1. Shortest Path

2. What is the shortest route that visits every desk?

3. What is the shortest route that visits every desk?

4. What is the shortest route that visits every desk?
Is there an optimal solution to reduce the amount of miles I travel in a day?

5. Minimum Connector Algorithms
Kruskal’s algorithm
Select the shortest edge in a network
Select the next shortest edge which does not create a cycle
Repeat step 2 until all vertices have been connected
Prim’s algorithm
Select any vertex
Select the shortest edge connected to that vertex
Select the shortest edge connected to any vertex already connected
Repeat step 3 until all vertices have been connected

6. Jan 2013 – Q1

8. 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 4 3 7 1 2 5 B F D A G E C

9. 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.
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.

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

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

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

13. Dijkstra’s Algorithm
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. A C D B F E G 4 3 7 1 2 5 4
6 < 7 so replace the t-label here 7 6 8 3

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

15. Dijkstra’s Algorithm
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. A C D B F E G 4 3 7 1 2 5 4
5 < 6 so replace the t-label here 8 7 6 5 8 3

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

17. Dijkstra’s Algorithm
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. A C D B F E G 4 3 7 1 2 5 4 8 7
7 < 8 so replace the t-label here 7 6 5
7 < 8 so replace the t-label here 12 8 7 3

18. Dijkstra’s Algorithm 4 1 2 7 3 5 B F D A G E C 3 4 4 8 7 4 5 1 0 7 6 5 4 8 7 7 6 5 12
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

19. Dijkstra’s Algorithm
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. A C D B F E G 4 3 7 1 2 5 4 8 7 7 6 5 12 9 8 7 3
9 < 12 so replace the t-label here

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

21. Dijkstra’s Algorithm
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. A C D B F E G 4 3 7 1 2 5 4 8 7 7 6 5 12 9 8 7
11 > 9 so do not replace the t-label here 3

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

23. The shortest path is ABDEG, with length 9.
Dijkstra’s Algorithm
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. A C D B F E G 4 3 7 1 2 5 4 8 7 7 6 5 12 9 8 7 3
The shortest path is ABDEG, with length 9.

24. Jan 2013 – Q1