Minimum Spanning Trees

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Presentation transcript:

Minimum Spanning Trees

Example Application for MST Minimum Spanning Tree for Electrical wiring

Spanning Trees Connected and having no cycles. A graph having more than V – 1 edges contains at least one cycle.

Minimum Spanning Tree (MST) What is minimum spanning tree? A tree that covers (spans) all the vertices of a connected graph which has the minimum total cost of edges in the tree. A MST exists for a graph iff the graph is connected. The same problem makes sense for directed graphs too, but the solution is more difficult.

Minimum Spanning Tree (MST) Our problem is finding a minimum spanning tree in an undirected and connected graph.

Example: Minimum Spanning Tree v1 2 v2 4 1 3 10 2 v3 v4 7 v5 8 4 5 6 v6 1 v7 Example undirected and connected graph v1 v2 v5 v3 v6 v7 2 1 4 6 v4 The corresponding minimum spanning tree

Properties: Minimum Spanning Tree If the number of vertices of a connected undirected graph is |V|, then its minimum spanning tree will have |V| vertices |V| - 1 edges. A MST does not contain any cycle, since it is a tree. If we add an extra edge to a MST, then it will have a cycle.

Example Application for MST power outlet or light Electrical wiring of a house using minimum amount of wires (cables)

Example Application for MST Minimum Spanning Tree for Electrical wiring

Prim’s Algorithm for MST MST is constructed in successive stages. At each stage: A new vertex is added to the tree by choosing the edge (u,v) such that the cost of (u,v), w(u,v), is the smallest among all edges where u is in the tree and v is not.

1 2 3 4 5 6 7

6 8 5 3 7 9 15

Prim’s Algorithm: Example Start with vertex v1. It is the initial current tree which we will grow to an MST V1 0 V4 1 V2 2 V3 4 V6  v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 Content of the priority queue A connected, undirected graph G is given above.

Prim’s Algorithm: Example Select an edge from graph: that is not in the current tree, that has the minimum cost, and that can be connected to the current tree. Step 1 v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V4 1 V2 2 V3 4 Content of the priority queue

Prim’s Algorithm: Example The edges that can be connected are: (v1,v2): cost 2 (v1,v4): cost 1 (v1,v3): cost 2 Step 1 v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V4 1 V2 2 V3 4 Content of the priority queue

Prim’s Algorithm: Example We include the vertex v4, that is connected to the selected edge, to the current tree. In this way we grow the tree. Step 2 v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V2 2 V3 2 V7 4 V5 7 V6 8 Content of the priority queue

Content of the priority queue Prim’s Algorithm Repeat steps: 1, and 2 Add either edge (v4, v3) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V3 2 V7 4 V5 7 V6 8 Content of the priority queue

Content of the priority queue Prim’s Algorithm Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V7 4 V6 5 V5 7 Content of the priority queue

Content of the priority queue Prim’s Algorithm Add edge (v4, v7) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V7 4 V6 5 V5 7 Content of the priority queue

Content of the priority queue Prim’s Algorithm Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V6 1 V5 6 Content of the priority queue

Content of the priority queue Prim’s Algorithm Add edge (v7, v6) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V6 1 V5 6 Content of the priority queue

Content of the priority queue Prim’s Algorithm Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V5 6 Content of the priority queue

Content of the priority queue Prim’s Algorithm Add edge (v7, v5) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 V5 6 Content of the priority queue

Prim’s Algorithm: Example Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4 Content of the priority queue

Prim’s Algorithm: Example v1 v2 v5 v3 v6 v7 2 1 4 6 v4 Finished! The resulting MST is shown below!

PRIM’s algorithm is similar to the Dijkstra’s shortest path algorithm PRIM’s algorithm is similar to the Dijkstra’s shortest path algorithm. Complexities are the same.