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### Linear Verification For Spanning Trees

J. Komlos
1984 25th Annual Symposium onFoundations of Computer Science, 1984.
This leads to a spanning tree verification algorithm using O(n -e ) comparisons in a graph with n vertices and e edges. No implementation is offered.  ...  Given a rooted tree with values associated with the n pertices and a set A of 'directed paths (queries), we describe an algorithm which finds the maximum value of every one of the given paths, and which  ...  A 1 for 1 A ~ : , > n. thus the linear term O(e) for the tree verification problem ( j A ! en + l ) comes from the comparisons made between the maxima in A and the outside edges.  ...

### Linear verification for spanning trees

J. Komlós
1985 Combinatorica
' Here we describe an algorithm which finds maxima over wlrious paths o f a tree, which leads to a minimal spanning tree verification algorithm with a linear number o f comparisons.  ...  Whatever cost we obtain for this directed path problem, we only need 2 ( e -~, + I) extra comparisons for the spanning tree verification problem.  ...

### Randomized minimum spanning tree algorithms using exponentially fewer random bits

Seth Pettie, Vijaya Ramachandran
2008 ACM Transactions on Algorithms
The prominent exception -and the main focus of this paper -is a linear-time randomized minimum spanning tree algorithm that is not derived from the well known Karger-Klein-Tarjan algorithm.  ...  Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set  ...  Randomized Minimum Spanning Tree Algorithms · 27 ACKNOWLEDGMENTS We would like to thank David Zuckerman for his helpful suggestions.  ...

### Recognition of DFS trees: sequential and parallel algorithms with refined verifications

Ephraim Korach, Zvi Ostfeld
1993 Discrete Mathematics
Therefore, the question: Given an undirected graph G = (V, E) and an undirected spanning tree r, is T a DFS tree (T-DFS) in G? was naturally raised and answered by sequential linear-time algorithms.  ...  In a previous work we have shown that the family of graphs in which every spanning tree is a DFS tree is quite limited.  ...  This also enables us to simplify the optimal negative verification we had for this algorithm. We thank Michael Kaminski for pointing out some references on the complexity of matrix multiplication.  ...

### Page 4973 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews
(H-AOS) Linear verification for spanning trees. Combinatorica 5 (1985), no. 1, 57-65.  ...  This leads to a spanning tree verification algorithm using O(n + e) comparisons in a graph with n vertices and e edges. Fan R. K.  ...

### Sensitivity Analysis of Minimum Spanning Trees in Sub-Inverse-Ackermann Time [article]

Seth Pettie
2014 arXiv   pre-print
We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(mα(m,n)) time, where α is the inverse-Ackermann function.  ...  Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algoritm.  ...  I thank Bob Tarjan for encouraging me to finally publish the complete version of this paper.  ...

### Inheritance Operations in Massively Parallel Knowledge Representation [chapter]

James Geller
1994 Machine Intelligence and Pattern Recognition
We present a massively parallel representation of transitive relations, emphasizing the subclass relation, which extends our previous linear tree representation of class hierarchies.  ...  A linear graph representation is generated by a left-to-right preorder traversal of the spanning tree.  ...  Lemma 3b: Once the spanning tree has been established, the non-tree arcs of a graph have no influence on the tree pairs.  ...

### Sensitivity Analysis of Minimum Spanning Trees in Sub-Inverse-Ackermann Time

Seth Pettie
2015 Journal of Graph Algorithms and Applications
We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m log α(m, n)) time, where α is the inverse-Ackermann function.  ...  Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algorithm.  ...  Acknowledgements I thank Bob Tarjan for encouraging me to publish the complete version of this paper.  ...

### Sensitivity Analysis of Minimum Spanning Trees in Sub-inverse-Ackermann Time [chapter]

Seth Pettie
2005 Lecture Notes in Computer Science
We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m log α(m, n)) time, where α is the inverse-Ackermann function.  ...  Together with the randomized linear time MST algorithm of Karger, Klein, and Tarjan, this gives another randomized linear time MST sensitivity algorithm.  ...  Acknowledgements I thank Bob Tarjan for encouraging me to publish the complete version of this paper.  ...

### An Inverse-Ackermann Type Lower Bound For Online Minimum Spanning Tree Verification*

Seth Pettie†
2006 Combinatorica
Given a spanning tree T of some graph G, the problem of minimum spanning tree verification is to decide whether T = MST (G).  ...  Somewhat unexpectedly, MST verification turns out to be useful in actually computing minimum spanning trees from scratch.  ...  I thank Vijaya Ramachandran for many helpful comments, and Stephen Alstrup, Theis Rauhe, and Uri Zwick for reintroducing me to this problem.  ...

### A randomized linear-time algorithm to find minimum spanning trees

David R. Karger, Philip N. Klein, Robert E. Tarjan
1995 Journal of the ACM
The algorithm uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree.  ...  We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights.  ...  We thank Rajeev Motwani, Satish Rae, and David Zuckerman for fruitful discussions.  ...

### Extending the Limits of Ensemble Forecast Verification with the Minimum Spanning Tree

Leonard A. Smith, James A. Hansen
2004 Monthly Weather Review
spaces, including those of the verifications for 10 6 dimensional numerical weather prediction forecasts.  ...  Current rank histogram ensemble verification techniques can only evaluate scalars drawn from ensembles and associated verification; a new method is presented that allows verification in high-dimensional  ...  Smith for valuable conversations about ensemble assessment, and thank the Newton Institute at Cambridge University. The comments of two anonymous reviewers greatly enhanced the manuscript.  ...

### On Distributed Verification [chapter]

Amos Korman, Shay Kutten
2006 Lecture Notes in Computer Science
This talk was intended to give a partial survey and to motivate further studies of distributed verification.  ...  In this paper we explain some motivations for specific definitions, survey some very related notions and their motivations in the literature, survey some examples for problems and solutions, and mention  ...  In the simple model for local verification, both the spanning tree and the MST verification problems require Ω(n) time rounds. Sketch of Proof: We show the result for the spanning tree case.  ...

### Tight Bounds For Distributed MST Verification

Liah Kor, Amos Korman, David Peleg, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science
This paper establishes tight bounds for the Minimum-weight Spanning Tree (MST) verification problem in the distributed setting.  ...  Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of Ω(|E|) messages and Ω( √ n+D) time hold,  ...  The distributed MST Verification problem Formally, the minimum-weight spanning tree (MST) verification problem can be stated as follows.  ...

### Page 2263 of Mathematical Reviews Vol. , Issue 94d [page]

1994 Mathematical Reviews
Therefore, the question: Given an undirected graph G = (V, E) and an undirected spanning tree T, is T a DFS tree (J-DFS) in G? was naturally raised and answered by sequential linear-time algorithms.  ...  “In a previous work we showed that the family of graphs in which every spanning tree is a DFS tree is quite limited.  ...
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