Discrete Math Notes:
Chapter 6: Graphs and Trees
6.12 Minimum spanning trees
weighted graph, a graph G = (V ,E), along with a function w: E -> R.
minimum spanning tree, a weighted graph, is a spanning tree T of G whose weight is no larger than any other spanning tree of G.
overview:
3
11
Eligible edges invellow Select min eligible edge. Update eligible edges
4.1
The set of all vertices in the Minimum Spanning Tree is complete!
Captions ^
1. Start with T consisting of a single initial vertex. The edges incident to the initial vertex are the eligible edges. 2. Select the eligible edge with min weight. Add the selected edge and its endpoint to T. 3. The eligible edges have one endpoint in T and one outside of T.Edges become eligible if one endpoint is T's new vertex and the other is outside T. 4. T now has 2 vertices. Select the min eligible edge. Add the selected edge and its endpoint to T. T now consists of 3 vertices. 5. Update the eligible edges and select the min eligible edge from the new set of eligible edges. Add the selected edge and its endpoint to T. T has 4 vertices. 6.After the next iteration,T has 5 vertices. In updating the set of eligible edges,an edge is removed because both endpoints of the edges are now in T. 7. In each remaining iteration, a new vertex and edge are added to T. 8. The Minimum Spanning Tree is complete when all the vertices are in T.
A graph's spanning tree describes a subset of the edges that connects every pair of vertices while
1 edges, there's no reason to choose one over the other if the primary purpose is to have as few edges as feasible. A weighted graph's minimum spanning tree (MST) is a spanning tree T of G whose weight is equal to or less than that of any other spanning tree of G. Because there can be multiple spanning trees
with the same weight, a graph can have multiple minimum spanning trees. A whole order is an example of a partial order (Z, ). If every edge has a weight of 1, then every spanning tree has a weight of n - 1 and is likewise a minimum spanning tre