CS3240: Data Structures and Algorithms

Spanning Tree

spanning tree

spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some (or perhaps all) of the edges of G.

forms a tree spanning every vertex.

every vertex lies in the tree, but no cycles (or loops) are formed.

Minimum Spanning Tree

a spanning tree with weight less than or equal to the weight of every other spanning tree

Application of Minimum Spanning Tree --an Example

What is the best way to run cable in a neighborhood?

If it is constrained to bury the cable only along certain paths, then there would be a graph representing which points are connected by those paths.

Some of those paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights.

A spanning tree for that graph would be a subset of those paths that has no cycles but still connects to every house. There might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost.

CS3240: Data Structures and Algorithms

a spanning tree with weight less than or equal to the weight of every other spanning tree

What is the best way to run cable in a neighborhood?

If it is constrained to bury the cable only along certain paths, then there would be a graph representing which points are connected by those paths.

Some of those paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights.

A spanning tree for that graph would be a subset of those paths that has no cycles but still connects to every house. There might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost.