On Constructing Minimum Spanning Trees in R k 1

We give an algorithm to find a minimum spanning tree in the k-dimensional space under rectilinear metric. The running time is for k≥ 3. This improves the previous bound by a factor . Received January 10, 1995; revised December 21, 1995.     

基于最小代价和生成树的算法研究

最小生成树问题是一类经典的组合优化问题,已有许多快速有效的算法。但是在实际中更多存在这样的网络,除边有权值外,结点也有权值,并且结点的权值有多种情况,这就产生了基于代价和最小的生成树问题。根据结点权值的取值情况,对几种最小代价和生成树问题进行分类和求解,得到了一些有价值的性质和算法,有一定的实际应用背景。     

Lower bounds for testing Euclidean Minimum Spanning Trees

The Euclidean Minimum Spanning Tree problem is to decide whether a given graph G=(P,E) on a set of points in the two-dimensional plane is a minimum spanning tree with respect to the Euclidean distance. Czumaj et al. [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] gave a 1-sided-error non-adaptive property-tester for this task of query complexity . We show that every non-adaptive (not necessarily 1-sided-error) property-tester for this task has a query complexity of , implying that the test in [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] is of asymptotically optimal complexity. We further prove that every adaptive property-tester has query complexity of Ω(n1/3). Those lower bounds hold even when the input graph is promised to be a bounded degree tree.     

On the Complexity of Approximating Euclidean Traveling Salesman Tours and Minimum Spanning Trees

We consider the problems of computing r-approximate traveling salesman tours and r-approximate minimum spanning trees for a set of n points in , where d≥ 1 is a constant. In the algebraic computation tree model, the complexities of both these problems are shown to be , for all n and r such that r < n and r is larger than some constant. In the more powerful model of computation that additionally uses the floor function and random access, both problems can be solved in O(n) time if . Received January 8, 1996; revised July 15, 1996.     

Hierarchical Segmentations with Graphs: Quasi-flat Zones,Minimum Spanning Trees,and Saliency Maps

Hierarchies of partitions are generally represented by dendrograms (direct representation). They can also be represented by saliency maps or minimum spanning trees. In this article, we precisely study the links between these three representations. In particular, we provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as often used in image processing and we characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results make up a toolkit for designing new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies. More precisely, we show the practical interest of the proposed framework for: (i) hierarchical watershed image segmentations, (ii) combinations of different hierarchical segmentations, (iii) hierarchicalizations of some non-hierarchical image segmentation methods based on regional dissimilarities, and (iv) hierarchical analysis of geographic data.     

最小生成树用于基因表示数据的聚类算法

在生物学研究中，需要对植物和动物分类，对基因进行分类，以获得对种群固有结构的认识．使用聚类分析方法，有效地鉴别基因表示数据的模式，将它们分组成为由类似对象组成的多个类，对研究基因的结构、功能以及不同种类基因之间的关系都具有重要意义．将图论的最小生成树理论引入分子生物学中基因表示数据的聚类分析方法，设计了生成树的表示和基于最小生成树的聚类算法，证明了该方法对于一些准则函数能够产生全局最优簇，并根据实验结果对算法进行了讨论和评价．     

Distributed Algorithms for Constructing Approximate Minimum Spanning Trees in Wireless Sensor Networks

While there are distributed algorithms for the MST problem, these algorithms require relatively large number of messages and time; this makes these algorithms impractical for resource-constrained networks such as ad hoc wireless sensor networks. In such networks, a sensor has very limited power, and any algorithm needs to be simple, local, and energy efficient for being practical. Motivated by these considerations, we design and analyze a class of simple and local distributed algorithms called Nearest Neighbor Tree (NNT) algorithms for energy-efficient construction of MSTs in a wireless ad hoc setting. We assume that the nodes are uniformly distributed in a unit square and show provable bounds on the performance with respect to both the quality of the spanning tree produced and the energy needed to construct them. In particular, we show that NNT produces a close approximation to the MST, and they can be maintained dynamically with polylogarithmic number of rearrangements under node insertions/deletions. We also perform extensive simulations of our algorithms. We tested our algorithms on both uniformly random distributions of nodes, and on a realistic distributions of nodes in an urban setting. Simulations validate the theoretical results and show that the bounds are much better in practice.     

Using Bounded Diameter Minimum Spanning Trees to Build Dense Active Appearance Models

We present a method for producing dense active appearance models (AAMs), suitable for video-realistic synthesis. To this end we estimate a joint alignment of all training images using a set of pairwise registrations and ensure that these pairwise registrations are only calculated between similar images. This is achieved by defining a graph on the image set whose edge weights correspond to registration errors and computing a bounded diameter minimum spanning tree. Dense optical flow is used to compute pairwise registration and a flow refinement method to align small scale texture is introduced. Further, given the registration of training images, vertices are added to the AAM to minimise the error between the observed flow fields and the flow fields interpolated between the AAM mesh points. We demonstrate a significant improvement in model compactness.     

On Cartesian Trees and Range Minimum Queries

We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cache-oblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree for the bottleneck edge query problem on trees and undirected graphs, and a proof that no Cartesian tree exists for the two-dimensional version of the range minimum query problem.     

On the History of the Minimum Spanning Tree Problem

It is standard practice among authors discussing the minimum spanning tree problem to refer to the work of Kruskal(1956) and Prim (1957) as the sources of the problem and its first efficient solutions, despite the citation by both of Boruvka (1926) as a predecessor. In fact, there are several apparently independent sources and algorithmic solutions of the problem. They have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century. We shall explore and compare these works and their motivations, and relate them to the most recent advances on the minimum spanning tree problem.     

一种最小生成树聚类算法

现有的聚类算法都不能在输入较少参数的情况下得到任意形状任意密度的类.提出一种最小生成树的聚类算法,该算法不但能解决上述问题,还能处理高维数据,发现异常点,且具有扩展性.针对该算法提出一个目标函数,该函数根据对象的类属情况和相似度统计信息来判别聚类效果的质量.最后,通过实验验证了该算法的聚类质量很好,目标函数具有良好的收敛性.     

Maintaining Spanning Trees of Small Diameter

Given a graph G with m edges and n nodes, a spanning tree T of G , and an edge e that is being deleted from or inserted into G , we give efficient O(n) algorithms to compute a possible swap for e that minimizes the diameter of the new spanning tree. This problem arises in high-speed networks, particularly in optical networks. Received January 1995; revised February 1997.     

最小生成树是否唯一

最小生成树是图论的经典问题,求最小生成树以及求最小生成树的权值和得到了足够关注,而很少人去研究最小生成树是否唯一.对于给定的图而言,因为最小生成树的权值和是确定的,所以最小生成树不唯一当且仅当最小生成树的形状不唯一.本文提出判断最小生成树是否唯一的三种方法并且对它们给予分析和评价.     

The Stackelberg Minimum Spanning Tree Game

We consider a one-round two-player network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor??s prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the well-studied Stackelberg shortest-path game. We analyze the complexity and approximability of the first player??s best strategy in StackMST. In particular, we prove that the problem is APX-hard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most min?{k,1+ln?b,1+ln?W}, where k is the number of distinct red costs, b is the number of blue edges, and W is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm.     

Spanning Trees and the Complexity of Flood-Filling Games

We consider problems related to the combinatorial game (Free-) Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time.     

Optimal Independent Spanning Trees on Odd Graphs

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and bandwidth. The designs of multiple ISTs on several classes of networks have been widely investigated. In this paper we show a construction algorithm of ISTs on odd graphs, and we analyze that all the lengths of the paths in the ISTs are less than or equal to the length of the shortest path+4, which is optimal. We also prove that the heights of the ISTs we constructed are d+1, which again is optimal, since the fault diameter of an odd graph is d+1.     

一种求解最小生成树问题的算法

基于节点合并和反向追踪的思想,提出一种求解最小生成树问题的算法。该算法依据网络邻接矩阵,将与源节点相邻的节点逐步合并为新的源节点,使网络中的所有节点合并为一个点,借助引入的前点标号数组得到网络的最小生成树,对算法正确性与算法复杂度进行分析。将该算法应用于某高速公路网工程建设方案,结果证明了算法的有效性。