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Minimum Spanning Trees Subgraph • A graph G is a subgraph of graph H if – The vertices of G are a subset of the vertices of H, and – The edges of G are a subset of the edges of H. Spanning Tree • A spanning tree of a graph G is a subgraph of G that is a tree and includes all the vertices of G. • Every connected graph has a spanning tree – Start with G. If G is a tree then it is a ST of G. If not, G has a cycle. Remove any edge of the cycle. The result is a connected subgraph of G. Repeat. Process must terminate since the number of edges is finite. Spanning Trees Spanning Trees a spanning tree Spanning Trees another spanning tree (can have many) Weighted Graphs • A weighted graph is a graph with a mapping of edges to R+ assigning a weight to each edge • A minimum spanning tree of a weighted graph is a spanning tree of minimum total weight (sum of the weights of the edges in the tree) Minimum Spanning Trees 3 4 1 4 6 3 7 1 2 Minimum Spanning Trees 3 4 1 4 6 3 7 1 1. Choose any vertex 2 Minimum Spanning Trees 3 4 1 4 6 3 7 1 2 2. Choose minimum weight edge connected to that vertex Minimum Spanning Trees 3 4 1 4 6 3 7 1 2 3. Choose minimum weight edge connected to chosen vertices Minimum Spanning Trees 3 4 1 4 6 3 7 1 2 4. Choose minimum weight edge connected to chosen vertices Minimum Spanning Trees 3 4 1 4 6 3 7 1 2 5. Choose minimum weight edge connected to chosen vertices without creating a cycle Minimum Spanning Trees 3 4 1 4 6 3 7 1 2 6. Choose minimum weight edge connected to chosen vertices without creating a cycle Prim’s algorithm • To find a MST of a connected weighted graph G, we build a tree one edge at a time – Start with T being any node of G – Pick a minimum weight edge that does not create a cycle and that connects a node in T to a node not in T. Add that edge to T. – Repeat until all nodes of G have been added to T. • A “Greedy algorithm” Proof that Prim’s algorithm works • Invariant: At every stage, T is a subgraph of a MST • True initially since T is just a single node • The algorithm runs until all nodes have been added to T, so in the end the invariant says that T is a MST. • So we just need to show that each stage of the algorithm preserves the invariant. Proof that the algorithm maintains the invariant • Proof by contradiction. Suppose there is a stage at which the invariant is true before and false after. • Let e be the edge added when the invariant is first violated. Suppose e joins node v in T to node w not in T. • So T is a subgraph of a MST but T+e is not. • Let T’ be a MST of which T is a subgraph. • Edge e is not in T’. Proof that the algorithm maintains the invariant • Since T’ is a spanning tree, there is a path p from v to w in T’. Path p does not go through e since e is not part of T’. So p+e is a cycle consisting of two different paths from u to v. • Path p must include an edge f which has one end in T and the other not in T. • Edge f must have weight ≥ the weight of e, otherwise the algorithm would have chosen to add edge f to T instead of e. • Removing f from T’ and adding e results in a spanning tree of weight ≤ the weight of T’ and including T as a subgraph, contradicting the assumption that T+e was not a subgraph of a MST. ✔ Finis