1. Graph Representation (23.1/22.1)
HW: problem 23.3, p.496
G=(V, E) -graph:
V = V(G) - vertices;
E = E(G) - edges (connecting pairs of vertices) 1 2 3 4 5
Adjacency-list
Adjacency-matrix 1 3 4 2 3 3 5 4 5 5 2

2. Depth-First Search (23.3/22.3)
Methodically explore all vertices and edges
All vertices are White, t = 0
For each vertex u do if u is White then Visit(u)
Procedure Visit(u)
color u Gray; d[u]  t  t +1
for each v adjacent to u do
if v is White then Visit(v)
color u Black
f [u]  t  t +1
Gray vertices =
stack of recursive calls

3. Depth-First Search 5 6 1 3 8 7 4 2

4. Depth-First Search 1 5 6 1 3 8 7 4 2

5. Depth-First Search 1 5 6 1 3 8 2 7 4 2

6. Depth-First Search 1 5 6 1 3 8 3 2 7 4 2

7. Depth-First Search 1 5 6 1 3 8 3 2 7 4 2

8. Depth-First Search 1 5 6 1 3 8 3 2 7 4 2

9. Depth-First Search 1 5 6 1 3 8 3 2 7 4 2

10. Depth-First Search 1 5 6 1 4 3 8 3 2 7 4 2

11. Depth-First Search 1 5 6 1 4 3 8 3 2 7 4 2

12. Depth-First Search 1 5 6 1 4 3 8 3 2 7 4 2

13. Depth-First Search 1 5 5 6 1 4 3 8 3 7 4 2 2

14. Depth-First Search 1 5 5 6 1 4 3 8 3 7 4 2 2

15. Depth-First Search 1 5 5 6 1 4 6 3 8 3 4 2 7 2

16. Depth-First Search 1 5 5 6 1 4 6 3 8 3 4 2 7 2

17. Depth-First Search 1 5 5 6 1 4 6 3 8 3 4 2 7 2

18. Depth-First Search 1 5 5 6 1 4 6 3 8 3 4 2 7 2

19. Depth-First Search 1 5 5 6 1 4 6 3 8 3 4 2 7 2

20. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 7 2

21. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 7 2

22. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 8 7 2

23. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 8 7 2

24. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 8 7 2

25. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 8 7 2

26. Depth-First Search 1 5 5 7 6 1 4 6 3 8 3 4 2 8 7 2

27. Depth-First Search (23.3/22.3)
Runtime of DFS = O(V+E)
once per vertex
once per edge
Kinds of edges:
tree edge (gray to gray)
back edge (gray to gray)
forward edge (gray to black)
cross edge (gray to black)
G is undirected  only tree and back
Undirected G is acyclic  no back edges
If a graph is acyclic can be found in O(V) time

28. Topological Sort (23.4/22.4) DAG = directed acyclic graph
has levels or depth
cannot return up
Topological Sort(G)
call DFS(G) to compute f[v]
sort according to finishing times
Directed graph G is acyclic  no back edges

29. Breadth-First Search (23.2/22.2)
BFS discovers all vertices at distance k before any vertices at distance k+1.
Initialization
assigns  to all vertices: w(v)= 
color all white
Color s gray, w(s)=0, enqueue s in Q
for the head u of Q
for each v adjacent to u,
d[v]=d[u]+1
enqueue v in Q
dequeue Q
color u black

30. Breadth-First Search r s t u         v w x y

31. Breadth-First Search r s t u        v w x y Q s

32. Breadth-First Search r s t u 1    1   v w x y Q r w 1 1

33. Breadth-First Search r s t u 1   2 1   v w x y Q w v 1 2

34. Breadth-First Search r s t u 1 2  2 1 2  v w x y Q v t x 2 2 2

35. Breadth-First Search r s t u 1 2  2 1 2  v w x y Q t x 2 2

36. Breadth-First Search r s t u 1 2 3 2 1 2  v w x y Q x u 2 3

37. Breadth-First Search r s t u 1 2 3 2 1 2 3 v w x y Q u y 3 3

38. Breadth-First Search r s t u 1 2 3 2 1 2 3 v w x y Q y 3

39. Breadth-First Search r s t u 1 2 3 2 1 2 3 v w x y Q 

40. Breadth-First Search (23.2/22.2)
Run-time = O(V+E)
Shortest-path tree
the final weight is the minimum distance
keep predecessors and get the shortest path
all BFS shortest paths make a tree