Section 4.6 – Shortest Path Algorithm
A
2
B
5
6
3
6
6
4
6
D
2
E
1
F
4
1
1
2
6
5
3
H
4
J
2
C
G
K
Problem: Find the shortest path from A to K
A
B
C
D
E
F
G
H
J
K
vertex w
with smallest
t value
vertex used
when t
updated
There are three values associated with each vertex:
temp(u) = temporary shortest distance from the start to vertex u; these are the values written in the table
done(u) = yes if temp(u) is the actual shortest distance, no otherwise. When yes the value has been circled in
the table and that vertex is finished with
last(u) = the vertex used when temp(u) was last changed – this is stored in an array at the foot of the table,
update entry when temp is changed
Algorithm:
for each vertex u set temp(u) = sum of all the weights, done(u) = no, last(u) = empty
set temp(start) = 0 and done(start) = yes
set current vertex (denote by u) to start
repeat until done(finish) = yes
loop look at each vertex v which is adjacent to the current vertex, u, and has done = no (i.e. those for which
the shortest distance has not yet been found)
calculate temp(u) + weight of edge uv
if smaller than temp(v) then use it to update temp(v) and set last(v) to u
end loop
determine the vertex with the smallest temporary distance and done = no and set its done to yes and set
current vertex to this vertex
end repeat
The last values are then used to find the route from start to finish by tracking backwards.