An upper bound for the probability of a union

The problem of bounding P(∪ Ai ) given P(A i) and P(A i A j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

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An upper bound for the probability of a union

Journal of Applied Probability ◽

10.2307/3212481 ◽

1976 ◽

Vol 13 (3) ◽

pp. 597-603 ◽

Cited By ~ 179

Author(s):

David Hunter

Keyword(s):

Graph Theory ◽

Upper Bound ◽

Spanning Tree ◽

Upper Bounds ◽

Minimal Spanning Tree ◽

Multivariate Normal ◽

Tail Probabilities ◽

Tree Algorithm ◽

New Family ◽

The Family

The problem of bounding P(∪ Ai) given P(Ai) and P(AiAj) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

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A Piecewise Solution to the Reconfiguration Problem by a Minimal Spanning Tree Algorithm

International Journal of Emerging Electric Power Systems ◽

10.1515/ijeeps-2013-0094 ◽

2014 ◽

Vol 15 (5) ◽

pp. 419-427

Author(s):

Juan M. Ramirez ◽

Diana P. Montoya

Keyword(s):

Graph Theory ◽

Combinatorial Optimization ◽

Radial Distribution ◽

Spanning Tree ◽

Distribution Systems ◽

Optimization Problem ◽

Combinatorial Optimization Problem ◽

Minimal Spanning Tree ◽

Power Losses ◽

Tree Algorithm

Abstract This paper proposes a minimal spanning tree (MST) algorithm to solve the networks’ reconfiguration problem in radial distribution systems (RDS). The paper focuses on power losses’ reduction by selecting the best radial configuration. The reconfiguration problem is a non-differentiable and highly combinatorial optimization problem. The proposed methodology is a deterministic Kruskal’s algorithm based on graph theory, which is appropriate for this application generating only a feasible radial topology. The proposed MST algorithm has been tested on an actual RDS, which has been split into subsystems.

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Optimal feeder reconfiguration in distributed generation environment under time-varying loading condition

SN Applied Sciences ◽

10.1007/s42452-021-04557-w ◽

2021 ◽

Vol 3 (6) ◽

Author(s):

Yanrenthung Odyuo ◽

Dipu Sarkar ◽

Lilika Sumi

Keyword(s):

Graph Theory ◽

Distributed Generation ◽

Spanning Tree ◽

Voltage Stability ◽

Electrical Network ◽

List Type ◽

Network Reconfiguration ◽

Time Varying ◽

Electrical Networks ◽

Tree Algorithm

Abstract The development and planning of optimal network reconfiguration strategies for electrical networks is greatly improved with proper application of graph theory techniques. This paper investigates the application of Kruskal's maximal spanning tree algorithm in finding the optimal radial networks for different loading scenarios from an interconnected meshed electrical network integrated with distributed generation (DG). The work is done with an objective to assess the prowess of Kruskal's algorithm to compute, obtain or derive an optimal radial network (optimal maximal spanning tree) that gives improved voltage stability and highest loss minimization from among all the possible radial networks obtainable from the DG-integrated mesh network for different time-varying loading scenarios. The proposed technique has been demonstrated on a multiple test systems considering time-varying load levels to investigate the performance and effectiveness of the suggested method. For interconnected electrical networks with the presence of distributed generation, it was found that application of Kruskal's algorithm quickly computes optimal radial configurations that gives the least amount of power losses and better voltage stability even under varying load conditions. Article Highlights Investigated network reconfiguration strategies for electrical networks with the presence of Distributed Generation for time-varying loading conditions. Investigated the application of graph theory techniques in electrical networks for developing and planning reconfiguration strategies. Applied Kruskal’s maximal spanning tree algorithm to obtain the optimal radial electrical networks for different loading scenarios from DG-integrated meshed electrical network.

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Improved Visualization of Frequent Itemset Relationships Using the Minimal Spanning Tree Algorithm

Tehnicki vjesnik - Technical Gazette ◽

10.17559/tv-20171109130510 ◽

2019 ◽

Vol 26 (2) ◽

Keyword(s):

Spanning Tree ◽

Frequent Itemset ◽

Minimal Spanning Tree ◽

Tree Algorithm

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A Minimal Spanning Tree algorithm for source detection in γ-ray images

Monthly Notices of the Royal Astronomical Society ◽

10.1111/j.1365-2966.2007.12616.x ◽

2007 ◽

Vol 383 (3) ◽

pp. 1166-1174 ◽

Cited By ~ 27

Author(s):

Riccardo Campana ◽

Enrico Massaro ◽

Dario Gasparrini ◽

Sara Cutini ◽

Andrea Tramacere

Keyword(s):

Spanning Tree ◽

Minimal Spanning Tree ◽

Source Detection ◽

Tree Algorithm ◽

Γ Ray

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A Computational Perspective on Network Coding

Mathematical Problems in Engineering ◽

10.1155/2010/436354 ◽

2010 ◽

Vol 2010 ◽

pp. 1-11

Author(s):

Qin Guo ◽

Mingxing Luo ◽

Lixiang Li ◽

Yixian Yang

Keyword(s):

Graph Theory ◽

Computational Complexity ◽

Lower Bound ◽

Network Coding ◽

Upper Bound ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Multicast Networks ◽

Computational Perspective ◽

Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

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