MST2.ppt

1. Minimum Spanning Tree

2. A spanning tree of a graph is just a subgraph that contains all the vertices and is a tree. A graph may have many spanning trees. or or or Some Spanning Trees from Graph A Graph A Spanning Trees

3. All 16 of its Spanning Trees Complete Graph

4. Minimum Spanning Trees The Minimum Spanning Tree for a given graph is the Spanning Tree of minimum cost for that graph. 5 7 2 1 3 4 2 1 3 Complete Graph Minimum Spanning Tree

5. Algorithms for Obtaining the Minimum Spanning Tree • Kruskal's Algorithm • Prim's Algorithm

6. Kruskal's Algorithm This algorithm creates a forest of trees. Initially the forest consists of n single node trees (and no edges). At each step, we add one edge (the cheapest one) so that it joins two trees together. If it were to form a cycle, it would simply link two nodes that were already part of a single connected tree, so that this edge would not be needed.

7. The steps are: 1. The forest is constructed - with each node in a separate tree. 2. The edges are placed in a priority queue. 3. Until we've added n-1 edges, 1. Extract the cheapest edge from the queue, 2. If it forms a cycle, reject it, 3. Else add it to the forest. Adding it to the forest will join two trees together. Every step will have joined two trees in the forest together, so that at the end, there will only be one tree in T.

8. 4 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J Complete Graph

9. 1 4 2 5 2 5 4 3 4 4 4 10 1 6 3 3 2 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J A A B D B B B C D J C C E F D D H J E G F F G I G G I J H J J I

10. 2 5 2 5 4 3 4 4 10 1 6 3 3 2 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J B B D J C C E F D D H J E G F F G I G G I J H J J I 1 A D 4 B C 4 A B Sort Edges (in reality they are placed in a priority queue - not sorted - but sorting them makes the algorithm easier to visualize)

11. 2 5 2 5 4 3 4 4 10 1 6 3 3 2 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J B B D J C C E F D D H J E G F F G I G G I J H J J I 1 A D 4 B C 4 A B Add Edge

16. 2 5 2 5 4 3 4 4 10 1 6 3 3 2 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J B B D J C C E F D D H J E G F F G I G G I J H J J I 1 A D 4 B C 4 A B Cycle Don’t Add Edge

20. 2 5 2 5 4 3 4 4 10 1 6 3 3 2 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J B B D J C C E F D D H J E G F F G I G G I J H J J I 1 A D 4 B C 4 A B Cycle Don’t Add Edge

22. 4 1 2 2 1 3 3 2 4 A B C D E F G H I J 4 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J Minimum Spanning Tree Complete Graph

23. Prim's Algorithm This algorithm starts with one node. It then, one by one, adds a node that is unconnected to the new graph, each time selecting the node whose connecting edge has the smallest weight out of the available nodes’ connecting edges.

24. The steps are: 1. The new graph is constructed - with one node from the old graph. 2. While new graph has fewer than n nodes, 1. Find the node from the old graph with the smallest connecting edge to the new graph, 2. Add it to the new graph Every step will have joined one node, so that at the end we will have one graph with all the nodes and it will be a minimum spanning tree of the original graph.

25. 4 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J Complete Graph

26. 4 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J Old Graph New Graph 4 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J

36. 4 1 2 2 1 3 3 2 4 A B C D E F G H I J Complete Graph Minimum Spanning Tree 4 1 2 3 2 1 3 5 3 4 2 5 6 4 4 10 A B C D E F G H I J

MST2.ppt

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MST2.ppt