1. Chapter 23 Minimum Spanning Tree

2. A minimum spanning tree (MST) for a connected graph
MST not unique: replace (b,c) by (a,h)

3. Theorem 23.1
Let G(V,E) be a connected, undirected graph. For every e in E, w(e) is its (real-valued) ‘weight’. Let A be a subset of E that is included in some MST for G. Let (S,V-S) be any cut of G that respects A. [A cut (S,V-S) of G respects A if no edge of A crosses the cut.] Let (u,v) be a light edge crossing (S,V-S) [w(u,v) is minimal among the edges crossing S,V-S]. Then, A UNION {(u,v)} is included in some MST for G.

4. 2 views of a cut (S,V-S)
(d,c): unique lightest edge that crosses the cut

5. Proof of Theorem 23.1 S: black vertices. V-S: white vertices.
Edges shown are in T, which is an MST. Other edges in G, are not shown.
Shaded edges are in A.
Suppose that T does not contain the light edge (u,v).
(x,y) is an edge in unique path from u to v that the cut (S,V-S) does not respect.
The edge (u,v) can replace the edge (x,y) so that the total weight of T does not increase and it is still a tree. QED

6. Kruskal’s algorithm
Sort edge by increasing weight. Initially T is empty.
Add next edge (u,v) to T, unless there is already a path from u to v in T. [How to determine that?]

7. Data structures and complexity
Union-Find Enough for current algorithm: how to do Union & Find in O(log n) time each? Complexity: O(m log n)

8. Kruskal’s algorithm (cont’d)

9. Prim’s algorithm
T is empty. Put some vertex in T. Among the vertices not in T, pick one that has a light a vertex in T and add it to T. [How to determine that?] Repeat.

10. Data structures and complexity
Extract-min Decrease-key Complexity: O(m log n). Cab be enhanced to O(m + n log n)