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07 - Graphs
May 10, 2015
I used this set of slides for the lecture on Graphs I gave at the University of Zurich for the 1st year students following the course of Formale Grundlagen der Informatik.
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b
a
c d
e
f
g
G = (V, E)E = { {u,v} | u,v ∈ V}
b
a
c d
e
f
g
V = { a, b, c, d, e, f, g }
E = { {a,b}, {a,c}, {b,c}, {c,d}, {d,e}, {d,f}, {e,g}, {f,g} }
G = (V, E)E = { {u,v} | u,v ∈ V}
b
a
c d
e
f
g
G = (V, E)E = { {u,v} | u,v ∈ V}
V = { a, b, c, d, e, f, g }
E = { {a,b}, {a,c}, {b,c}, {c,d}, {d,e}, {d,f}, {e,g}, {f,g} }
a b c d e f g
a
b
c
d
e
f
g
0 1 1 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 1 0
0 0 0 0 0 0 1
0 0 0 0 0 0 1
0 0 0 0 0 0 0
b
a
c d
e
f
g
a b c d e f g
a
b
c
d
e
f
g
0 1 1 0 0 0 0
1 0 1 0 0 0 0
1 1 0 1 0 0 0
0 0 1 0 1 1 0
0 0 0 1 0 0 1
0 0 0 1 0 0 1
0 0 0 0 1 1 0
b
a
c d
e
f
g
a b c d e f g
a
b
c
d
e
f
g
0 1 1 0 0 0 0
1 0 1 0 0 0 0
1 1 0 1 0 0 0
0 0 1 0 1 1 0
0 0 0 1 0 0 1
0 0 0 1 0 0 1
0 0 0 0 1 1 0
b
a
c d
e
f
g
Degree of a node
2
2
3 3
2
2
2
2 3 3 2 2 2
a b c d e f g
a
b
c
d
e
f
g
0 2 3 0 0 0 0
0 0 1 0 0 0 0
0 0 0 2 0 0 0
0 0 0 0 5 4 0
0 0 0 0 0 0 3
0 0 0 0 0 0 3
0 0 0 0 0 0 0
1
3
22
5 3
34
Weighted graphs
b
a
c d
e
f
g
b
a
c
b
a
c
Not complete Complete
b
a
c
b
a
c
Bipartite Not bipartite
G = (V, E)∀ e={v,w} ∈ E, v ∈ V and w ∈ W.
b
a
c d
e
f
g
Path
Path: (b, a, c); Length (b, a, c) = 2Path: (b, d, f)
Cycle
Cycle: (f, g, e, d, f); Length (f, g, e, d, f) = 4
b
a
c d
e
f
g
Path
Path: (b, a, c); Length (b, a, c) = 2Path: (b, d, f)
Cycle
Cycle: (f, g, e, d, f); Length (f, g, e, d, f) = 4
b
a
c d
e
f
g
Path
Path: (b, a, c); Length (b, a, c) = 2Path: (b, d, f)
Cycle
Cycle: (f, g, e, d, f); Length (f, g, e, d, f) = 4
b
a
c d
e
f
g
b
a
c d
e
f
g
Loop-free Loop
b
a
c d
e
f
g
b
a
c d
e
f
g
Eulerian path
b
a
c d
e
f
g
b
a
c d
e
f
g
Eulerian path
b
a
c d
e
f
g
Hamiltonian path
b
a
c d
e
f
g
Spanning tree
G = (V, E).T ⊆ E.
d
e
f
g
a
c
Components
b
a
c d
e
f
g
Critical node
Critical edge
b
a
c d
e
f
g
Biconnected components
b
a
c d
e
f
gSubgraph Not subgraph
G = (V, E)G1 = (V1, E1)E1 = {{u,v}∈ E | u,v ∈ V1} ⊆ E.
b
a
c d
e
f
g
Weakly reachable = exists undirected path
Strongly reachable = exists directed path
E
F
DC
B
A
14
9
2
11
6
9
7
1015
http://scg.unibe.ch/download/lectures/ei/01ComputationalThinking.pptxExample: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
Example: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
∞
0
∞
∞
∞∞
Example: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
7
0
14
∞
9 ∞
Example: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
7
0
14
∞
9 < 7 + 10 7 + 15 = 22
Example: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
7
0
14 > 9 + 2
∞
9 22 > 9 + 11
Example: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
7
0
11
20
9 20
Example: Dijkstra algorithm
E
F
DC
B
A
14
9
2
11
6
9
7
1015
7
0
11
20 < 20 + 6
9 20
Example: Dijkstra algorithm
a b c d e f gabcdefg
0 2 3 0 0 0 00 0 1 0 0 0 00 0 0 2 0 0 00 0 0 0 5 4 00 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 0
1
3
22
5 3
34b
a
c d
e
f
g
Example: Floyd Warshall
a b c d e f gabcdefg
0 2 3 0 0 0 00 0 1 0 0 0 00 0 0 2 0 0 00 0 0 0 5 4 00 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 0
1
3
22
5 3
34b
a
c d
e
f
g
procedure FloydWarshall () for k := 1 to n for i := 1 to n for j := 1 to n path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );
Example: Floyd Warshall
Example: Traveling salesman
Tudor Gîrbawww.tudorgirba.com
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