1. 1 2/23/2016 ITCS 6114 Topological Sort Minimum Spanning Trees

2. 2 2/23/2016 Directed Acyclic Graphs ● A directed acyclic graph or DAG is a directed graph with no directed cycles:

3. 3 2/23/2016 DFS and DAGs ● Argue that a directed graph G is acyclic iff a DFS of G yields no back edges: ■ Forward: if G is acyclic, will be no back edges ○ Trivial: a back edge implies a cycle ■ Backward: if no back edges, G is acyclic ○ Argue contrapositive: G has a cycle   a back edge  Let v be the vertex on the cycle first discovered, and u be the predecessor of v on the cycle  When v discovered, whole cycle is white  Must visit everything reachable from v before returning from DFS-Visit()  So path from u  v is yellow  yellow, thus (u, v) is a back edge

4. 4 2/23/2016 Topological Sort ● Topological sort of a DAG: ■ Linear ordering of all vertices in graph G such that vertex u comes before vertex v if edge (u, v)  G ● Real-world example: getting dressed

5. 5 2/23/2016 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket

6. 6 2/23/2016 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket SocksUnderwearPantsShoesWatchShirtBeltTieJacket

7. Topological Sort Algorithm Topological-Sort() { Run DFS When a vertex is finished, output it Vertices are output in reverse topological order } ● Time: O(V+E) ● Correctness: Want to prove that (u,v)  G  u  f > v  f

8. 8 2/23/2016 Correctness of Topological Sort ● Claim: (u,v)  G  u  f > v  f ■ When (u,v) is explored, u is yellow ○ v = yellow  (u,v) is back edge. Contradiction (Why?) ○ v = white  v becomes descendent of u  v  f < u  f (since must finish v before backtracking and finishing u) ○ v = black  v already finished  v  f < u  f

9. 9 2/23/2016 Minimum Spanning Tree ● Problem: given a connected, undirected, weighted graph: 14 10 3 64 5 2 9 15 8

10. 10 2/23/2016 Minimum Spanning Tree ● Problem: given a connected, undirected, weighted graph, find a spanning tree using edges that minimize the total weight 14 10 3 64 5 2 9 15 8

11. 11 2/23/2016 Minimum Spanning Tree ● Which edges form the minimum spanning tree (MST) of the below graph? HBC GED F A 14 10 3 64 5 2 9 15 8

12. 12 2/23/2016 Minimum Spanning Tree ● Answer: HBC GED F A 14 10 3 64 5 2 9 15 8

13. 13 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);

14. 14 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 14 10 3 64 5 2 9 15 8 Run on example graph

15. 15 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);     14 10 3 64 5 2 9 15 8 Run on example graph

16. 16 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  0    14 10 3 64 5 2 9 15 8 Pick a start vertex r r

17. 17 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  0    14 10 3 64 5 2 9 15 8 Red vertices have been removed from Q u

18. 18 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  0  3  14 10 3 64 5 2 9 15 8 Red arrows indicate parent pointers u

19. 19 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 14  0  3  10 3 64 5 2 9 15 8 u

20. 20 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 14  0  3  10 3 64 5 2 9 15 8 u

21. 21 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 14  08  3  10 3 64 5 2 9 15 8 u

22. 22 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 10  08  3  14 10 3 64 5 2 9 15 8 u

23. 23 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 10  08  3  14 10 3 64 5 2 9 15 8 u

24. 24 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 102  08  3  14 10 3 64 5 2 9 15 8 u

25. 25 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 102  0815 3  14 10 3 64 5 2 9 15 8 u

26. 26 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 102  0815 3  14 10 3 64 5 2 9 15 8 u

27. 27 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 1029 0815 3  14 10 3 64 5 2 9 15 8 u

28. 28 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 1029 0815 3 4 14 10 3 64 5 2 9 15 8 u

29. 29 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 529 0815 3 4 14 10 3 64 5 2 9 15 8 u

30. e 30 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 529 0815 3 4 14 10 3 64 5 2 9 15 8 u

31. 31 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 529 0815 3 4 14 10 3 64 5 2 9 15 8 u

32. 32 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 529 0815 3 4 14 10 3 64 5 2 9 15 8 u

33. 33 2/23/2016 Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 529 0815 3 4 14 10 3 64 5 2 9 15 8 u

34. 34 2/23/2016 Review: Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); What is the hidden cost in this code?

35. 35 2/23/2016 Review: Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v));

36. ke 36 2/23/2016 Review: Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v)); How often is ExtractMin() called? How often is DecreaseKey() called?

37. 37 2/23/2016 Review: Prim’s Algorithm MST-Prim(G, w, r) Q = V[G]; for each u  Q key[u] =  ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v  Adj[u] if (v  Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); What will be the running time? A: Depends on queue binary heap: O(E lg V) Fibonacci heap: O(V lg V + E)

38. 38 2/23/2016 Single-Source Shortest Path ● Problem: given a weighted directed graph G, find the minimum-weight path from a given source vertex s to another vertex v ■ “Shortest-path” = minimum weight ■ Weight of path is sum of edges ■ E.g., a road map: what is the shortest path from Chapel Hill to Charlottesville?