1. Graphs - Definition G(V,E) - graph with vertex set V and edge set E
E  {(a,b)| aV and bV} - for directed graphs
E  {{a,b}| aV and bV} - for undirected graphs
w: E  R - weight function
|V| - number of vertices
|E| - number of edges
Often we will assume that V = {1, ,n}

2. Graphs - Examples 6 1 6 1 2 2 3 4 5 3 4 5

3. Graphs - Trees 6 1 6 1 2 4 2 4 3 5 3 5

4. Graphs - Directed Acyclic Graphs (DAG)6 1 2 4 3 5

5. Graphs - Representations - Adjacency matrix1 2 3 4 5 6 6 1 2 3 4 5 6 1 1 2 3 4 5

6. Graphs - Representations - Adjacency lists6 1 1 2 3 4 5 6 6 2 6 2 3 3 4 5 1 3 5 2 1

7. Breadth-First Search - Algorithm
BreadthFirstSearch(graph G, vertex s)
for u  V[G]  {s} do
colour[u]  white; d[u] ; p[u]  0
colour[s]  gray; d[s]  0; p[s]  0
Q  {s}
while Q  0 do
u  Head[Q]
for v  Adj[u] do
if colour[v] = white then
colour[v]  gray; d[v]  d[u] + 1; p[v]  u
EnQueue(Q,v)
DeQueue(Q)
colour[u]  black

8. Breadth-First Search - Exampleg

9. Breadth-First Search - Example       d e f g s Q

10. Breadth-First Search - Example1    1   d e f g e a Q

11. Breadth-First Search - Example1 2   1 2  d e f g a b f Q

12. Breadth-First Search - Example1 2  2 1 2  d e f g b f d Q

13. Breadth-First Search - Example1 2 3 2 1 2  d e f g f d c Q

14. Breadth-First Search - Example1 2 3 2 1 2 3 d e f g d c g Q

15. Breadth-First Search - Example1 2 3 2 1 2 3 d e f g c g Q

16. Breadth-First Search - Example1 2 3 2 1 2 3 d e f g g Q

17. Breadth-First Search - Example1 2 3 2 1 2 3 d e f g
Q = 

18. Breadth-First Search - Complexity
BreadthFirstSearch(graph G, vertex s)
for u  V[G]  {s} do
colour[u]  white; d[u] ; p[u]  0
colour[s]  gray; d[s]  0; p[s]  0
Q  {s}
while Q  0 do
u  Head[Q]
for v  Adj[u] do
if colour[v] = white then
colour[v]  gray; d[v]  d[u] + 1; p[v]  u
EnQueue(Q,v)
DeQueue(Q)
colour[u]  black
(V)
Thus T(V,E)=(V+E)
(V) without for cycle
(E) for all while cycles together

19. Breadth-First Search - Shortest Distances
Theorem
After BreadthFirstSearch algorithm terminates
d[v] is equal with shortest distance from s to v for all
vertices v
for all vertices v reachable from s the one of the
shortest paths from s to v contains edge (p[v], v)

20. Depth-First Search - Algorithm
DepthFirstSearch(graph G)
for u  V[G] do
colour[u]  white
p[u]  0
time  0
if colour[v] = white then
DFSVisit(v)

21. Depth-First Search - Algorithm
DFSVisit(vertex u)
time  time + 1
d[u]  time
colour[u]  gray
for v  Adj[u] do
if colour[v] = white then
p[v]  u
DFSVisit(v)
colour[u]  black
f[u]  time

22. Depth-First Search - Exampleb c d e

23. Depth-First Search - Exampleb 1/ c d e

24. Depth-First Search - Exampleb 1/ 2/ c d e

25. Depth-First Search - Exampleb 1/ 2/ 3/ c d e

26. Depth-First Search - Exampleb 1/ 2/ 4/ 3/ c d e

27. Depth-First Search - Exampleb 1/ 2/ B 4/ 3/ c d e

28. Depth-First Search - Exampleb 1/ 2/ B
4/5 3/ c d e

29. Depth-First Search - Exampleb 1/ 2/ B
4/5
3/6 c d e

30. Depth-First Search - Exampleb 1/
2/7 B
4/5
3/6 c d e

31. Depth-First Search - Exampleb 1/
2/7 B F
4/5
3/6 c d e

32. Depth-First Search - Exampleb
1/8
2/7 B F
4/5
3/6 c d e

33. Depth-First Search - Exampleb
1/8
2/7 9/ B F
4/5
3/6 c d e

34. Depth-First Search - Exampleb
1/8
2/7 9/ B C F
4/5
3/6 c d e

35. Depth-First Search - Exampleb
1/8
2/7 9/ B C F
4/5
3/6
10/ c d e

36. Depth-First Search - Exampleb
1/8
2/7 9/ B C F B
4/5
3/6
10/ c d e

37. Depth-First Search - Exampleb
1/8
2/7 9/ B C F B
4/5
3/6
10/11 c d e

38. Depth-First Search - Exampleb
1/8
2/7
9/12 B C F B
4/5
3/6
10/11 c d e

39. Depth-First Search - Complexity
DepthFirstSearch(graph G)
for u  V[G] do
colour[u]  white
p[u]  0
time  0
if colour[v] = white then
DFSVisit(v)
(V)
executed (V) times
DFSVisit(vertex u)
time  time + 1
d[u]  time
colour[u]  gray
for v  Adj[u] do
if colour[v] = white then
p[v]  u
DFSVisit(v)
colour[u]  black
f[u]  time
(E) for all DFSVisit calls together
Thus T(V,E)=(V+E)

40. Depth-First Search - Classification of Edges
Trees edges - edges in depth-first forest
Back edges - edges (u, v) connecting vertex u to an
v in a depth-first tree (including self-loops)
Forward edges - edges (u, v) connecting vertex u to a
descendant v in a depth-first tree
Cross edges - all other edges

41. Depth-First Search - Classification of Edges
Theorem
In a depth-first search of an undirected graph G, every
edge of G is either a tree edge or a back edge.

42. Depth-First Search - White Path Theorem
If during depth-first search a “white” vertex u is reachable
from a “grey” vertex v via path that contains only “white”
vertices, then vertex u will be a descendant on v in
depth-first search forest.

43. Depth-First Search - Timestamps
Parenthesis Theorem
After DepthFirstSearch algorithm terminates for any two
vertices u and v exactly one from the following three conditions holds
the intervals [d[u],f[u]] and [d[v],f[v]] are entirely disjoint
the intervals [d[u],f[u]] is contained entirely within the interval [d[v],f[v]] and u is a descendant of v in depth-
first tree
the intervals [d[v],f[v]] is contained entirely within the interval [d[u],f[u]] and v is a descendant of u in depth-

44. Depth-First Search - Timestampsb s c
3/6
2/9
1/10
11/16 B F C B
4/5
7/8
12/13
14/15 C C C d e f g

45. Depth-First Search - Timestampsb f g e a d
(s (b (a (d d) a) (e e) b) s) (c (f f) (g g) c)

46. Depth-First Search - TimestampsB F b f g C a e C B C d

47. DFS - Checking for cycles
[Adapted from M.Golin]

48. DFS - Checking for cycles
[Adapted from M.Golin]

49. DFS - Checking for cycles
[Adapted from M.Golin]

50. DFS - Checking for cycles
[Adapted from M.Golin]

51. DFS - Topological Sorting
undershorts
socks
watch
pants
shoes
belt
shirt
tie
jacket

52. DFS - Topological Sorting
[Adapted from M.Golin]

53. DFS - Topological Sorting
[Adapted from M.Golin]

54. DFS - Topological Sorting
TopologicalSort(graph G)
call DFS(G) to compute f[v] for all vertices v
as f[v] for vertex v is computed, insert onto the front of a linked list
return the linked list of vertices

55. DFS - Topological Sorting - Example 1
undershorts
11/16
socks
17/18
watch
9/10
pants
12/15
shoes
13/14
belt
shirt
6/7
1/8
tie
2/5
jacket
3/4

56. DFS - Topological Sorting - Example 1
socks
undershorts
pants
shoes
watch
17/18
11/16
12/15
13/14
9/10
shirt
belt
tie
jacket
1/8
6/7
2/5
3/4

57. DFS - Topological Sorting - Example 2
[Adapted from M.Golin]

58. DFS - Topological Sorting
Theorem
TopologicalSort(G) produces a topological sort of a
directed acyclic graph G.

59. DFS - Strongly Connected Components

60. DFS - Strongly Connected Components

61. DFS - Strongly Connected Components
[Adapted from L.Joskowicz]

62. DFS - Strongly Connected Components
[Adapted from L.Joskowicz]

63. DFS - Strongly Connected Components
[Adapted from L.Joskowicz]

64. DFS - Strongly Connected Components
[Adapted from L.Joskowicz]

65. DFS - Strongly Connected Components
StronglyConnectedComponents(graph G)
call DFS(G) to compute f[v] for all vertices v
compute GT
call DFS(GT) consider vertices in order of decreasing of f[v]
output the vertices of each tree in the depth-first forest as a separate strongly connected component

66. DFS - Strongly Connected Components
13/14
11/16
1/10
8/9
12/15
3/4
2/7
5/6

67. DFS - Strongly Connected Components
13/14
11/16
1/10
8/9
12/15
3/4
2/7
5/6

68. DFS - Strongly Connected Components
13/14
11/16
1/10
8/9
12/15
3/4
2/7
5/6

69. DFS - SCC - Correctness
13/14
11/16
1/10
8/9
12/15
3/4
2/7
5/6

70. DFS - SCC - Correctness y' y x
C(x)
Assume that y preceded by y' is the closest vertex to x outside C(x). Then:
- d(y)<f(y)<d(x)<d(y) (otherwise we will have xy (in G).
- for all x'C(x): d(x)<d(x')<f(x')<f(x)
(the largest value of f(x) will have the vertex first "discovered" in C(x)).
- thus we have d(y)<f(y)<d(y')<f(y'),
however there is and edge (y,y') in G,
implying f(y)<d(y') d(y')<y(y).
Contradiction.

71. DFS - SCC - Correctness Lemma
If two vertices are in the same strongly connected,
then no path between them leaves this strongly connected
component.
Theorem
In any depth-first search, all vertices in the same strongly
connected component are placed in the same depth-first
tree.

72. DFS - SCC - Correctness Theorem
In a directed graph G = (V,E) the forefather (u) of any
vertex uV in any depth-first search of G is an
ancestor of u.
Corollary
In any depth-first search of a directed graph G = (V,E)
for all uV vertices u and (u) lie in the same strongly
connected component.

73. DFS - SCC - Correctness Theorem
In a directed graph G = (V,E) two vertices u,vV
lie in the same strongly connected component
if and only if they have the same forefather in a
depth-first search of G.
StronglyConnectedComponents(G) correctly computes
the strongly connected components of a directed graph G.

74. DFS - SCC - Correctness 2
[Adapted from S.Whitesides]

75. DFS - SCC - Correctness 2
[Adapted from S.Whitesides]

76. DFS - SCC - Applications
[Adapted from L.Joskowicz]