Section 4.2 Minimal Spanning Trees Prepared by Amanda Dargie and Michele Fretta

1. Applied Combinatorics, 4th Ed.Alan Tucker Section 4.2 Minimal Spanning Trees Prepared by Amanda Dargie and Michele Fretta Tucker, Sec. 4.2

2. Recall • Recall:A spanning tree of a graph G is a subgraph of G that is a tree containing all vertices of G. 3 2 3 7 2 4 2 6 3 6 9 1 5 2 1 11 Tucker, Sec. 4.2

3. Definition • A minimal spanning tree in a network is a spanning tree whose sum of edge lengths k(e) is as small as possible. 3 2 3 7 2 4 2 6 3 6 9 1 5 2 1 12 Tucker, Sec. 4.2

4. Kruskal’s Algorithm • Let T be a minimal spanning set (which is initially empty). • Add to T the shortest edge that does not form a circuit with edges already in T. • Repeat adding edges in this manner until set T has n-1 edges Tucker, Sec. 4.2

5. k 1 7 3 2 a u 4 3 2 5 7 q 5 1 2 l 3 g v b 6 7 3 2 8 3 h 4 m 10 r 1 c w 2 3 5 4 3 7 i 8 n 8 s 2 d x 4 2 5 4 3 9 5 6 2 e y Example f p • First include all 3 edges of length 1: (a, f), (l, q), (r, w) • Next add all the edges of length 2: (a, b), (e, j), (g, l), (h, i), (l, m),(p, u), (s, x), (x, y) • Next add almost all the edges of length 3: (c, h), (d, e), (k, l), (k, p),(q, v), (r, s), (v, w),but not (w, x) unless (r, s) were omitted [if both were present we would get a circuit containing these 2 edges together with edges (r, w)and (s, x) • Next add all the edges of length 4 (f, g), (h, m), (c, d), (n, o), (s, t) • Finally add either (m, n)or (o, t)to obtain a minimal spanning set X t o j Tucker, Sec. 4.2

6. Prim’s Algorithm • Let T be a minimal spanning tree, initially including any one edge of shortest length. • Add to T the shortest edge between a vertex in T and a vertex not in T. • Repeat adding edges in this manner until T has n-1 edges. Tucker, Sec. 4.2

7. k 1 7 3 2 a u 4 3 2 5 7 q 5 1 2 l 3 g v b 6 7 3 2 8 3 h 4 m 10 r 1 c w 2 3 5 4 3 7 i 8 n 8 s 2 d x 4 2 5 4 3 9 5 6 2 e y Example f p • There are 3 edges of length 1: (a, f), (l, q), (r, w) • Suppose we pick (a, f) • Next add (a, b), of length 2 • Then (f, g), of length 4 • Then (g, l), (l, q), (l, m) and so forth • The next-to-last addition can be either (m, n), (o, t) (both of length 5) – in this case we’ll use (m, n) • Then follow with (n, o) o t j Tucker, Sec. 4.2

8. Theorem • Prim’s algorithm yields a minimal spanning tree. Proof • Assume that all edges have different lengths. • Let T'be a minimal spanning tree chosen to have as many edges as possible in common with T* (which will be constructed by Prim’s algorithm). Tucker, Sec. 4.2

9. Proof (cont’d) Tucker, Sec. 4.2

10. b a Proof (cont’d) Tucker, Sec. 4.2

11. b a Proof (cont’d) Tucker, Sec. 4.2

12. b a Proof (cont’d) Tucker, Sec. 4.2

13. a d 3 c 7 b 4 8 3 2 g e 5 5 f For the class to try:The red numbers indicate the value of each edge.Find a minimal spanning tree. 5 Hint: use Kruskal’s Algorithm Tucker, Sec. 4.2

14. Solution First, add (c,f) to T Then, add (a,b) and (d,g) Then, add (a,e) Then add (e,f) and (f,g). Don’t add (b,e), because we can’t have a circuit! a d 3 c 7 b 4 8 3 5 2 g e 5 5 f Tucker, Sec. 4.2