Minimum Spanning Trees

1. Minimum Spanning Trees Section 8.3

2. Definition • For a undirected graph G=(V,E), a spanning tree T is a sub-graph that is a tree covering all vertices in V. • In an undirected weighted, a Minimum Spanning Tree (MST) is the spanning tree with the min sum of weights. • We shall assume that |V|=n and G is connected. • Note that MST may not be unique • And there is n-1 vertices in MST.

3. Problem • Given a weighted undirected graph, find a MST. • Solution • Kruskal’s Algorithm • Prim’s Algorithm

4. Kruskal’s Algorithm • Sort the edges non-decreasingly • add the edge with the min weight to the tree if it doesn’t make a cycle, otherwise skip it. • Repeat • Note:We can’t just take the first n-1 edges with the min weights. Choose as long as it’s a tree.

5. Kruskal’s Algorithm Input: weighted connected undirected G=(V,E) Output: MST T • Sort the edges in E non-decreasingly • For each vertex v V makeset({v}) End for • T  { } • While (|T| < n-1) • let (x,y) be the next edge in E • if find(x)  find(y) then • add(x,y) to T • union(x,y) • end if • End while

6. Analysis • Running Time= (m log n) • Because we need (m log m) for sorting the edges and (m log* n) for the find union operations

7. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

8. Prim’s Algorithm • Divide V into disjoint X & Y • Find xX and yY such that (x,y) has the min weight among all such pairs • Add the edge to the tree and move y to X • Repeat

9. Prim’s Algorithm Input: weighted connected undirected G=(V,E) Output: MST T T  { }; X  {1 }; Y  V - {1 }; While Y  { } let (x,y) be the edge of min weight such that xX & yY X  {y }  X; Y  Y - {y }; T  T  {(x,y)} End while

10. More Details X Y • N[y] is closest vertex adjacent to y in X • C[y]=c[N[y],y] cost of incident edge of y y C[y] N[y]

11. Prim’s Algorithm Input: weighted connected undirected G=(V,E) Output: MST T T  { }; X  {1 }; Y  V - {1 }; For y  2 to n if y adjacent to 1 then N[y]  1; C[y] c[1,y] else C[y]   end if End for For j  1 to n-1 let yY be s.t. C[y] is min T  T  {y, N[y] }; X  X  {y }; Y  Y - {y }; for each vertex w Y if c[y,w]  C[w] then N[w]  y; C[w] c[y,w] end if end for End for

12. Analysis • Running Time = (m+ n2) • But can be improved by using the min-heap to find min C[y]. • Improved running time = (m log n) which is good if m=o(n2/log n)

13. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

14. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

15. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

16. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

17. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

18. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

19. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

20. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

21. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

22. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

23. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

24. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

25. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

26. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4

27. Example 7 2 4 1 3 5 4 2 6 1 7 1 3 2 3 5 4