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Kruskal’s Algorithm for Computing MSTs

Kruskal’s Algorithm for Computing MSTs. Section 9.2. The Minimum Spanning Tree. Give a graph G = (V, E), the minimum spanning tree (MST) is a weighted graph G’ = (V, E’) such that: E’  E G’ is connected G’ has the minimum cost. Example. 12. E. A. 12. 20. E. A. B. 6. D. 20. C.

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Kruskal’s Algorithm for Computing MSTs

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  1. Kruskal’s Algorithm for Computing MSTs • Section 9.2

  2. The Minimum Spanning Tree • Give a graph G = (V, E), the minimum spanning tree (MST) is a weighted graph G’ = (V, E’) such that: • E’  E • G’ is connected • G’ has the minimum cost

  3. Example 12 E A 12 20 E A B 6 D 20 C 13 B 6 D C 13 12 Which is the MST? E A 20 Why MST is a tree? 20 B 6 D C 13

  4. Naïve Algorithm for MST(Using Exhaustive Search) MST  Empty-Graph cost   For each subgraph G’ = (V, E’) of G = (V, E) do { 1. Check that G’ is connected 2. Compute cost c of G’ 3. If c < cost then { 3.1 MST  G’ 3.2 cost  c }} What is the complexity “in words” of the algorithm? “the number of subgraphs”  “cost of computing Steps 1-3” How many subgraphs are there? 2|E| This is bad!

  5. The Prim Algorithm // input: a graph G // output: E: a MST for G • Select a starting node, v • T  {v} //the nodes in the MST • E  {} //the edges in the MST • 4. While not all nodes in G are in the T do • 3.1 Choose the edge v’ in G − T such that there is a v in T: • weight(v,v’) is the minimum in • {weight(u,w) : w in G − T and u in T} • 3.2 T  T  {v’} • 3.3 E  E  {(v,v’)} • 5. return E Complexity: - O((|E|+|V|)log2|E|) - In class we show O(|E||V|)

  6. T G-T Why does it works? Property: Suppose that we divide the nodes of a graph G = (V, E) in two groups V, V’: 12 22 7 20 2 5 6 66 10 13 If there is an edge e between the 2 groups with minimum cost, then there is an MST containing e

  7. Visited Non visited Example of a “Greedy” Algorithm Dijkstra-Primis an example of a greedy algorithm since it looks at a subset of the larger problem 12 22 7 20 2 5 6 66 10 13 Unfortunately, not all problems can be solved with greedy algorithms Example of a non greedy algorithm? Our naïve algorithm for MST

  8. Kruskal Algorithm 12 E A 20 12 3 2 B 6 D C 13 6 F • |V| = 6 • |E| = 8 • A tree with |V| nodes has edges |V|-1

  9. The Kruskal Algorithm // input: a graph G with n nodes and m edges // output: E: a MST for G 1. 2. E  {} //the edges in the MST 3. i  1 //counter for EG 4. While do if addingEG[i] to E does not add a cycle then E  E  {EG[i]} i  i + 1 5. return E EG[1..m]  Sort the m edges in G in increasing weight order |E| < n - 1 Is this algorithm Greedy? Yes Complexity: O(|E|log2 |E|)

  10. Why Does the Kruskal Algorithm Works? Property: If a connected graph G has n nodes, then any acyclic subgraph G’ of G with n-1 edges is also connected and includes all n nodes (G’ is a tree) Corollary: Kruskal’s algorithm returns an MST of G

  11. Homework • 010: • 9.1: 3, 6.b, 9 (just explain) • 9.2: 1.a, 2.a, 5 • 011: • 9.1: 2, 6.a, 9.a • 9.2: 1.b, 2.d, 4

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