1. Premaster Course Algorithms 1 Chapter 5: Basic Graph Algorithms
Christian Scheideler
SS 2019

2. Basic Graph Algorithms
Overview:
Basic notation
Graph representations
Breadth-First-Search
Depth-First-Search
Minimum spanning trees
SS 2019
Chapter 5

3. Basic Notation
A directed graph G is a pair (V,E), where V is a finite set and E⊆VV.
Elements in V are called nodes or vertices, elements in E are called edges. Correspondingly, V is the node set and E the edge set of G.
Edges are ordered pairs of nodes. Edges of the form (u,u), uV, are allowed and called loops.
If (u,v)E, we say that the edge leads from u to v. We also say that u and v are adjacent.
The degree of a node v is the number of edges (v,w) in E.
SS 2019
Chapter 5

4. Directed Graph 1 2 5 3 4 6
SS 2019
Chapter 5

5. Basic Notation
An undirected graph G is a pair (V,E), where V is a finite set and E is a set of subsets of V of size two.
Elements in V are called nodes or vertices, elements in E are called edges. Correspondingly, V is the node set and E the edge set of G.
Edges have the form {u,v}. Edges of the form {u,u} are usually not allowed.
If {u,v}E, then we say that u and v are adjacent.
The degree of u is the number of edges {u,v} in E.
SS 2019
Chapter 5

6. Undirected Graph V={1,2,3,4,5,6} E={ {1,2}, {1,5}, {2,5}, {3,6} } 1 2
SS 2019
Chapter 5

7. Graph Theory D=4 D B C A E n: number of nodes, m: number of edges
(v,w): distance of w to v in G - directed graph: number of edges of a shortest directed path from v to w - undirected draph: number of edges of a shortest path from v to w
D=maxv,w (v,w): diameter of G D B C A E
D=4
SS 2019
Chapter 5

8. Graph Theory G undirected:
G is connected: diameter D is finite (i.e., there is a path from every node to every other node in G)
Example: graph that is not connected D B C A E
SS 2019
Chapter 5

9. Graph Theory G directed:
G is weakly connected: diameter D is finite, if all edges are seen as undirected
G is strongly connected: D is finite D B C A E
SS 2019
Chapter 5

10. large diameter (n-1 for n nodes)  bad for data structures
Graph Theory
Linear list:
degree 2
large diameter (n-1 for n nodes)  bad for data structures
SS 2019
Chapter 5

11. Diameter is 2d ~ 2 log2 n  good for data structures
Graph Theory
Complete binary tree:
n=2d+1-1 nodes, degree 3
Diameter is 2d ~ 2 log2 n  good for data structures
depth d d
SS 2019
Chapter 5

12. n = k2 nodes, maximum degree 4 diameter is 2(k-1) ~ 2 n
Graph Theory
2-dimensional grid:
n = k2 nodes, maximum degree 4
diameter is 2(k-1) ~ 2 n 1
side length k k
SS 2019
Chapter 5

13. Graph Representations
1: Sequence of edges 1 2
Dummy 4 3 /
(1,2)
(2,3)
(3,4)
(4,1)
SS 2019
Chapter 5

14. Graph Representations
2: Adjacency list 1 n 1 2 V 2 3 4 1 4 3 3 4
SS 2019
Chapter 5

15. Graphrepräsentationen
3: Adjacency matrix
A[i,j]{0,1}
A[i,j]=1 if and only if (i,j)E 1 2
A = 4 3
SS 2019
Chapter 5

16. Graph Traversal
Central question: How can we traverse the nodes of the graph so that every node is visited at least once?
SS 2019
Chapter 5

17. Graph Traversal
Central question: How can we traverse the nodes of the graph so that every node is visited at least once?
Basic strategies:
Breadth-first search
Depth-first search
SS 2019
Chapter 5

18. Explore graph distance by distance from s
Breadth-First Search
Start at some node s
Explore graph distance by distance from s s
SS 2019
Chapter 5

19. Depth-First Search Start from some node s
Explorate graph in depth-first manner ( : current, : still active, : done) s
SS 2019
Chapter 5

20. Breadth-First Search
Breadth-first search algorithm forms the base of many other graph algorithms.
Given a graph G=(V,E) and a source sV, the goal of breadth-first search is to find all nodes vV that are reachable from s. A node v is reachable from s if there is a (directed) path in G from s to v.
The breadth-first search algorithm can also be used to compute the distance d(s,v) of v from s.
SS 2019
Chapter 5

21. Breadth-First Search
If a new node v is discovered while searching for new nodes in Adj[u], then u is called the predecessor of v. (Adj[u]: set of neighbors or adjacency list of u, i.e., the nodes vV with {u,v}E resp. (u,v)E)
In the algorithm: nodes are white, gray, or black.
White nodes are nodes that have not been discovered yet.
Gray nodes are nodes that have already been discovered but whose adjacency list has not completely been explored yet.
Black nodes are nodes that have already been discovered and fully processed.
SS 2019
Chapter 5

22. Pseudo-code for Breadth-First Search
BFS(G,S)
1 for each node uV\{s} do
color[u]WHITE
d[u]  
p[u]  NIL
color[s]  GRAY
d[s]  0
p[s]  NIL
Q  { }
Enqueue(Q,s)
while Q  { } do
u  Dequeue(Q)
for each vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1
p[v]  u
Enqueue(Q,v)
color[u]  BLACK
SS 2019
Chapter 5

23. Pseudo-code for Breadth-First Search
BFS(G,S)
1 for each node uV\{s} do
color[u]WHITE
d[u]  
p[u]  NIL
color[s]  GRAY
d[s]  0
p[s]  NIL
Q  { }
Enqueue(Q,s)
while Q  { } do
u  Dequeue(Q)
for each vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1
p[v]  u
Enqueue(Q,v)
color[u]  BLACK
initialize BFS
SS 2019
Chapter 5

24. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK c g s b f i Q: s
SS 2019
Chapter 5 24

25. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK c g s b f i Q: s
SS 2019
Chapter 5 25

26. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK c g s u b f i Q:
SS 2019
Chapter 5 26

27. Breadth-First Search Example
v=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK c g s u b f i Q:
SS 2019
Chapter 5 27

28. Breadth-First Search Example
v=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK c g s u b f i Q:
SS 2019
Chapter 5 28

29. Breadth-First Search Example
v=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK c g s u b f i Q:
SS 2019
Chapter 5 29

30. Breadth-First Search Example
v=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u b f i Q:
SS 2019
Chapter 5 30

31. Breadth-First Search Example
v=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u b f i Q: a
SS 2019
Chapter 5 31

32. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u
v=b f i Q: a
SS 2019
Chapter 5 32

33. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u
v=b f i Q: a
SS 2019
Chapter 5 33

34. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u
v=b f i Q: a
SS 2019
Chapter 5 34

35. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u
v=b f i 1 Q: a
SS 2019
Chapter 5 35

36. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u
v=b f i 1 Q:
a, b
SS 2019
Chapter 5 36

37. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u b f i 1 Q:
a, b
SS 2019
Chapter 5 37

38. Breadth-First Search Example
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s u b f i 1 Q:
a, b
SS 2019
Chapter 5 38

39. Breadth-First Search Example
u=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s b f i 1 Q: b
SS 2019
Chapter 5 39

40. Breadth-First Search Example
v=d
u=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s b f i 1 Q: b
SS 2019
Chapter 5 40

41. Breadth-First Search Example
v=d
u=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s b f i 1 Q: b
SS 2019
Chapter 5 41

42. Breadth-First Search Example
v=d
u=a e h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s b f i 1 Q: b
SS 2019
Chapter 5 42

43. Breadth-First Search Example
v=d
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s b f i 1 Q: b
SS 2019
Chapter 5 43

44. Breadth-First Search Example
v=d
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s b f i 1 Q:
b, d
SS 2019
Chapter 5 44

45. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s b f i 1 Q:
b, d
SS 2019
Chapter 5 45

46. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s b f i 1 Q:
b, d
SS 2019
Chapter 5 46

47. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s b f i 1 Q:
b, d
SS 2019
Chapter 5 47

48. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2 b f i 1 Q:
b, d
SS 2019
Chapter 5 48

49. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2 b f i 1 Q:
b, d, c
SS 2019
Chapter 5 49

50. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2 b f i 1 Q:
b, d, c
SS 2019
Chapter 5 50

51. Breadth-First Search Example
u=a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2 b f i 1 Q:
b, d, c
SS 2019
Chapter 5 51

52. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2
u=b f i 1 Q:
d, c
SS 2019
Chapter 5 52

53. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2
u=b f i 1 Q:
d, c
SS 2019
Chapter 5 53

54. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2
u=b f i 1 Q:
d, c
SS 2019
Chapter 5 54

55. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2
u=b f i 1 Q:
d, c
SS 2019
Chapter 5 55

56. Breadth-First Search Example
u=d a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1
v=c g s 2 b f i 1 Q: c
SS 2019
Chapter 5 56

57. Breadth-First Search Example
u=d a
v=e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 c g s 2 b f i 1 Q: c
SS 2019
Chapter 5 57

58. Breadth-First Search Example
u=d a
v=e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g s 2 b f i 1 Q:
c, e
SS 2019
Chapter 5 58

59. Breadth-First Search Example
u=d a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g s 2 b
v=f i 1 Q:
c, e
SS 2019
Chapter 5 59

60. Breadth-First Search Example
u=d a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g s 2 b
v=f i 1 3 Q:
c, e, f
SS 2019
Chapter 5 60

61. Breadth-First Search Example
u=d a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 b f i 1 3 Q:
c, e, f
SS 2019
Chapter 5 61

62. Breadth-First Search Example
u=d a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 b f i 1 3 Q:
c, e, f
SS 2019
Chapter 5 62

63. Breadth-First Search Example
u=d a e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 b f i 1 3 Q:
c, e, f
SS 2019
Chapter 5 63

64. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3
u=c g
v=s 2 b f i 1 3 Q:
e, f
SS 2019
Chapter 5 64

65. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3
u=c g
v=s 2 b f i 1 3 Q:
e, f
SS 2019
Chapter 5 65

66. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3
u=c g
v=s 2 b f i 1 3 Q:
e, f
SS 2019
Chapter 5 66

67. Breadth-First Search Example
u=e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 b f i 1 3 Q: f
SS 2019
Chapter 5 67

68. Breadth-First Search Example
u=e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 4 b f i 1 3 Q:
f, g
SS 2019
Chapter 5 68

69. Breadth-First Search Example
u=e 2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 4 b f i 1 3 Q:
f, g
SS 2019
Chapter 5 69

70. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 4 b
u=f i 1 3 Q: g
SS 2019
Chapter 5 70

71. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 4 b
u=f i 1 3 4 Q:
g, i
SS 2019
Chapter 5 71

72. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c g
v=s 2 4 b
u=f i 1 3 4 Q:
g, i
SS 2019
Chapter 5 72

73. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c
u=g
v=s 2 4 b f i 1 3 4 Q: i
SS 2019
Chapter 5 73

74. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5
u=g
v=s 2 4 b f i 1 3 4 Q:
i, h
SS 2019
Chapter 5 74

75. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5
u=g
v=s 2 4 b f i 1 3 4 Q:
i, h
SS 2019
Chapter 5 75

76. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5 g
v=s 2 4 b f
u=i 1 3 4 Q: h
SS 2019
Chapter 5 76

77. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5 g
v=s 2 4 b f
u=i 1 3 4 Q: h
SS 2019
Chapter 5 77

78. Breadth-First Search Example2 h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5 g
v=s 2 4 b f
u=i 1 3 4 Q: h
SS 2019
Chapter 5 78

79. Breadth-First Search Example2
u=h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5 g
v=s 2 4 b f i 1 3 4 Q:
SS 2019
Chapter 5 79

80. Breadth-First Search Example2
u=h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5 g
v=s 2 4 b f i 1 3 4 Q:
SS 2019
Chapter 5 80

81. Breadth-First Search Example2
u=h
BFS(G,s)
1. „initialize BFS“
2. while Q do
u  Dequeue(Q)
for vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
d[v]  d[u]+1; p[v]  u
Enqueue(Q,v)
color[u]  BLACK 1 3 c 5 g
v=s 2 4 b f i 1 3 4 Q:
SS 2019
Chapter 5 81

82. Breadth-First Search Runtime
Theorem 5.1: Given a graph G=(V,E) and a source s, algorithm BFS has a runtime of
SS 2019
Chapter 5

83. Depth-First Search BFS(G,S) 1 for each node uV\{s} do color[u]WHITE
p[u]  NIL
color[s]  GRAY
p[s]  NIL
Q  { }
Enqueue(Q,s)
while Q  { } do
u  Dequeue(Q)
for each vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
p[v]  u
Enqueue(Q,v)
color[u]  BLACK
DFS(G,S)
1 for each node uV\{s} do
color[u]WHITE
p[u]  NIL
color[s]  GRAY
p[s]  NIL
Q  { }
Push(Q,s)
while Q  { } do
u  Pop(Q)
for each vAdj[u] do
if color[v]=WHITE then
color[v]  GRAY
p[v]  u
Push(Q,v)
color[u]  BLACK
SS 2019
Chapter 5

84. Depth-First Search
Instead of an iterative approach we will look at a recursive approach of realizing depth-first search.
Recursive approach:
Initially: all nodes are white
Discovered nodes become gray
Fully processed nodes become black
Two time stamps: d[v] and f[v] (lie between 1 and 2|V|)
d[v]: time at which v is discovered
f[v]: time at which v has been processed
SS 2019
Chapter 5

85. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
SS 2019
Chapter 5 85

86. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
SS 2019
Chapter 5 86

87. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
SS 2019
Chapter 5 87

88. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=0
SS 2019
Chapter 5 88

89. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=0 u
SS 2019
Chapter 5 89

90. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=0 u
SS 2019
Chapter 5 90

91. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=0 u
SS 2019
Chapter 5 91

92. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=1 u 1
SS 2019
Chapter 5 92

93. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=1 v u 1
SS 2019
Chapter 5 93

94. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=1 v u 1
SS 2019
Chapter 5 94

95. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=1 u 1
SS 2019
Chapter 5 95

96. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=2 u 2 1
SS 2019
Chapter 5 96

97. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=2 v u 2 1
SS 2019
Chapter 5 97

98. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=2 v u 2 1
SS 2019
Chapter 5 98

99. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=2 u 2 1
SS 2019
Chapter 5 99

100. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=3 u 3 2 1
SS 2019
Chapter 5
100

101. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=3 u 3 2 1
SS 2019
Chapter 5
101

102. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=3 u 3 2 1
SS 2019
Chapter 5
102

103. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=4 4 3 2 1
SS 2019
Chapter 5
103

104. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=4 4 u 3 2 1 v
SS 2019
Chapter 5
104
104

105. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=4 4 u 3 2 1 v
SS 2019
Chapter 5
105
105

106. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=4 4 3 2 1 u
SS 2019
Chapter 5
106
106

107. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=5 4 3 2 1 u 5
SS 2019
Chapter 5
107
107

108. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=5 4 3 2 v 1 u 5
SS 2019
Chapter 5
108
108

109. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=5 4 3 2 v 1 u 5
SS 2019
Chapter 5
109
109

110. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=5 4 3 2 v 1 u 5
SS 2019
Chapter 5
110
110

111. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=6 4 3 2 u 6 1 5
SS 2019
Chapter 5
111
111

112. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=6 4 v 3 2 u 6 1 5
SS 2019
Chapter 5
112
112

113. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=6 4 v 3 2 u 6 1 5
SS 2019
Chapter 5
113
113

114. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=6 4 3 2 u v 6 1 5
SS 2019
Chapter 5
114
114

115. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=6 4 3 2 u v 6 1 5
SS 2019
Chapter 5
115
115

116. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=6 4 3 2 u v 6 1 5
SS 2019
Chapter 5
116
116

117. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=7 4 3 2 7 v 6 1 5
SS 2019
Chapter 5
117
117

118. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=7 4 3 2 7 6 1 u 5
SS 2019
Chapter 5
118
118

119. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=8 4 3 2 7 6 1 8 5
SS 2019
Chapter 5
119
119

120. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=8 4 u 3 2 7 6 1 8 5
SS 2019
Chapter 5
120
120

121. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=9 4 9 3 2 7 6 1 8 5
SS 2019
Chapter 5
121
121

122. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=9 4 9 3 2 7 u 6 1 v 8 5
SS 2019
Chapter 5
122
122

123. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=9 4 9 3 2 7 u 6 1 v 8 5
SS 2019
Chapter 5
123
123

124. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=9 4 9 3 2 7 6 1 u 8 5
SS 2019
Chapter 5
124
124

125. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=10 4 9 3 2 7 6 1 u 8 10 5
SS 2019
Chapter 5
125
125

126. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=10 4 v 9 3 2 7 6 1 u 8 10 5
SS 2019
Chapter 5
126
126

127. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=10 4 v 9 3 2 7 6 1 u 8 10 5
SS 2019
Chapter 5
127
127

128. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=10 4 9 3 2 7 6 1 u 8 v 10 5
SS 2019
Chapter 5
128
128

129. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=10 4 9 3 2 7 6 1 u 8 v 10 5
SS 2019
Chapter 5
129
129

130. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=10 4 9 3 2 7 6 1 u 8 v 10 5
SS 2019
Chapter 5
130
130

131. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=11 4 9 3 2 7 6 1 11 8 v 10 5
SS 2019
Chapter 5
131
131

132. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=11 4 9 3 2 7 6 1 11 8 10 5
SS 2019
Chapter 5
132
132

133. Depth-First Search DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=12 4 9 3 2 7 12 6 1 11 8 10 5
SS 2019
Chapter 5
133
133

134. Depth-First Search runtime: O(|V|+|E|) DFS(G)
1. for each vertex uV do color[u]  WHITE ; p[u]  nil
2. time  0
3. for each vertex uV do
if color[u]=WHITE then DFS-Visit(u)
DFS-Visit(u)
1. color[u]  GRAY
2. time  time +1; d[u]  time
4. for each vAdj[u] do
if color[v] = WHITE then p[v]  u ; DFS-Visit(v)
6. color[u]  BLACK
7. time  time+1; f[u]  time
time=12 4 9 3 2 7 12 6 1 11 8 10 5
runtime:
O(|V|+|E|)
SS 2019
Chapter 5
134
134

135. Depth-First Search Runtime
SS 2019
Chapter 5

136. Classification of edges:
Depth-First Search
Classification of edges:
Tree edges are edges of the DFS-forest in G
Back edges are edges (u,v) that connect node u with one of its ancestors in the DFS-tree
Forward edges are non-tree edges (u,v) that connect node u with one of its descendents in the DFS-tree
Cross edges are the other edges 4 9 e b 3 2 7 12 a 6 1 11 8 c d 10 5
SS 2019
Chapter 5

137. Classification of edges:
Depth-First Search
Classification of edges:
Tree edges are edges of the DFS-forest in G
Back edges are edges (u,v) that connect node u with one of its ancestors in the DFS-tree
Forward edges are non-tree edges (u,v) that connect node u with one of its descendents in the DFS-tree
Cross edges are the other edges
back edge 4 9 e b 3 2 7 12 a 6 1 11 8 c d 10 5
SS 2019
Chapter 5

138. Classification of edges:
Depth-First Search
Classification of edges:
Tree edges are edges of the DFS-forest in G
Back edges are edges (u,v) that connect node u with one of its ancestors in the DFS-tree
Forward edges are non-tree edges (u,v) that connect node u with one of its descendents in the DFS-tree
Cross edges are the other edges 4
forward edge 9 e b 3 2 7 12 a 6 1 11 8 c d 10 5
SS 2019
Chapter 5

139. Classification of edges:
Depth-First Search
Classification of edges:
Tree edges are edges of the DFS-forest in G
Back edges are edges (u,v) that connect node u with one of its ancestors in the DFS-tree
Forward edges are non-tree edges (u,v) that connect node u with one of its descendents in the DFS-tree
Cross edges are the other edges 4 9 e b 3 2 7
cross edge 12 a 6 1 11 8 c d 10 5
SS 2019
Chapter 5

140. Classification of edges (v,w):
Depth-First Search
Classification of edges (v,w):
Edge type
d[v]< d[w]
f[v]>f[w]
tree & forward
yes
back no
cross
SS 2019
Chapter 5

141. Recognition of acyclic directed graphs (DAG)
Depth-First Search
Application:
Recognition of acyclic directed graphs (DAG)
Property: no directed cycles
SS 2019
Chapter 5

142. Depth-First Search Theorem 5.3: The following statements are
equivalent:
G is a DAG
The DFS-tree does not contain a back edge
(v,w)∈E: f[v]>f[w]
Topological sort: Assign numbers s(v) to the nodes v so that for all edges (v,w)E, s(v)<s(w).
Theorem 5.3 implies: When sorting the nodes in decreasing order of their finishing times, we obtain a topological sort.
SS 2019
Chapter 5

143. Minimum Spanning Trees
Definition 5.4:
A weighted undirected graph (G,w) is an undirected graph G=(V,E) with a weight function w:Eℝ
For any subgraph H=(U,F) (i.e., U⊆V, F⊆E) of G, the weight w(H) of H is defined as
w(H) = S w(e)
e∈F
SS 2019
Chapter 5

144. Minimum Spanning Trees
Definition 5.5:
A subgraph H of an undirected graph G is called a spanning tree of G=(V,E) if H is a tree over all nodes of G.
A spanning tree S of a weighted undirected graph G is called a minimum spanning tree (MST) of G if the weight of S is minimal among all spanning trees of G.
SS 2019
Chapter 5

145. Minimum Spanning Treesd c b a 1 4 3 2
Graph G=(V,E) d c b a 1 3 2
Spanning tree of G
(minimal)
(not minimal) d c b a 1 3 2
Spanning tree of G 3 d c b a 2
Spanning tree of G
SS 2019
Chapter 5

146. Minimum Spanning Trees8 7 b c d 4 9 2 4 a 11 i 14 e 6 7 8 10 h g f 1 2
SS 2019
Chapter 5

147. Minimum Spanning Trees
Goal: Given a weighted undirected graph (G,w) with G=(V,E), find a minimum spanning tree of (G,w).
Approach: We successively extend the edge set A⊆E to a minimum spanning tree.
Initially, A={ }.
In each step, we extend A to A∪{ {u,v} }, where {u,v} is an A-safe edge, until |A|=|V|-1.
Definition 5.6: {u,v} is called A-safe if under the assumption that A can be extended to an MST, also A∪{ {u,v} } can be extended to an MST.
SS 2019
Chapter 5

148. Generic MST-Algorithm
Generic-MST(G,w)
1. A  { }
2. while A is not a spanning tree do
find an A-safe edge {u,v}
A  A ∪ { {u,v} }
5. return A
Correctness: results from the definition of A-safe edges
SS 2019
Chapter 5

149. Cuts in Graphs V\C C A Definition 5.7:
A cut (C,V\C) in a graph G=(V,E) is a partition of the node set V of the graph.
An edge of G is crossing a cut (V,V\C) if one node of it is in C and the other node in V\C.
A cut (C,V\C) respects a subset A⊆E if no edge in A crosses the cut.
An edge crossing the cut (C,V\C) is called light if it has a minimal weight among all edges crossing (C,V\C).
V\C C A
SS 2019
Chapter 5

150. Cuts in Graphs 8 7 b c d 4 9 2 4 a 11 i 14 e 6 7 8 10 h g f 1 2 light
V-C
V-C h g f 1 2
crossing edges
SS 2019
Chapter 5

151. Characterization of Safe Edges
Theorem 5.8: Let (G,w) with G=(V,E) be a weighted undirected graph, and let A⊆E be contained in a minimum spanning tree of (G,w). Moreover, let (C,V\C) be a cut respecting A and {u,v} be a light edge crossing (C,V\C). Then {u,v} is an A-safe edge. Corollary 5.9: Let (G,w) with G=(V,E) be a weighted undirected graph, and let A⊆E be contained in a minimum spanning tree of (G,w). If {u,v} is an edge of mimimal weight that connects a connected component C of GA=(V,A) with the rest of the graph GA, then {u,v} is an A-safe edge.
SS 2019
Chapter 5

152. Prim‘s Algorithm
At any time of the algorithm, GA=(V,A) consists of a tree TA and a set of isolated nodes IA.
A edge of minimal weight that connects IA with TA is added to A. TA u
Node v with minimum w(u,v) over all edges crossing the cut
SS 2019
Chapter 5

153. Prim‘s Algorithm
At any time of the algorithm, GA=(V,A) consists of a tree TA and a set of isolated nodes IA.
A edge of minimal weight that connects IA with TA is added to A.
The nodes in IA are organized in a Min-Heap. The key d[v] of a node vIA is given by the minimal weight of an edge that connects v with TA.
SS 2019
Chapter 5

154. Prim‘s Algorithm Prim(G,w,s)
1. for each vertex vV do d[v]; p[v]nil
2. d[s]0; QBuildHeap(V)
3. while Q do
udeleteMin(Q)
for each vertex vAdj[u] do
if vQ and d[v]>w(u,v) then
DecreaseKey(Q,v,w(u,v)); p[v]u
DecreaseKey(Q,v,d) operation:
Reduces d[v] of v in Q to d and performs heapifyUp from the current position of v to repair heap property.
SS 2019
Chapter 5

155. Prim‘s Algorithm 8 7 b c d 4 9 2 4 a 11 i 14 e 6 7 8 10 h g f 1 2 4 d
SS 2019
Chapter 5

156. Prim‘s Algorithm 4 d e g c f b h a i 8 7 9 10 2 1 6 14 11 4 d e g c f
SS 2019
Chapter 5

157. Prim‘s Algorithm 4 d e g c f b h a i 8 7 9 10 2 1 6 14 11 4 d e g c f
SS 2019
Chapter 5

158. Prim‘s Algorithm 4 d e g c f b h a i 8 7 9 10 2 1 6 14 11 4 d e g c f
SS 2019
Chapter 5

159. Prim‘s Algorithm Correctness of Prim‘s algorithm for general instances4 d e g c f b h a i 8 7 9 10 2 1 6 14 11
Correctness of Prim‘s algorithm for general instances
(G,w): results from correctness of Generic-MST and
Corollary 5.9.
SS 2019
Chapter 5