44 demo dijk stra
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Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2011 · November 11, 2011 5:56:51 AM
AlgorithmsF O U R T H E D I T I O N
R O B E R T S E D G E W I C K K E V I N W A Y N E
4.4 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
4
4
7
1 3
5
2
6
relax all edges incident from 0
9
8
5
v distTo[] edgeTo[]
0 0.0 -
1
2
3
4
5
6
7
0
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 incident from that vertex.
∞
Dijkstra's algorithm
5
4
7
1 3
5
2
6
relax all edges incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
8
0
4
7
1 3
5
2
6
relax all edges incident 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 incident from that vertex.
Dijkstra's algorithm
9
0
4
7
1 3
5
2
6
relax all edges incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
∞
Dijkstra's algorithm
12
0
4
7
1 3
5
2
6
relax all edges incident 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 incident from that vertex.
Dijkstra's algorithm
13
0
4
7
1 3
5
2
6
relax all edges incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
15
0
4
7
1 3
5
2
6
select vertex 4
s
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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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 incident 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 incident from that vertex.
Dijkstra's algorithm
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 incident from that vertex.
Dijkstra's algorithm
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