Unit 5 graphs minimum spanning trees

1. Minimum Spanning Tree Kalyani Neve Asst.Prof. GHRIBM,Jalgaon

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 AGraph A Spanning Trees

3. All 16 of its Spanning TreesComplete 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 AB D B B B C D J C C E F D D H J E G F FG I G GI J H J JI

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 JI 1A D 4B C 4A 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 JI 1A D 4B C 4A 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 JI 1A D 4B C 4A 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 JI 1A D 4B C 4A B Cycle Don’t Add Edge

22. 4 1 2 2 1 3 32 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. Analysis of Kruskal's Algorithm Running Time = O(m log n) (m = edges, n = nodes) Testing if an edge creates a cycle can be slow unless a complicated data structure called a “union-find” structure is used. It usually only has to check a small fraction of the edges, but in some cases (like if there was a vertex connected to the graph by only one edge and it was the longest edge) it would have to check all the edges. This algorithm works best, of course, if the number of edges is kept to a minimum.

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

25. 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.

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 Complete Graph

27. 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

37. 4 1 2 2 1 3 32 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

38. Analysis of Prim's Algorithm Running Time = O(m + n log n) (m = edges, n = nodes) If a heap is not used, the run time will be O(n^2) instead of O(m + n log n). However, using a heap complicates the code since you’re complicating the data structure. A Fibonacci heap is the best kind of heap to use, but again, it complicates the code. Unlike Kruskal’s, it doesn’t need to see all of the graph at once. It can deal with it one piece at a time. It also doesn’t need to worry if adding an edge will create a cycle since this algorithm deals primarily with the nodes, and not the edges. For this algorithm the number of nodes needs to be kept to a minimum in addition to the number of edges. For small graphs, the edges matter more, while for large graphs the number of nodes matters more.

Unit 5 graphs minimum spanning trees

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Unit 5 graphs minimum spanning trees