Step by step
The question
Question: find the shortest path from A to E.
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 A so far, and, small, the vertex we came from.
A is 0 from itself.
The others are not reached yet: ∞, as far as can be.
Careful!
A number can still go down later.
It is final only when it is the smallest open one and gets its ring.
3 Round 1
The smallest open number is A: 0.
Nothing can beat it any more, so A is final.
B: 0 plus 6 is 6.
C: 0 plus 2 is 2.
The next row: the new numbers, the rest copied down.
4 Round 2
The smallest open number is C: 2.
Nothing can beat it any more, so C is final.
D: 2 plus 3 is 5.
The next row: the new numbers, the rest copied down.
5 Round 3
The smallest open number is D: 5.
Nothing can beat it any more, so D is final.
B: 5 plus 1 is 6, not less than 6.
Keep 6.
E: 5 plus 6 is 11.
The next row: the new numbers, the rest copied down.
6 Round 4
The smallest open number is B: 6.
Nothing can beat it any more, so B is final.
E: 6 plus 2 is 8, less than 11.
Change it.
The next row: the new numbers, the rest copied down.
7 Round 5
The smallest open number is E: 8.
Nothing can beat it any more, so E is final.
8 The route back
Go back with the small letters: E from B, B from A.
That is the route.
The shortest path is A, B, E, length 8.
9 Check
Add the lengths along the route: 6 + 2 = 8.
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: round 5.
Then: the route back.
Last: check.