enumerate all UPP2 graphs of small orders 3^2 and 4^2. We attempt to use the direct graph formalism and find that the algorithm is inefficient. We introduce a generalised problem and derive algebraic and combinatoric structures with appropriate structure. Then we are able to design an orderly algorithm to determine all UPP2 graphs of order 3^2, which runs fast enough. We hope to be able to determine the UPP2 graphs of order 4^2 in the near future.     

Optimal design of hierarchical networks with free main path extremes

We propose an optimal, two-stage procedure for the optimal design of minimum cost hierarchical spanning networks, consisting of a main path and secondary trees. The optimal location of the origin and destination nodes of the path is also found. We test our procedure and compare it with a known method.     

Protected Branches in Ordered Trees

In this paper, we consider the class of ordered trees and its two subclasses, bushes and planted trees, which consist of the ordered trees with root degree at least $2$ and with root degree $1$ respectively. In these three classes, we study the number of trees of size $n$ with $k$ protected (resp. unprotected) branches, and the total number of branches (resp. protected branches, unprotected branches) among all trees of size $n$. The explicit formulas as well as the generating functions are obtained. Furthermore, we find that, in each class, as $n$ goes to infinity, the proportion of protected branches among all branches in all trees of size $n$ approaches $ 1/3$.     

A new ranking algorithm and an evolutionary model based on comprehensive weighted clique degree for complex network

This paper proposes a new ranking algorithm based on comprehensive weighted clique degree (CWCDR) for ranking importance of nodes in complex network. Simulation results show that CWCDR algorithm can overcome the limitation of degree ranking algorithm and find important nodes in complex networks more precisely and effectively. To the shortage of small-world model and BA model, this paper proposes an evolutionary model of complex network based on CWCDR algorithms, named CWCDR model. Simulation results show that the CWCDR model accords with power-law distribution. Compared with BA model, this model has better average shortest path length and clustering coefficient. Therefore, the CWCDR model is more consistent with the real network.     

Some typical properties of large AND/OR Boolean formulas

In this paper typical properties of large random Boolean AND/OR formulas are investigated. Such formulas with n variables are viewed as rooted binary trees chosen from the uniform distribution of all rooted binary trees on m nodes, where n is fixed and m tends to infinity. The leaves are labeled by literals and the inner nodes by the connectives AND/OR, both uniformly at random. In extending the investigation to infinite trees, we obtain a close relation between the formula size complexity of any given Boolean function f and the probability of its occurrence under this distribution, i.e., the negative logarithm of this probability differs from the formula size complexity of f only by a polynomial factor. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 337–351 (1997)     

On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,C_max)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length C_max, where O(f(n,m,C_max)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.     

Extensive facility location problems on networks with equity measures

This paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case.     

On a Poset of Trees II

In this article, we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the so‐called generalized tree shift which seems to be a powerful tool to attack the problems of the above‐mentioned kind. We show that the generalized tree shift increases the largest eigenvalue of the adjacency matrix and Laplacian matrix, decreases the coefficients of the characteristic polynomials of these matrices in absolute value. We will prove similar theorems for the independence polynomial and the edge cover polynomial. The generalized tree shift induces a partially ordered set on trees having fixed number of vertices. The smallest element of this poset is the path, largest element is the star. Hence, the above‐mentioned results imply the extremality of the path and the star for these parameters.