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The Minimum Spanning Tree Problem on a Planar Graph
 Discrete Appl. Math
, 1994
"... cle. A spanning forest of G is called a spanning tree when it is connected. In this note, we present a spanning forest as its edge set. Given a graph G and its edge e; Gne denotes the graph obtained by deleting the edge e and G=e denotes the graph obtained by contracting e: For each edge e 2 E; w( ..."
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cle. A spanning forest of G is called a spanning tree when it is connected. In this note, we present a spanning forest as its edge set. Given a graph G and its edge e; Gne denotes the graph obtained by deleting the edge e and G=e denotes the graph obtained by contracting e: For each edge e 2 E; w(e) denotes the weight of the edge e: The weight of an edge subset F; denoted be w(F ); is the sum of the weights of edges in F: A maximal spanning forest F is called a minimum (maximum) weight spanning forest, when F minimizes (maximizes) the weight w(F ): A graph is called planar if it can be drawn in th
PROXIMITY PROBLEMS FOR POINTS ON A RECTILINEAR PLANE WITH RECTANGULAR OBSTACLES
, 1993
"... We consider the following four problems for a set S of k points on a plane, equipped with the rectilinear metric and containing a set R of n disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles in R): (a) nd a closest pair of points in S, (b) ..."
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We consider the following four problems for a set S of k points on a plane, equipped with the rectilinear metric and containing a set R of n disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles in R): (a) nd a closest pair of points in S, (b) nd a nearest neighbor for each point inS, (c) compute the rectilinear Voronoi diagram of S, and (d) compute a rectilinear minimal spanning tree of S. Wedescribe O((n + k) log(n + k)) time sequential algorithms for (a) and (b) based on planesweep, and the consideration of geometrically special types of shortest paths, socalled z rst paths. For (c) we present an O((n + k) log(n + k)logn) time sequential algorithm that implements a sophisticated divideandconquer scheme with an added extension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular socalled zdiagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved in O((n + k) log(n + k)logn) time as well. All our algorithms are nearoptimal, as well as easy to implement.
A lineartime approximation scheme for TSP for planar weighted graphs
 In Proceedings, 46th IEEE Symposium on Foundations of Computer Science
, 2005
"... edgeweights ..."
How to Find a Minimum Spanning Tree in Practice
 results and New Trends in Computer Science, volume 555 of Lecture Notes in Computer Science
, 1991
"... We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through lowlevel implementation choices, to purely algor ..."
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We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through lowlevel implementation choices, to purely algorithmic issues. We discuss how to design a careful experimental comparison between various alternatives. Finally, we present some results from an ongoing study in which we are using: multiple languages, compilers, and machines; all the major variants of the comparisonbased algorithms; and eight varieties of graphs with sizes of up to 130,000 vertices (in sparse graphs) or 750,000 edges (in dense graphs). 1 Introduction Finding spanning trees of minimum weight (minimum spanning trees or MSTs) is one of the best known graph problems; algorithms for this problem have a long history, for which see the article of Graham and Hell [6]. The best comparisonbased algorithm to date, due to Gabow...
LinearTime Algorithms for Parametric Minimum Spanning Tree Problems on Planar Graphs
, 1995
"... A lineartime algorithm for the minimumratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having s ..."
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A lineartime algorithm for the minimumratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. 1 Introduction Suppose we are given an undirected graph G where each edge e has two weights a e and b e ; the b e 's are assumed to be either all negative or all positive. The minimum ratio spanning tree problem (MRST) [Cha77] is to find a spanning tree T of G such that the ratio P e2T a e = P e2T b e is minimized. One application of MRST arises in the design of communication networks. The number a e represents the cost of building link e, while b e represents the time required to build that link. The goal is to find a tree that minimizes the ratio of total cost over construction time. Other applications of MRST are given elsewhere [CMV89, Meg83]. The main result of thi...
Minimum Spanning Tree Pose Estimation
"... The extrinsic camera parameters from video stream images can be accurately estimated by tracking features through the image sequence and using these features to compute parameter estimates. The poses for long video sequences have been estimated in this manner. However, the poses of large sets of sti ..."
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The extrinsic camera parameters from video stream images can be accurately estimated by tracking features through the image sequence and using these features to compute parameter estimates. The poses for long video sequences have been estimated in this manner. However, the poses of large sets of still images cannot be estimated using the same strategy because widebaseline correspondences are not as robust as narrowbaseline feature tracks. Moreover, video pose estimation requires a linear or hierarchicallylinear ordering on the images to be calibrated, reducing the image matches to the neighboring video frames. We propose a novel generalization to the linear ordering requirement of video pose estimation by computing the Minimum Spanning Tree of the camera adjacency graph and using the tree hierarchy to determine the calibration order for a set of input images. We validate the pose accuracy using an error metric that is functionally independent of the estimation process. Because we do not rely on feature tracking for generating feature correspondences, our method can use internally calibrated wide or narrowbaseline images as input, and can estimate the camera poses from multiple video streams without special preprocessing to concatenate the streams. 1
Optimal Algorithms to Find the Most Vital Edge of a Minimum Spanning Tree
, 1995
"... The problem of finding the most vital edge with respect to a minimum spanning tree of a given connected and weighted graph (with m edges and n vertices) is considered. New sequential and parallel algorithms (3 each) for the problem are proposed, and a lower bound\Omega\Gamma m) is established. We c ..."
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The problem of finding the most vital edge with respect to a minimum spanning tree of a given connected and weighted graph (with m edges and n vertices) is considered. New sequential and parallel algorithms (3 each) for the problem are proposed, and a lower bound\Omega\Gamma m) is established. We characterize the set of entering edges and show that the cardinality of this set is O(n). We show the connection between most vital edge problem and the minimum spanning tree update problems and exploit this idea in developing one of the proposed sequential algorithms. Two of our sequential algorithms are optimal. One of our parallel algorithms is optimal if the underlying graph is dense, or planar. We also consider a related problem for weighted matroids. Keywords: Data structures, design of algorithms, parallel algorithms, minimum spanning trees, most vital edge, complexity, matroids. 1 INTRODUCTION Networks are ubiquitous in many scientific and technological applications. A few examples ...
Design and Analysis of CacheConscious Programs
, 1999
"... algorithms are presented in some examples. This work is about experimental algorithmics, and the methodology is therefore based on experiments. All important theory is experimentally evaluated. All experiments are made by the author; when I refer to other experimental work, it is not in direct compa ..."
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algorithms are presented in some examples. This work is about experimental algorithmics, and the methodology is therefore based on experiments. All important theory is experimentally evaluated. All experiments are made by the author; when I refer to other experimental work, it is not in direct comparison. Preprocessing input data to some required data structure is considered a part of the algorithm, that is, the time for reading the input stream to some data structure is measured as part of a program's execution. The aim is the construction of an analytical model for predicting the behaviour of the memory hierarchy, and attempts are made to discriminate the "random noise" from the execution of programs. This noise is considered to partly, but heavily, depend on the pattern of memory references of a program. With the knowledge of when memory references happen in the program and where the references are made in the different levels of the memory hierarchy, an analytical method is propos...
CPlanarity of CConnected Clustered Graphs  Part I – Characterization
, 2006
"... We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2 ..."
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We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2] we exploit such a characterization to give a linear time cplanarity testing and embedding algorithm.
Computational Morphology of Implicit Curves
, 1992
"... We divide the computation of polygonal approximations of implicit objects into two phases: sampling and structuring. Unlike classical treatments, we study each phase separately. Classical sampling methods are reviewed, and a new sampling method is proposed; this method uses physicallybased particle ..."
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We divide the computation of polygonal approximations of implicit objects into two phases: sampling and structuring. Unlike classical treatments, we study each phase separately. Classical sampling methods are reviewed, and a new sampling method is proposed; this method uses physicallybased particle systems, and is more robust than classical enumeration and continuation methods. We describe a broad taxonomy of structuring problems. According to this taxonomy, the presence of an independent structuring phase casts the modeling problem as a problem in nonparametric scattered data interpolation, which we propose to solve using computational morphology. The use of global structuring is described in detail for curves: we prove that minimal spanning trees are good morphological tools for arcs. A couple of methods are also suggested for surfaces.