INPUT OUTPUT

**Problem:**
The subset of *E* of *G* of minimum weight which forms a
tree on *V*.

**Excerpt from**
The Algorithm Design Manual:
The minimum spanning tree (MST) of a graph defines the cheapest subset of edges that keeps the graph in one
connected component. Telephone companies are particularly interested in minimum spanning trees, because the minimum
spanning tree of a set of sites defines the wiring scheme that connects the sites using as little wire as possible.
It is the mother of all network design problems.

Minimum spanning trees prove important for several reasons:

- They can be computed quickly and easily, and they create a sparse subgraph that reflects a lot about the original graph.
- They provide a way to identify clusters in sets of points. Deleting the long edges from a minimum spanning tree leaves connected components that define natural clusters in the data set, as shown in the output figure above.
- They can be used to give approximate solutions to hard problems such as Steiner tree and traveling salesman.
- As an educational tool, minimum spanning tree algorithms provide graphic evidence that greedy algorithms can give provably optimal solutions.

Geometric Spanner Networks by G. Narasimhan and M. Smid | Spanning Trees and Optimization Problems by B. Wu and K Chao | Algorithms in Java, Third Edition (Parts 1-4) by Robert Sedgewick and Michael Schidlowsky |

Computational Geometry : Algorithms and Applications by Mark De Berg, Marc Van Kreveld, Mark Overmars, and O. Schwartskopf | Network Flows : Theory, Algorithms, and Applications by Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin | Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica by S. Pemmaraju and S. Skiena |

Computational Geometry by F. Preparata and M. Shamos | Combinatorial Optimization: Algorithms and Complexity by C. Papadimitriou and K. Steiglitz | Combinatorial Optimization: Networks and Matroids by E. Lawler |

This page last modified on 2008-07-10 . www.algorist.com