1 Foundations of Software Design Fall 2002 Marti Hearst Lecture 25: Graph Traversal, DAGs, and Weighted Graphs

2 Graph Traversal

3 Slide adapted from Goodrich & Tamassia

4 Exploring a Labyrinth Without Getting Lost A depth-first search in an undirected graph G is like wandering in a labyrinth with a string and a can of red paint without getting lost. –Start at vertex s –Tie the string to s –Paint s red (visited) –Set u to be s (the current vertex) –Travel on an arbitrary edge (u,v), trailing string behind –If v is already visited, follow the string back to u –Else if v is unvisited, paint v red –Set u (current vertex) to v and repeat –Keep visiting and unrolling (backtracking) until we finally arrive back at s

5 Slide adapted from Goodrich & Tamassia DFS Starting at vertex A Discovery edges solid Back edges dashed Unvisited nodes black a)Initial graph b)Path of edges from A until node linking back to A found c)Start from B, reach F, a dead end (because E and I have already been visited) d)After backtracking to C, going on to G and hitting J, another dead end.

6 Slide adapted from Goodrich & Tamassia e)After backtracking to G f)After backtracking to N (which had been sitting on the stack waiting)

7 Slide adapted from Goodrich & Tamassia

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10 Slide adapted from Goodrich & Tamassia

11 Slide adapted from Goodrich & Tamassia

12 Slide adapted from Goodrich & Tamassia

13 Slide adapted from Goodrich & Tamassia

14 Digraphs (Directed Graphs)

15 Slide adapted from Goodrich & Tamassia (Directed Graphs)

16 Slide adapted from Goodrich & Tamassia

17 Slide adapted from Goodrich & Tamassia

18 DAGs Directed Acyclic Graphs

19 Slide adapted from Goodrich & Tamassia

20 Slide adapted from Goodrich & Tamassia

21 Slide adapted from Goodrich & Tamassia

22 Slide adapted from Goodrich & Tamassia

23 Slide adapted from Goodrich & Tamassia

24 Weighted Graphs

25 Slide adapted from Goodrich & Tamassia

26 Slide adapted from Goodrich & Tamassia

27 Greedy Algorithms An algorithm which always takes the best immediate, or local, solution in looking for an answer. Greedy algorithms sometimes find less-than- optimal solutions Some greedy algorithms always find the optimal solution –Dijkstra’s shortest paths algorithm –Prim’s algorithm for minimum spanning trees

28 Slide adapted from Goodrich & Tamassia An animation

29 Slide adapted from Goodrich & Tamassia

30 Slide adapted from Goodrich & Tamassia

31 Slide adapted from Goodrich & Tamassia

32 Slide adapted from Goodrich & Tamassia

33 Slide adapted from Goodrich & Tamassia

34 Slide adapted from Goodrich & Tamassia

35 Slide adapted from Goodrich & Tamassia

36 Slide adapted from Goodrich & Tamassia

37 This algorithm uses a forest of trees. – Initially the forest consists of n single node trees (and no edges). –At each step, we add one (the cheapest one) edge so that it joins two trees together. –If it were to form a cycle, it would simply link two nodes that were already part of a single connected tree, so we don’t use the edge in this case. Kruskal’s Minimum Spanning Tree

38 Kruskal’s Minimum Spanning Tree

39 Kruskal’s Minimum Spanning Tree (cont)

40 Kruskal’s Minimum Spanning Tree (cont) An animation of Kruskal’s