Toronto. 650. 700. Boston. Chicago. 200. 600. New York. Shortest Path Problems. We can assign weights to the edges of graphs, for example to represent the distance between cities in a railway network:. Shortest Path Problems.
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650
700
Boston
Chicago
200
600
New York
Shortest Path ProblemsApplied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
d
a
z
c
e
Dijkstra’s Algorithm
5
6
4
8
1
2
0
3
2
10
Step 0
Applied Discrete Mathematics Week 14: Trees
d
5
6
4
8
a
1
z
2
0
3
2
10
c
e
Dijkstra’s Algorithm4 (a)
2 (a)
Step 1
Applied Discrete Mathematics Week 14: Trees
d
5
6
4
8
a
1
z
2
0
3
2
10
c
e
Dijkstra’s Algorithm
3 (a, c)
4 (a)
10 (a, c)
2 (a)
12 (a, c)
Step 2
Applied Discrete Mathematics Week 14: Trees
d
5
6
4
8
a
1
z
2
0
3
2
10
c
e
Dijkstra’s Algorithm3 (a, c)
4 (a)
10 (a, c)
8 (a, c, b)
2 (a)
12 (a, c)
Step 3
Applied Discrete Mathematics Week 14: Trees
d
5
6
4
8
a
1
z
2
0
3
2
10
c
e
Dijkstra’s Algorithm3 (a, c)
4 (a)
10 (a, c)
8 (a, c, b)
14 (a, c, b, d)
2 (a)
12 (a, c)
10 (a, c, b, d)
Step 4
Applied Discrete Mathematics Week 14: Trees
d
5
6
4
8
a
1
z
2
0
3
2
10
c
e
Dijkstra’s Algorithm
4 (a)
3 (a, c)
8 (a, c, b)
10 (a, c)
14 (a, c, b, d)
13 (a, c, b, d, e)
2 (a)
12 (a, c)
10 (a, c, b, d)
Step 5
Applied Discrete Mathematics Week 14: Trees
d
5
6
4
8
a
1
z
2
0
3
2
10
c
e
Dijkstra’s Algorithm
4 (a)
3 (a, c)
8 (a, c, b)
10 (a, c)
14 (a, c, b, d)
13 (a, c, b, d, e)
2 (a)
12 (a, c)
10 (a, c, b, d)
Step 6
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
650
550
700
Boston
700
Chicago
200
600
New York
The Traveling Salesman ProblemSolution: The shortest path is Boston, New York, Chicago, Toronto, Boston (2,000 miles).
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Yes.
No.
No.
Yes.
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees
James
Christine
Bob
Frank
Joyce
Petra
Applied Discrete Mathematics Week 14: Trees
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Applied Discrete Mathematics Week 14: Trees
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y
z
x
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This tree represents the expression (y + z)(x  y).
Applied Discrete Mathematics Week 14: Trees
Applied Discrete Mathematics Week 14: Trees