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CSCI 3160 Design and Analysis of Algorithms Tutorial 3

CSCI 3160 Design and Analysis of Algorithms Tutorial 3. Yang LIU. Outline. Union-find data structure Minimum spanning tree (MST ) Kruskal’s algorithm. Union-find. A data structure for disjoint sets n = number of members, forming disjoint groups

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CSCI 3160 Design and Analysis of Algorithms Tutorial 3

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  1. CSCI 3160 Design and Analysis of AlgorithmsTutorial 3 Yang LIU

  2. Outline Union-find data structure Minimum spanning tree (MST) Kruskal’s algorithm

  3. Union-find • A data structure for disjoint sets • n = number of members, forming disjoint groups • Two members are in the same group if and only if they have a common leader • Operations: • Union: merge two groups • Find: name the leader of the group • We do not really care about who the leader is; we only want to tell one group from another

  4. Union-find • Idea: use a forest (= collection of trees) • a group → a tree • leader → root of the tree • Example: • Group 1: Alice, Bob • Group 2: Carol, Dave, Eve • Store the height of each tree at the root A 1 D 1 B C E

  5. Union-find • Find: return the root of the group • Union: make the leader of one group the boss of the other • Our heuristic: make the root of the shorter tree point to the root of the other tree • If both trees are of the same height h, then the resulting tree has height h+1

  6. Union-find in action • Initialize • Everyone is his/her own boss A C E B D 0 0 0 0 0

  7. Union-find in action • Union(A, B) • Find(A) returns A and Find(B) returns B • Make Alice the boss of Bob • Increase the height of the tree A C E B D 1 0 0 0 0

  8. Union-find in action • Union(C, D) • Find(C) returns C and Find(D) returns D • Make Carol the boss of Dave • Increase the height of the tree A C E B D 1 1 0 0 0

  9. Union-find in action • Union(B, D) • Find(B) returns A and Find(D) returns C • Make Alice the boss of Carol • Increase the height of the tree A C E B D 2 1 0 0 0

  10. Your turn • Find(C)? A C E B D 2 1 0 0 0

  11. Your turn • Find(C)? • Find(C) returns A A C E B D 2 1 0 0 0

  12. Your turn • Union(B, E)? A C E B D 2 1 0 0 0

  13. Your turn • Union(B, E)? • Find(B) returns A and Find(E) returns E • Make Alice the boss of Eve • No increase in the tree height (Why?) A C E B D 2 1 0 0 0

  14. Final result • Does this look more like a tree? 2 A C 1 E 0 B 0 D 0

  15. Analysis • Height of each tree = O(log n) • Cost of Find(): O(log n) • Cost of Union(): O(log n) • Dominated by the cost of Find()

  16. Minimum spanning tree • G = (V, E): undirected, connected, (non-negatively) weighted • Problem: find a subset of edges with minimum total weight such that all vertices in V are connected using these edges 1 B A 4 5 3 6 C D E 2 7 Total weight = 1 + 2 + 3 + 6 = 12

  17. Kruskal’s algorithm • In words: • Sort the edges in ascending order of weights • Initialize: set T = Ø • While T is not a spanning tree • Consider the next edge in the sorted list • If adding it to T does not cause a cycle, add it • The union-find structure helps us check for cycles • Adding an edge corresponds to putting the two endpoints into the same group • Connecting two vertices in the same group causes a cycle

  18. Dry run • 1: Sort the edges 1 B A 4 5 3 6 C D E 2 7

  19. Dry run • 2: Set T = Ø B A C D E 1 B A 4 5 3 6 C D E 2 7

  20. Dry run • 3: Consider the first edge B A C D E 1 B A 4 5 3 6 C D E 2 7

  21. Dry run • 4: Consider the second edge B A C D E 1 B A 4 5 3 6 C D E 2 7

  22. Dry run • 5: Consider the third edge B A C D E 1 B A 4 5 3 6 C D E 2 7

  23. Dry run • 6: B A C D E 1 B A 4 5 3 6 C D E 2 7

  24. Dry run • 7: B A C D E 1 B A 4 5 3 6 C D E 2 7

  25. Dry run • 8: B A C D E 1 B A 4 5 3 6 C D E 2 7

  26. Dry run • 9: We have find an MST! B A C D E 1 B A 4 5 3 6 C D E 2 7

  27. Analysis • Correctness • In each step, we keep a set of edges T that is a subset of an MST • Theorem: Let S be any tree in the forest (V, T). We can add the lightest edge from S to V-S to T, and the resulting set of edges is also a subset of an MST • Space complexity = O(|V|+|E|) • Time complexity = O(|E|log|V|) • Sorting alone takes time O(|E|log|V|) (Note that |E| = O(|V|2)) • Cycle checking takes O(log|V|) operations (Find()) • Adding an edge also takes O(log|V|) operations (Union())

  28. Dry run revisited • 2: Set T = Ø B A C D E A C E B D 0 0 0 0 0

  29. Dry run revisited • 3: Consider the first edge B A C D E A C E B D 1 0 0 0 0

  30. Dry run revisited • 4: Consider the second edge B A C D E A C E B D 1 1 0 0 0

  31. Dry run revisited • 5: Consider the third edge B A C D E A C E B D 2 1 0 0 0

  32. Dry run revisited • 6: B A C D E A C E B D 2 1 0 0 0

  33. Dry run revisited • 7: B A C D E A C E B D 2 1 0 0 0

  34. Dry run revisited • 8: B A C D E A C E B D 2 1 0 0 0

  35. Dry run revisited • 9: We have find an MST! B A C D E A C E B D 2 1 0 0 0

  36. End • Questions

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