1. Minimum Spanning Tree Sarah Brubaker Tuesday 4/22/8

2. Minimum Spanning Tree • Input: graph G with weights on the edges • Output: connected sub graph G’ of G that includes all the vertices of G, of minimum total weight

3. Exhaustive Search • List all connected sub-graphs of G • Return sub-graph with least weight • Number of vertices, N, and number of edges, M • O(mn-1)

4. Greedy Algorithm • Start with a graph containing just the vertices, G’={V,Ø} • Add edges with the least weight until the graph is connected but no cycles are created • To do this: • Order the edges in increasing weight • While the graph is not connected, add an edge • End when all vertices are connected

5. Greedy Algorithm - Example Initial Graph: List of Edges:

6. Greedy Algorithm - Example Start with G’={V,Ø} and sorted list of edges Graph: Sorted List of Edges:

7. Greedy Algorithm - Example Add edge from list to graph s.t. no cycles are created Remove edge from list Graph: Sorted List of Edges:

8. Greedy Algorithm - Example Add edge from list to graph s.t. no cycles are created Remove edge from list Graph: Sorted List of Edges:

9. Greedy Algorithm - Example Add edge from list to graph s.t. no cycles are created Remove edge from list Graph: Sorted List of Edges: • Check for cycles using Depth First Search starting at a vertex in the added edge: • Start at C • C-> D -> F -> C • C gets visited twice => a cycle exists and CF should not be added to the graph

10. Greedy Algorithm - Example Add edge from list to graph s.t. no cycles are created Remove edge from list End when all vertices can be visited (graph is connected) Graph: Sorted List of Edges: • Check for cycles using Depth First Search starting at a vertex in the added edge: • Start at B • B ->D ->C -> F -> A • If all vertices are visited by DFS, then the graph is connected and we are done

11. Greedy Algorithm - Example Add edge from list to graph s.t. no cycles are created Remove edge from list End when all vertices can be visited (graph is connected) Graph: Sorted List of Edges: • Check for cycles using Depth First Search starting at a vertex in the added edge: • Start at E • E -> F -> D -> B -> A ->C • All vertices were visited and there were no cycles => we have found a minimum spanning tree with weight 12.

12. Greedy Algorithm- Psuedocode • Given G = (V, E) • G’ <- (V,Ø) • While G’ is not connected • Add e Є E to G’ s.t. G’ is acyclic and e is minimum • Remove e from E

13. Greedy Algorithm – Run Time • Initialization – constant • While loop – O(n-1) • Connected graph has n-1 edges • Must add n-1 edges to the graph for it to be connected • Find a minimum e ЄE – O(m) • Make sure G’ is acyclic – O(2n) • DFS is O(m+n)<O(2n) where m<=n-1 • Test connectivity of G’ – O(n) • Can use DFS; could be done in same step as testing acyclicity • Remove e from E - constant • Total Runtime: O(n*(m+2n+n)) ~ O(nm+ n2) -> POLYNOMIAL

14. Error? • Let X be any subset of the vertices of G, and let edge e be the smallest edge connecting X to G-X. Then e is part of the minimum spanning tree.

15. Error? • T does not contain e and is a spanning tree • e is smallest edge connecting X = {C,D} to G-X ={A,B,F,E} • Adding e creates a cycle in T • Another spanning tree exists, T2 = T+e-f, where f is another edge that could connect X to G-x • Because e < f, T was not the MST and e must be part of the MST 1 2 4

16. Error? • In the greedy algorithm, the edge added was always the smallest. • Because the smallest edge to connect two parts of the original graph must be included, each edge added must be a part of the final minimum spanning tree. • In this case, the greedy algorithm is always correct (but we know this is not the case for ALL greedy algorithms).