An Algorithm for Finding <i>K</i> Minimum Spanning Trees

Katoh, Naoki; Ibaraki, Toshihide; Mine, Hisashi

doi:10.1137/0210017

Cited by 71 publications

(44 citation statements)

References 10 publications

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“…From this paper, we conclude that the K shortest path trees problem has the same difficulty as the K minimum spanning trees problem (see Katoh et al [16]). This result is possible since in both problems, the kth best solution is adjacent to at least one of the k−1 best previous solutions.…”

Section: Discussionmentioning

confidence: 86%

On the K shortest path trees problem

González-Martín

2010

European Journal of Operational Research

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We address the problem for finding the K best path trees connecting a source node with any other nonsource node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k−1) shortest path trees. In this paper, we consider the K shortest path trees problem as the problem to determine the K best basis trees (solutions) of the classical mathematical formulation of the SP problem.The determination of the K shortest paths in a network has a wide range of applications. Some of them are cited in Eppstein [8]. We are unaware of any previous references to this problem in the literature. A proof is offered which shows that the kth best basis tree is adjacent to at least one of the previous k−1 best basis trees. In other words, the kth best solution is obtained from one of the previous best solutions by exchanging an arc in the basis tree for an arc outside of the basis tree. This results allows an algorithm to be designed running in

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Section: Discussionmentioning

confidence: 86%

On the K shortest path trees problem

González-Martín

2010

European Journal of Operational Research

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“…Recently, Frederickson's technique for maintaining a MST in a dynamic planar graph has been improved [12,17], leading to a bound for the k best spanning trees problem of O(n + k 2 log n); this improves the bound of [18] whenever k = O(n/ log n).…”

Section: Introductionmentioning

confidence: 90%

“…Then, we show that many edges not in the MST can also not be in any of the k best spanning trees; therefore, we can remove them from the graph, leaving only O(k) edges. Finally, we apply the algorithm of [18], which takes time O(k 2 ) on the reduced graph. Our algorithms for geometric spanning trees similarly rely on reducing the problem to a small graph, but are more complicated.…”

Section: Whenever K = O(m)mentioning

confidence: 99%

“…This can be solved in time O(n log n), using the fact that all MST edges must appear in the Delauney triangulation of the points [3]. Strangely, the problem of finding the k best minimum spanning trees of a set of points in the plane does not seem to have been studied; however it can clearly be solved in time O(kn 2 ) using the technique of [18] on the complete graph of all pairs of points.…”

Section: Introductionmentioning

confidence: 99%

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Finding the k smallest spanning trees

Eppstein

1990

Swat 90

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We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log β(m, n) + k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of points in the plane can be computed in time O(min(k 2 n + n log n, k 2 + kn log(n/k))). The k best orthogonal spanning trees in the plane can be found in time O(n log n + kn log log(n/k) + k 2 ).

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“…The problem of computing the k MST in order is well known and can be solved in O(k · n 2 ) for a complete graph (Katoh, Ibaraki, & Mine, 1981). It is easy to see that if instead of computing the MST we compute the k MST and their relative weights then we have an algorithm that finds the k MAP undirected tree structures and their relative probabilities.…”

Section: Calculating the K Map Undirected Tree Structures And Their Rmentioning

confidence: 99%

TAN Classifiers Based on Decomposable Distributions

Cerquides

Mántaras

2005

Mach Learn

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Abstract. In this paper we present several Bayesian algorithms for learning Tree Augmented Naive Bayes (TAN) models. We extend the results in Meila & Jaakkola (2000a) to TANs by proving that accepting a prior decomposable distribution over TAN's, we can compute the exact Bayesian model averaging over TAN structures and parameters in polynomial time. Furthermore, we prove that the k-maximum a posteriori (MAP) TAN structures can also be computed in polynomial time. We use these results to correct minor errors in Meila & Jaakkola (2000a) and to construct several TAN based classifiers. We show that these classifiers provide consistently better predictions over Irvine datasets and artificially generated data than TAN based classifiers proposed in the literature.

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An Algorithm for Finding K Minimum Spanning Trees

Cited by 71 publications

References 10 publications

On the K shortest path trees problem

On the K shortest path trees problem

Finding the k smallest spanning trees

TAN Classifiers Based on Decomposable Distributions

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