ROBERT SEDGEWICK | KEVIN WAYNE

F O U R T H E D I T I O N

Algorithms

http://algs4.cs.princeton.edu

Algorithms ROBERT SEDGEWICK | KEVIN WAYNE

DIJKSTRA'S ALGORITHM DEMO

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

2

0

4

7

1 3

5

2

6

s

69

8

4

5

7

1

54

15

312

20

13

11

9

an edge-weighted digraph

0→1 5.0

0→4 9.0

0→7 8.0

1→2 12.0

1→3 15.0

1→7 4.0

2→3 3.0

2→6 11.0

3→6 9.0

4→5 4.0

4→6 20.0

4→7 5.0

5→2 1.0

5→6 13.0

7→5 6.0

7→2 7.0

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

3

0

4

7

1 3

5

2

6

choose source vertex 0

v distTo[] edgeTo[]

0 0.0 -

1

2

3

4

5

6

7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

4

4

7

1 3

5

2

6

relax all edges adjacent from 0

9

8

5

0

0

∞

∞

∞

v distTo[] edgeTo[]

0 0.0 -

1

2

3

4

5

6

7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

∞

Dijkstra's algorithm demo

5

4

7

1 3

5

2

6

relax all edges adjacent from 0

9

8

5

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2

3

4 9.0 0→4

5

6

7 8.0 0→7

0

∞

5

0

∞

8

9

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

6

0

4

7

3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2

3

4 9.0 0→4

5

6

7 8.0 0→7

1

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

7

0

4

7

1 3

5

2

6

choose vertex 1

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2

3

4 9.0 0→4

5

6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

8

0

4

7

1 3

5

2

6

relax all edges adjacent from 1

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2

3

4 9.0 0→4

5

6

7 8.0 0→7

4

15

12

5

∞

∞

8

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

9

0

4

7

1 3

5

2

6

relax all edges adjacent from 1

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 17.0 1→2

3 20.0 1→3

4 9.0 0→4

5

6

7 8.0 0→7

4

15

12

✔

∞

∞5

17

20

8

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

10

0

4

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 17.0 1→2

3 20.0 1→3

4 9.0 0→4

5

6

7 8.0 0→7

7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

11

0

4

7

1 3

5

2

6

choose vertex 7

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 17.0 1→2

3 20.0 1→3

4 9.0 0→4

5

6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

∞

Dijkstra's algorithm demo

12

0

4

7

1 3

5

2

6

relax all edges adjacent from 7

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 17.0 1→2

3 20.0 1→3

4 9.0 0→4

5

6

7 8.0 0→7

6

78

17

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

13

0

4

7

1 3

5

2

6

relax all edges adjacent from 7

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 14.0 7→5

6

7 8.0 0→7

6

78

17

∞ 14

15

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

14

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 14.0 7→5

6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

15

0

4

7

1 3

5

2

6

select vertex 4

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 14.0 7→5

6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

16

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 14.0 7→5

6

7 8.0 0→7

relax all edges adjacent from 4

4

5

20

8

14

9 ∞

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

17

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 29.0 4→6

7 8.0 0→7

relax all edges adjacent from 4

4

5

20

✔

∞ 29

8

14

9

13

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

18

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 29.0 4→6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

19

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 29.0 4→6

7 8.0 0→7

select vertex 5

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

20

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 15.0 7→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 29.0 4→6

7 8.0 0→7

relax all edges adjacent from 5

1

13

29

13

15

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

21

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 26.0 5→6

7 8.0 0→7

relax all edges adjacent from 5

1

13

29

13

15 14

26

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

22

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 26.0 5→6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

23

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 26.0 5→6

7 8.0 0→7

select vertex 2

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

24

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 20.0 1→3

4 9.0 0→4

5 13.0 4→5

6 26.0 5→6

7 8.0 0→7

relax all edges adjacent from 2

3

11

26

14

20

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

25

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

relax all edges adjacent from 2

3

11

26

14

20 17

25

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

26

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

27

0

4

7

1 3

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

select vertex 3

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

28

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

relax all edges adjacent from 3

9

3

25

20

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

29

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

relax all edges adjacent from 3

9

✔

3

25

20

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

30

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

3

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

31

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

3

select vertex 6

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

32

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

3

relax all edges adjacent from 6

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

33

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

3

・Consider vertices in increasing order of distance from s

(non-tree vertex with the lowest distTo[] value).

・Add vertex to tree and relax all edges adjacent from that vertex.

Dijkstra's algorithm demo

34

0

4

7

1

5

2

6

v distTo[] edgeTo[]

0 0.0 -

1 5.0 0→1

2 14.0 5→2

3 17.0 2→3

4 9.0 0→4

5 13.0 4→5

6 25.0 2→6

7 8.0 0→7

3

shortest-paths tree from vertex s

s