Algorithms R OBERT S EDGEWICK | K EVIN W AYNE D IJKSTRA ' S A - - PowerPoint PPT Presentation

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Nov 23, 2022 429 likes •789 views

Algorithms R OBERT S EDGEWICK | K EVIN W AYNE D IJKSTRA ' S A LGORITHM D EMO Algorithms F O U R T H E D I T I O N R OBERT S EDGEWICK | K EVIN W AYNE http://algs4.cs.princeton.edu Dijkstra's algorithm demo Consider vertices in increasing

SLIDE 1

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

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

4 7 1 3 5 2 6 s 6 9 8 4 5 7 1 5 4 15 3 12 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

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

4 7 1 3 5 2 6 choose source vertex 0 v distTo[] edgeTo[] 0 0.0 - 1 2 3 4 5 6 7

SLIDE 4

・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 pointing from that vertex.

Dijkstra's algorithm demo

4 7 1 3 5 2 6 relax all edges pointing from 0 9 8 5

∞ ∞ ∞

v distTo[] edgeTo[] 0 0.0 - 1 2 3 4 5 6 7

SLIDE 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 pointing from that vertex.

∞

Dijkstra's algorithm demo

4 7 1 3 5 2 6 relax all edges pointing 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

∞

8 9

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

4 7 1 3 5 2 6 relax all edges pointing 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

∞ ∞

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

4 7 1 3 5 2 6 relax all edges pointing 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

SLIDE 10

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 11

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 12

・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 pointing from that vertex.

∞

Dijkstra's algorithm demo

4 7 1 3 5 2 6 relax all edges pointing 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 7 8 17

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

4 7 1 3 5 2 6 relax all edges pointing 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 7 8 17

∞

14 15

SLIDE 14

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 16

・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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 4 4 5 20 8 14 9

∞

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 4 4 5 20 ✔

∞

29 8 14 9 13

SLIDE 18

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 19

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 5 1 13 29 13 15

SLIDE 21

・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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 5 1 13 29 13 15 14 26

SLIDE 22

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 23

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 24

・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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 2 3 11 26 14 20

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 2 3 11 26 14 20 17 25

SLIDE 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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 27

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 28

・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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 3 9 3 25 20

SLIDE 29

・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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 3 9 ✔ 3 25 20

SLIDE 30

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 31

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 32

・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 pointing from that vertex.

Dijkstra's algorithm demo

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 pointing from 6

SLIDE 33

・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 pointing from that vertex.

Dijkstra's algorithm demo

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

SLIDE 34

・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 pointing from that vertex.

Dijkstra's algorithm demo

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