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CS 332: Algorithms

CS 332: Algorithms. Review of MST Algorithms Disjoint-Set Union Amortized Analysis. Review: MST Algorithms. In a connected, weighted, undirected graph, will the edge with the lowest weight be in the MST? Why or why not? Yes:

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CS 332: Algorithms

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  1. CS 332: Algorithms Review of MST Algorithms Disjoint-Set Union Amortized Analysis David Luebke 16/9/2014

  2. Review: MST Algorithms • In a connected, weighted, undirected graph, will the edge with the lowest weight be in the MST? Why or why not? • Yes: • If T is MST of G, and A  T is a subtree of T, and (u,v) is the min-weight edge connecting A to V-A, then (u,v)  T • The lowest-weight edge must be in the tree (A=) David Luebke 26/9/2014

  3. Review: MST Algorithms • What do the disjoint sets in Kruskal’s algorithm represent? • A: Parts of the graph we have connected up together so far David Luebke 36/9/2014

  4. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 46/9/2014

  5. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1? David Luebke 56/9/2014

  6. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 66/9/2014

  7. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2? 19 9 14 17 8 25 5 21 13 1 David Luebke 76/9/2014

  8. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 86/9/2014

  9. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5? 21 13 1 David Luebke 96/9/2014

  10. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 106/9/2014

  11. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8? 25 5 21 13 1 David Luebke 116/9/2014

  12. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 126/9/2014

  13. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9? 14 17 8 25 5 21 13 1 David Luebke 136/9/2014

  14. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 146/9/2014

  15. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13? 1 David Luebke 156/9/2014

  16. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 166/9/2014

  17. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14? 17 8 25 5 21 13 1 David Luebke 176/9/2014

  18. Kruskal’s Algorithm Run the algorithm: Kruskal() { T = ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 14 17 8 25 5 21 13 1 David Luebke 186/9/2014

  19. Review: Shortest-Path Algorithms • How does the Bellman-Ford algorithm work? • How can we do better for DAGs? • Under what conditions can we use Dijkstra’s algorithm? David Luebke 196/9/2014

  20. Review: Running Time ofKruskal’s Algorithm • Expensive operations: • Sort edges: O(E lg E) • O(V) MakeSet()’s • O(E) FindSet()’s • O(V) Union()’s • Upshot: • Comes down to efficiency of disjoint-set operations, particularly Union() David Luebke 206/9/2014

  21. Disjoint Set Union • So how do we represent disjoint sets? • Naïve implementation: use a linked list to represent elements, with pointers back to set: • MakeSet(): O(1) • FindSet(): O(1) • Union(A,B): “Copy” elements of A into set B by adjusting elements of A to point to B: O(A) • How long could n Union()s take? David Luebke 216/9/2014

  22. Disjoint Set Union: Analysis • Worst-case analysis: O(n2) time for n Union’s Union(S1, S2) “copy” 1 element Union(S2, S3) “copy” 2 elements … Union(Sn-1, Sn) “copy” n-1 elements O(n2) • Improvement: always copy smaller into larger • How long would above sequence of Union’s take? • Worst case: n Union’s take O(n lg n) time • Proof uses amortized analysis David Luebke 226/9/2014

  23. Amortized Analysis of Disjoint Sets • If elements are copied from the smaller set into the larger set, an element can be copied at most lg n times • Worst case: Each time copied, element in smaller set 1st time resulting set size  2 2nd time  4 … (lg n)th time  n David Luebke 236/9/2014

  24. Amortized Analysis of Disjoint Sets • Since we have n elements each copied at most lg n times, n Union()’s takes O(n lg n) time • Therefore we say the amortized cost of a Union() operation is O(lg n) • This is the aggregate method of amortized analysis: • n operations take time T(n) • Average cost of an operation = T(n)/n David Luebke 246/9/2014

  25. Amortized Analysis: Accounting Method • Accounting method • Charge each operation an amortized cost • Amount not used stored in “bank” • Later operations can used stored money • Balance must not go negative • Book also discusses potential method • But we won’t worry about it here David Luebke 256/9/2014

  26. The End David Luebke 266/9/2014

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