Step by step
The question
Question: find the shortest path from C to A.
1 Draw the graph
The numbers on the edges are lengths: how far it is along each edge.
2 Start the table
A table: a column for each vertex.
Each cell holds how far it is from C so far, and, small, the vertex we came from.
C is 0 from itself.
The others are not reached yet: ∞, as far as can be.
Careful!
Fewer edges is not the same as shorter.
We must add the lengths.
3 Round 1
The smallest open number is C: 0.
Nothing can beat it any more, so C is final.
A: 0 plus 10 is 10.
B: 0 plus 5 is 5.
D: 0 plus 4 is 4.
The next row: the new numbers, the rest copied down.
4 Round 2
The smallest open number is D: 4.
Nothing can beat it any more, so D is final.
A: 4 plus 6 is 10, not less than 10.
Keep 10.
B: 4 plus 2 is 6, not less than 5.
Keep 5.
The next row: the new numbers, the rest copied down.
5 Round 3
The smallest open number is B: 5.
Nothing can beat it any more, so B is final.
A: 5 plus 3 is 8, less than 10.
Change it.
The next row: the new numbers, the rest copied down.
6 Round 4
The smallest open number is A: 8.
Nothing can beat it any more, so A is final.
7 The route back
Go back with the small letters: A from B, B from C.
That is the route.
The shortest path is C, B, A, length 8.
8 Check
Add the lengths along the route: 5 + 3 = 8.
Right!
The route with the fewest edges, C, A, is 10: longer.
Right!
Let's look back.
First: draw the graph.
Then: start the table.
Then: round 1.
Then: round 2.
Then: round 3.
Then: round 4.
Then: the route back.
Last: check.