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Lecture 27

Lecture 27. CSE 331 Nov 6, 2009. Homework related stuff. Solutions to HW 7 and HW 8 at the END of the lecture. Turn in HW 7. Kruskal’s Algorithm. Input: G=(V,E) , c e > 0 for every e in E. T = Ø. Sort edges in increasing order of their cost. Joseph B. Kruskal.

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Lecture 27

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  1. Lecture 27 CSE 331 Nov 6, 2009

  2. Homework related stuff Solutions to HW 7 and HW 8 at the END of the lecture Turn in HW 7

  3. Kruskal’s Algorithm Input: G=(V,E), ce> 0 for every e in E T = Ø Sort edges in increasing order of their cost Joseph B. Kruskal Consider edges in sorted order If an edge can be added to T without adding a cycle then add it to T

  4. Prim’s algorithm Similar to Dijkstra’s algorithm Input: G=(V,E), ce> 0 for every e in E S = {s}, T = Ø While S is not the same as V Among edges e= (u,w) with u in S and w not in S, pick one with minimum cost Add w to S, e to T

  5. Crucial Lemma for MSTs Condition: S and V\S are non-empty V \ S S Cheapest crossing edge is in all MSTs Assumption: All edge costs are distinct

  6. Optimality of Kruskal’s Algorithm V \ S S S Nodes connected to red in (V,T) Input: G=(V,E), ce> 0 for every e in E S is non-empty T = Ø V\S is non-empty Sort edges in increasing order of their cost First crossing edge considered Consider edges in sorted order If an edge can be added to T without adding a cycle then add it to T

  7. Kruskal’s algo outputs MST Group talk time!

  8. Is (V,T) a spanning tree? No cycles by design G is disconnected! Just need to show that (V,T) is connected V \ S’ S’ No edges here

  9. Crucial Lemma for MSTs Condition: S and V\S are non-empty V \ S S Cheapest crossing edge is in all MSTs

  10. Optimality of Prim’s algorithm S is already defined for us! By induction, show that (S,T) is a tree S not Ø Input: G=(V,E), ce> 0 for every e in E S = {s}, T = Ø While S is not the same as V Among edges e= (u,w) with u in S and w not in S, pick one with minimum cost Add w to S, e to T

  11. Prove the Crucial Lemma Condition: S and V\S are non-empty V \ S S Cheapest crossing edge is in all MSTs

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