1 / 17

Shortest Path

Shortest Path. 4. 4. 1. 2. 4. 7. 7. 3. 2. 3. 2. 5. Dijkstra’s Algorithm finds the shortest path from the start vertex to every other vertex in the network. We will find the shortest path from A to G. B. F. D. A. G. E. C. Dijkstra’s Algorithm.

dorotheat
Download Presentation

Shortest Path

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Shortest Path

  2. 4 4 1 2 4 7 7 3 2 3 2 5 Dijkstra’s Algorithmfinds the shortest path from the start vertex to every other vertex in the network. We will find the shortest path from A to G B F D A G E C

  3. Dijkstra’s Algorithm • Label the start vertex with permanent label 0 and order label 1 • Assign temporary labels to all the vertices that can be reached directly from the start • Select the vertex with the smallest temporary label and make its label permanent. Add the correct order label. • Put temporary labels on each vertex that can be reached directly from the vertex you have just made permanent. The temporary label must be equal to the sum of the permanent label and the direct distance from it. If there is an existing temporary label at a vertex, it should be replaced only if the new sum is smaller. • Select the vertex with the smallest temporary label and make its label permanent. Add the correct order label. • Repeat until the finishing vertex has a permanent label. • To find the shortest paths(s), trace back from the end vertex to the start vertex. Write the route forwards and state the length.

  4. Order in which vertices are labelled. Distance from A to vertex Working 4 B F 4 1 10 2 4 7 D A 7 Label vertex A 1 as it is the first vertex labelled 3 2 3 G 2 E 5 C Dijkstra’s Algorithm

  5. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C We update each vertex adjacent to A with a ‘working value’ for its distance from A. Dijkstra’s Algorithm 4 1 0 7 3

  6. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C Dijkstra’s Algorithm 4 1 0 7 Vertex C is closest to A so we give it a permanent label 3. C is the 2nd vertex to be permanently labelled. 2 3 3

  7. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C We update each vertex adjacent to C with a ‘working value’ for its total distance from A, by adding its distance from C to C’s permanent label of 3. Dijkstra’s Algorithm 4 6 < 7 so replace the t-label here 1 0 7 6 2 3 8 3

  8. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C The vertex with the smallest temporary label is B, so make this label permanent. B is the 3rd vertex to be permanently labelled. Dijkstra’s Algorithm 3 4 4 1 0 7 6 2 3 8 3

  9. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C We update each vertex adjacent to B with a ‘working value’ for its total distance from A, by adding its distance from B to B’s permanent label of 4. Dijkstra’s Algorithm 3 4 4 5 < 6 so replace the t-label here 8 1 0 7 6 5 2 3 8 3

  10. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C The vertex with the smallest temporary label is D, so make this label permanent. D is the 4th vertex to be permanently labelled. Dijkstra’s Algorithm 3 4 4 8 4 5 1 0 7 6 5 2 3 8 3

  11. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C We update each vertex adjacent to D with a ‘working value’ for its total distance from A, by adding its distance from D to D’s permanent label of 5. Dijkstra’s Algorithm 3 4 4 8 7 7 < 8 so replace the t-label here 4 5 1 0 7 6 5 7 < 8 so replace the t-label here 12 2 3 8 7 3

  12. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C Dijkstra’s Algorithm 3 4 4 8 7 4 5 1 0 7 6 5 12 5 7 2 3 The vertices with the smallest temporary labels are E and F, so choose one and make the label permanent. E is chosen - the 5th vertex to be permanently labelled. 8 7 3

  13. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C We update each vertex adjacent to E with a ‘working value’ for its total distance from A, by adding its distance from E to E’s permanent label of 7. Dijkstra’s Algorithm 3 4 4 8 7 4 5 1 0 7 6 5 12 9 5 7 2 3 8 7 3 9 < 12 so replace the t-label here

  14. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C The vertex with the smallest temporary label is F, so make this label permanent.F is the 6th vertex to be permanently labelled. Dijkstra’s Algorithm 3 4 4 6 7 8 7 4 5 1 0 7 6 5 12 9 5 7 2 3 8 7 3

  15. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C We update each vertex adjacent to F with a ‘working value’ for its total distance from A, by adding its distance from F to F’s permanent label of 7. Dijkstra’s Algorithm 3 4 4 6 7 8 7 4 5 1 0 7 6 5 12 9 5 7 2 3 8 11 > 9 so do not replace the t-label here 7 3

  16. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C Dijkstra’s Algorithm 3 4 4 6 7 8 7 4 5 1 0 7 6 5 7 9 12 9 5 7 2 3 8 7 G is the final vertex to be permanently labelled. 3

  17. 4 B F 4 1 2 4 7 D A 7 3 2 3 G 2 E 5 C To find the shortest path from A to G, start from G and work backwards, choosing arcs for which the difference between the permanent labels is equal to the arc length. Dijkstra’s Algorithm 3 4 4 6 7 8 7 4 5 1 0 7 6 5 7 9 12 9 5 7 2 3 8 7 3 The shortest path is ABDEG, with length 9.

More Related