The document describes Dijkstra's shortest path algorithm to find the shortest path between a start node 's' and target node 't' in a graph. It initializes a priority queue with all nodes, assigns initial distances/labels, and then iteratively extracts the minimum distance node and relaxes edges until the target node 't' is added to the set of visited nodes, at which point its distance will be the shortest path.

1. 1
Dijkstra's Shortest Path Algorithm
Find shortest path from s to t.
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6

2. 2
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6







0
distance label
S = { }
PQ = { s, 2, 3, 4, 5, 6, 7, t }

3. 3
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6







0
distance label
S = { }
PQ = { s, 2, 3, 4, 5, 6, 7, t }
delmin

4. 4
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9



14

0
distance label
S = { s }
PQ = { 2, 3, 4, 5, 6, 7, t }
decrease key

X


X
X

5. 5
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9



14

0
distance label
S = { s }
PQ = { 2, 3, 4, 5, 6, 7, t }

X


X
X
delmin

6. 6
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9



14

0
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }

X


X
X

7. 7
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9



14

0
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }

X


X
X
decrease key
X 33

8. 8
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9



14

0
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }

X


X
X
X 33
delmin

9. 9
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9



14

0
S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }

X


X
X
X 33
44
X
X
32

10. 10
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }

X


X
X
44
X
delmin

X 33
X
32

11. 11
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }

X


X
X
44
X
35
X
59 X
24

X 33
X
32

12. 12
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }

X


X
X
44
X
35
X
59 X
delmin

X 33
X
32

13. 13
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }

X


X
X
44
X
35
X
59 X
X
51
X 34

X 33
X
32

14. 14
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }

X


X
X
44
X
35
X
59 X
X
51
X 34
delmin

X 33
X
32
24

15. 15
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }

X


X
X
44
X
35
X
59 X
X
51
X 34
24
X
50
X
45

X 33
X
32

16. 16
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }

X


X
X
44
X
35
X
59 X
X
51
X 34
24
X
50
X
45
delmin

X 33
X
32

17. 17
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }

X


X
X
44
X
35
X
59 X
X
51
X 34
24
X
50
X
45

X 33
X
32

18. 18
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }

X


X
X
44
X
35
X
59 X
X
51
X 34
X
50
X
45
delmin

X 33
X
32
24

19. 19
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }

X


X
X
44
X
35
X
59 X
X
51
X 34
X
50
X
45

X 33
X
32

20. 20
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }

X


X
X
44
X
35
X
59 X
X
51
X 34
X
50
X
45

X 33
X
32