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### A Linear Time Algorithm for Finding Minimum Spanning Tree Replacement Edges [article]

David A. Bader, Paul Burkhardt
2020 arXiv   pre-print
Given an undirected, weighted graph, the minimum spanning tree (MST) is a tree that connects all of the vertices of the graph with minimum sum of edge weights.  ...  Given the MST and sorted non-tree edges, our algorithm is the first that runs in O(m+n) time and O(m+n) space to find all replacement edges.  ...  Thus, our approach will also find the most vital edge in O(m + n) time, and is the first linear algorithm for finding the most vital edge of the minimum spanning tree given the non-tree edges sorted by  ...

### Linear Time Algorithms Based on Multilevel Prefix Tree for Finding Shortest Path with Positive Weights and Minimum Spanning Tree in a Networks [article]

David S. Planeta
2007 arXiv   pre-print
In this paper I present general outlook on questions relevant to the basic graph algorithms; Finding the Shortest Path with Positive Weights and Minimum Spanning Tree.  ...  It should be noticed that the algorithms which compute the Shortest Path and Minimum Spanning Tree problems not only analyze the weight of arcs (which is the main and often the only criterion of solution  ...  If Q is implemented as a binary min-heap, the total time for Jarnik-Prim's algorithm is O(|V |log|V | + |E|log|V |) = O(|E|log|V |).  ...

### Efficient Algorithms for Finding the k Most Vital Edges for the Minimum Spanning Tree Problem [chapter]

Cristina Bazgan, Sonia Toubaline, Daniel Vanderpooten
2011 Lecture Notes in Computer Science
We study in this paper the problem of finding in a graph a subset of k edges whose deletion causes the largest increase in the weight of a minimum spanning tree.  ...  We propose for this problem an explicit enumeration algorithm whose complexity, when compared to the current best algorithm, is better for general k but very slightly worse for fixed k.  ...  The sensitivity of a minimum spanning tree T , i.e. the allowable variation for each edge weight so that T remains a minimum spanning tree, can be computed in O(m log α(m, n))  .  ...

### An ant-based algorithm for finding degree-constrained minimum spanning tree

Thang N. Bui, Catherine M. Zrncic
2006 Proceedings of the 8th annual conference on Genetic and evolutionary computation - GECCO '06
In this paper we give an Ant-Based algorithm for finding low cost degree-constrained spanning trees.  ...  The problem of finding the degree-constrained spanning tree of minimum cost in an edge weighted graph is well known to be NP-hard.  ...  We also would like to thank the anonymous reviewers for their helpful comments.  ...

### A Distributed Algorithm for Finding All Best Swap Edges of a Minimum-Diameter Spanning Tree

Beat Gfeller, Nicola Santoro, Peter Widmayer
2011 IEEE Transactions on Dependable and Secure Computing
algorithms for two variants of this problem.  ...  Finally, we consider the computation of swap edges in an arbitrary spanning tree, where swap edges are chosen to minimize the time required to adapt routing in case of a failure, and give efficient distributed  ...  For shortest paths trees (as opposed to minimum diameter spanning trees), an earlier centralized algorithm  has been complemented by a distributed algorithm using techniques for finding all best swap  ...

### An Õ(n2) algorithm for minimum cuts

David R. Karger, Clifford Stein
1993 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing - STOC '93
The rst is an (m 2 =n)-processor N C algorithm for nding a (2 + )-approximation to the minimum cut.  ...  Another application of these techniques yield an N C algorithm for nding a sparse kconnectivity certi cate for all polynomially bounded values of k.  ...  For each edge e in G that connects trees T and U, we add an edge e F connecting v T and v U . Thus, the reduced graph is what we get if we start with G and contract all the forest edges.  ...

### An O(logn)-Competitive Algorithm for Online Constrained Forest Problems [chapter]

Jiawei Qian, David P. Williamson
2011 Lecture Notes in Computer Science
Agrawal, Klein, and Ravi  give a 2-approximation algorithm for the offline problem; Berman and Coulston  give an O(log n)-competitive algorithm for the online problem.  ...  In this paper, we show how to combine the ideas of Goemans and Williamson and those of Berman and Coulston to give an O(log n)-competitive algorithm for online constrained forest problems, including an  ...  The goal of the Steiner tree problem is to find a minimum-cost tree T that spans all the terminals R.  ...

### An O(log n/log log n)-Approximation Algorithm for the Asymmetric Traveling Salesman Problem

Arash Asadpour, Michel X. Goemans, Aleksander Mądry, Shayan Oveis Gharan, Amin Saberi
2017 Operations Research
We present a randomized O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem (ATSP).  ...  The key ingredient of our approach is a new connection between the approximability of the ATSP and the notion of so-called thin trees.  ...  Algorithm 1 1 An O(log n/ log log n)-approximation algorithm for the ATSP. Input: A set V consisting of n points and a cost function c : V × V → R + satisfying the triangle inequality.  ...

### An algorithmic approach for finding minimum spanning tree in a intuitionistic fuzzy graph

Kartick Mohanta, Arindam Dey, Narayan C. Debnath, Anita Pal
unpublished
We define this problem as intuitionistic minimum spanning tree (IMST) problem. We introduce an algorithmic approach for designing the IMST of a fuzzy.  ...  The minimum spanning tree (MST) problem is a well known optimization problem in graph theory that has been used to model many real life problems, e.g., telecommuni- cations,transportation network, routing  ...  Modified Boruvka's Algorithm for IMST The classical Boruvka's algorithm is a well known greedy method for finding the MST of a connected graph and the time complexity of this algorithm is O(m log n) time  ...

### An O(n2(m + n log n) log n) min-cost flow algorithm

Zvi Galil, Eva Tardos
1986 27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
Galil and E. Tardos, An O(n 2 (m + n log n) logn) min-cost flow algorithm.  ...  In this paper we design an O( n 2 ( m + n log n) log n) algorithm. The previous best algorithm had an O( m 2 ( m+n log n) log n) time bound.  ...  By Corollary 5.2 we know that Step 1 is executed at most 2m times, each time computing maximum weight spanning tree in time Oem log n) [Tn] for a total of O(m 2 logn).  ...

### An almost O (log k )-approximation for k -connected subgraphs

Zeev Nutov
2009 ACM-SIAM Symposium on Discrete Algorithms
We consider two cases of the Survivable Network Design (SND) problem: given a complete graph G n = (V, E n ) with costs on the edges and connectivity requirements {r(u, v) : u, vV }, find a minimum  ...  Our main result is an O log k • log n n−kapproximation algorithm for the k-Connected Subgraph problem (the case r(u, v) = k for all u, vV ), for both directed and undirected graphs, where n = |V |.  ...  Objective: Find a minimum cost k-connected spanning subgraph G of G n .  ...

### Construction and impromptu repair of an MST in a distributed network with o(m) communication [article]

Valerie King, Shay Kutten, Mikkel Thorup
2015 arXiv   pre-print
knows n, then there are Monte Carlo algorithms which succeed w.h.p. to determine a minimum spanning forest (MST) and a spanning forest (ST) using O(n ^2 n/ n) messages for MST and O(n n ) messages for  ...  Previous algorithms for this problem that use an amortized o(m) messages per update require substantial preprocessing and additional local storage between updates.  ...  We show that O(log n/ log log n) broadcast-and-echoes suffice. In the previous subsection, the log n-wise "pivots" were chosen obliviously. Here, we use pivots based on randomly chosen edges.  ...

### An O(n logn) heuristic for steiner minimal tree problems on the euclidean metric

J. Macgregor Smith, D. T. Lee, Judith S. Liebman
1981 Networks
An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented.  ...  to the Voronoi diagram and Minimum Spanning Tree (MST) of the point set.  ...  The authors would like to thank the referees for their valuable criticism and the Graduate Research Board of the University of Illinois for providing the necessary funds for conducting the computer experiments  ...

### A Distributed Algorithm for Finding All Best Swap Edges of a Minimum Diameter Spanning Tree [chapter]

Beat Gfeller, Nicola Santoro, Peter Widmayer
Lecture Notes in Computer Science
algorithms for two variants of this problem.  ...  Finally, we consider the computation of swap edges in an arbitrary spanning tree, where swap edges are chosen to minimize the time required to adapt routing in case of a failure, and give efficient distributed  ...  ACKNOWLEDGEMENTS We would like to thank the anonymous reviewers for their comments, which helped to improve the presentation of this work.  ...

### An O(1)-Approximation for Minimum Spanning Tree Interdiction [article]

Rico Zenklusen
2015 arXiv   pre-print
One of the oldest and best-studied interdiction problems is minimum spanning tree (MST) interdiction.  ...  Frederickson and Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction, where m is the number of edges.  ...  Ravi, and the anonymous reviewers for many helpful comments.  ...
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