# Finding thek smallest spanning trees

@article{Eppstein1992FindingTS, title={Finding thek smallest spanning trees}, author={David Eppstein}, journal={BIT Numerical Mathematics}, year={1992}, volume={32}, pages={237-248} }

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m logβ(m, n)=k2); for planar graphs this bound can be improved toO(n+k2). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k2n+n logn,k2+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k2).

## 8 Citations

### The Label-Constrained Minimum Spanning Tree Problem

- Computer Science
- 2008

It is proved that the label-constrained minimum spanning tree problem is NP-complete and a genetic algorithm is presented which gets comparable results, but is much faster.

### Sparsification-a technique for speeding up dynamic graph algorithms

- Computer ScienceProceedings., 33rd Annual Symposium on Foundations of Computer Science
- 1992

The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and…

### Primal-dual algorithms for computing weight-constrained shortest paths and weight-constrained minimum spanning trees

- Computer ScienceConference Proceedings of the 2000 IEEE International Performance, Computing, and Communications Conference (Cat. No.00CH37086)
- 2000

This work presents simple but effective primal-dual algorithms for computing approximate solutions for both the QoS shortest path problem and the minimum spanning tree problem, both of which are NP-hard.

### Geometric kth shortest paths

- Computer Science
- 2015

It is shown that the complexity of the kth shortest path map is O(kh + kn), which is tight, and a simple visibility-based algorithm to compute the kTH shortest path between two points in O(km(m+ kn) log kn) time is presented.

### k -Best Enumeration

- Mathematics
- 2015

We survey k-best enumeration problems and the algorithms for solving them, including in particular the problems of finding the k shortest paths, k smallest spanning trees, and k best matchings in…

### Exact and approximate solving approaches in multi-objective combinatorial optimization, application to the minimum weight spanning tree problem. (Approches de résolution exacte et approchée en optimisation combinatoire multi-objectif, application au problème de l'arbre couvrant de poids minimal)

- Mathematics
- 2014

This thesis deals with several aspects related to solving multi-objective problems, without restriction to the bi-objectives, and proposes a new hybrid approach for the determination of the nondominated set, which is instantiated on the minimum spanning tree problem.

### Graph Modeling of Metabolism

- Computer Science
- 2000

This paper proposes the graph modeling of metabolism, a network of chemical reactions catalyzed by enzymes that is possible to describe metabolism as a circulation of atoms by representing all reactions with the chemical structures of small compounds (metabolites).

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This paper studies methods for finding minimum spanning trees in graphs and results include relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.

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An efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications is described, which performs O(log n) work per modification, where n is the number of vertices in the graph.

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This paper presents an algorithm for finding K minimum spanning trees in an undirected graph based on three subroutines which obtains the kth minimum spanning tree in $O(m)$ steps when the jthminimum spanning trees for $j = 1,2, \cdots ,k - 1$ are given.

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This paper uses F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs and can be extended to allow a degree constraint at one vertex.

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Ambivalent data structures are presented for several problems on undirected graphs and used in finding the k smallest spanning trees of a weighted undirecting graph in O(m log beta (m,n)+min(k/sup 3/2/, km/sup 1/2/)) time, where m and n are understood to be the current number of edges and vertices, respectively.

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Two algorithms for generating spanning trees of a connected graph in order of increasing weight are presented. The first generates the K smallest weight trees, where K can be specified in advance or…

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Data structures are presented for the problem of maintaining a minimum spanning tree on-line under the operation of updating the cost of some edge in the graph. For the case of a general graph,…

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It is proved that every spanning tree of weightW1 is at mostk−1 edge swaps away from some spanning treeof weightWk, which is the increasing sequence of all possible distinct spanning tree weights.

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Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network.