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Journal ArticleDOI

On the History of the Minimum Spanning Tree Problem

01 Jan 1985-IEEE Annals of the History of Computing (IEEE)-Vol. 7, Iss: 1, pp 43-57
TL;DR: There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
Abstract: 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.
Citations
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Journal ArticleDOI
TL;DR: Using F-heaps, a new data structure for implementing heaps that extends the binomial queues proposed by Vuillemin and studied further by Brown, the improved bound for minimum spanning trees is the most striking.
Abstract: In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in O(log n) amortized time and all other standard heap operations in O(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms. In particular, we obtain the following worst-case bounds, where n is the number of vertices and m the number of edges in the problem graph: O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(mlog(m/n+2)n);O(n2log n + nm) for the all-pairs shortest path problem, improved from O(nm log(m/n+2)n);O(n2log n + nm) for the assignment problem (weighted bipartite matching), improved from O(nmlog(m/n+2)n);O(mβ(m, n)) for the minimum spanning tree problem, improved from O(mlog log(m/n+2)n); where β(m, n) = min {i | log(i)n ≤ m/n}. Note that β(m, n) ≤ log*n if m ≥ n.Of these results, the improved bound for minimum spanning trees is the most striking, although all the results give asymptotic improvements for graphs of appropriate densities.

2,484 citations


Cites background from "On the History of the Minimum Spann..."

  • ...Introduction The history of the minimum spanning tree (MST) problem is long and rich, going as far back as Borůvka’s work in 1926 [Borůvka 1926; Graham and Hell 1985; Nešetřil 1997]....

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Proceedings ArticleDOI
24 Oct 1984
TL;DR: The structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown to obtain improved running times for several network optimization algorithms.
Abstract: In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in 0(log n) amortized time and all other standard heap operations in 0(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms.

1,757 citations


Cites background from "On the History of the Minimum Spann..."

  • ...Introduction The history of the minimum spanning tree (MST) problem is long and rich, going as far back as Borůvka’s work in 1926 [Borůvka 1926; Graham and Hell 1985; Nešetřil 1997]....

    [...]

Journal ArticleDOI
TL;DR: A recently developed very efficient (linear time) hierarchical clustering algorithm is described, which can also be viewed as a hierarchical grid‐based algorithm.
Abstract: We survey agglomerative hierarchical clustering algorithms and discuss efficient implementations that are available in R and other software environments. We look at hierarchical self-organizing maps and mixture models. We review grid-based clustering, focusing on hierarchical density-based approaches. Finally, we describe a recently developed very efficient (linear time) hierarchical clustering algorithm, which can also be viewed as a hierarchical grid-based algorithm. This review adds to the earlier version, Murtagh F, Contreras P. Algorithms for hierarchical clustering: an overview, Wiley Interdiscip Rev: Data Mining Knowl Discov 2012, 2, 86–97. WIREs Data Mining Knowl Discov 2017, 7:e1219. doi: 10.1002/widm.1219 This article is categorized under: Algorithmic Development > Hierarchies and Trees Technologies > Classification Technologies > Structure Discovery and Clustering

977 citations

Book
31 Oct 1998
TL;DR: The Steiner Ratio of Banach-Minkowski Space and Probabilistic Verification and Non-Approximability and Network-Based Model and Algorithms in Data Mining and Knowledge Discovery are studied.
Abstract: A Unified Approach for Domination Problems on Different Network Topologies Advanced Techniques for Dynamic Programming Advances in Group Testing Advances in Scheduling Problems Algebrization and Randomization Methods Algorithmic Aspects of Domination in Graphs Algorithms and Metaheuristics for Combinatorial Matrices Algorithms for the Satisfiability Problem Bin Packing Approximation Algorithms: Survey and Classification Binary Unconstrained Quadratic Optimization Problem Combinatorial Optimization Algorithms for Probe Design and Selection Problems Combinatorial Optimization in Data Mining Combinatorial Optimization Techniques for Network-based Data Mining Combinatorial Optimization Techniques in Transportation and Logistic Networks Complexity Issues on PTAS Computing Distances between Evolutionary Trees Connected Dominating Set in Wireless Networks Connections between Continuous and Discrete Extremum Problems, Generalized Systems and Variational Inequalities Coverage Problems in Sensor Networks Data Correcting Approach for Routing and Location in Networks Dual Integrality in Combinatorial Optimization Dynamical System Approaches to Combinatorial Optimization Efficient Algorithms for Geometric Shortest Path Query Problems Energy Efficiency in Wireless Networks Equitable Coloring of Graphs Faster and Space Efficient Exact Exponential Algorithms: Combinatorial and Algebraic Approaches Fault-Tolerant Facility Allocation Fractional Combinatorial Optimization Fuzzy Combinatorial Optimization Problems Geometric Optimization in Wireless Networks Gradient-Constrained Minimum Interconnection Networks Graph Searching and Related Problems Graph Theoretic Clique Relaxations and Applications Greedy Approximation Algorithms Hardness and Approximation of Network Vulnerability Job Shop Scheduling with Petri Nets Key Tree Optimization Linear Programming Analysis of Switching Networks Map of Geometric Minimal Cuts with Applications Max-Coloring Maximum Flow Problems and an NP-complete variant on Edge Labeled Graphs Modern Network Interdiction Problems and Algorithms Network Optimization Neural Network Models in Combinatorial Optimization On Coloring Problems Online and Semi-online Scheduling Online Frequency Allocation and Mechanism Design for Cognitive Radio Wireless Networks Optimal Partitions Optimization in Multi-Channel Wireless Networks Optimization Problems in Data Broadcasting Optimization Problems in Online Social Networks Optimizing Data Collection Capacity in Wireless Networks Packing Circles in Circles and Applications Partition in High Dimensional Spaces Probabilistic Verification and Non-approximability Protein Docking Problem as Combinatorial Optimization Using Beta-complex Quadratic Assignment Problems Reactive Business Intelligence: Combining the Power of Optimization with Machine Learning Reformulation-Linearization Techniques for Discrete Optimization Problems Resource Allocation Problems Rollout Algorithms for Discrete Optimization: A Survey Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimizations Small World Networks in Computational Neuroscience Social Structure Detection Steiner Minimal Trees: An Introduction, Parallel Computation and Future Work Steiner Minimum Trees in E^3 Tabu Search Variations of Dominating Set Problem

921 citations

References
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Journal ArticleDOI
TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

22,704 citations

Journal ArticleDOI
01 Feb 1956
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings, Trans Amer Math Soc vol 74 (1953) pp 384-409

5,104 citations

Journal ArticleDOI
S. C. Johnson1
TL;DR: A useful correspondence is developed between any hierarchical system of such clusters, and a particular type of distance measure, that gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data.
Abstract: Techniques for partitioning objects into optimally homogeneous groups on the basis of empirical measures of similarity among those objects have received increasing attention in several different fields. This paper develops a useful correspondence between any hierarchical system of such clusters, and a particular type of distance measure. The correspondence gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data. In an explicitly defined sense, one method forms clusters that are optimally “connected,” while the other forms clusters that are optimally “compact.”

4,560 citations

Journal ArticleDOI
TL;DR: In this paper, the basic problem of interconnecting a given set of terminals with a shortest possible network of direct links is considered, and a set of simple and practical procedures are given for solving this problem both graphically and computationally.
Abstract: The basic problem considered is that of interconnecting a given set of terminals with a shortest possible network of direct links Simple and practical procedures are given for solving this problem both graphically and computationally It develops that these procedures also provide solutions for a much broader class of problems, containing other examples of practical interest

4,395 citations

Book
01 Jan 1963
TL;DR: The authors continued the story of psychology with added research and enhanced content from the most dynamic areas of the field, such as cognition, gender and diversity studies, neuroscience and more, while at the same time using the most effective teaching approaches and learning tools.
Abstract: This new edition continues the story of psychology with added research and enhanced content from the most dynamic areas of the field--cognition, gender and diversity studies, neuroscience and more, while at the same time using the most effective teaching approaches and learning tools

3,332 citations