# Dijkstra's algorithm

If debugging is the process of removing bugs, then programming must be the process of putting them in.

—Edsger W. Dijkstra (cannot find the citation, this may be fabricated)

Dijkstra’s algorithm is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. The more common algorithm fixes a single node as the initial node and finds shortest paths from the initial node to all other nodes in the graph, producing a shortest-path tree.

The steps for implementing Dijkstra’s algorithm are as follows:

- find the initial distance vector \(D\) of each nodes (the distance from initial node to other nodes)
- find the minimum distance between the initial node and one other node, set the distance to vector \(D\)
- update the initial node index, from the new initial node find the minimum distance again.

Here is the MATLAB code for calculate the distance from a graph, the output vector is the shortest paths from initial node to other nodes.

```
function dijkstra(V,o)
%V is adjacency matrix, o is the initial node index
A=V;
[m,n]=size(A);
d=zeros(1,m);% vector to store the distance of initial node and other nodes
d(:)=inf;
Q=A(o,:);%find the distance between two nodes (the initial node and others)
d(o)=0;% set the distance of start point to initial node is zero
p=10;%loop flag
while(min(Q)~=inf&p~=0)% stop loop when all distance is Inf or loop flag is zero
[a id]=min(Q);% find the minimum distance and the index of the node
Q(id)=inf;%set the minimum index to Inf
A(o,id)=inf;%set the minimum distance to Ind
o=id;%update the initial node
if d(id)>a
d(id)=a;
end
for j=1:m%change the distance of each node
if (Q(j)>d(id)+A(id,j))
Q(j)=d(id)+A(id,j);
end
end
p=0;%loop flat == 0
for i=1:m%check inf value, set the loop flag > 0 (looping)
if d(i)==inf
p=p+1;
end
end
end
d
end
```

The distance from initial `node 1`

to `node 2`

is Inf, from `node 2`

to `node 3`

is 10, from `node 3`

to `node 4`

is 50, from `node 4`

to `node 5`

is 20, from `node 5`

to `node 6`

is 60.

```
V=[0 inf 10 inf 30 100;
inf 0 5 inf inf inf;
inf inf 0 50 inf inf;
inf inf inf 0 inf 10;
inf inf inf 20 0 60;
inf inf inf inf inf 0];
octave:3> dijkstra(V,1)
warning: Matlab-style short-circuit operation performed for operator &
warning: called from
dijkstra at line 25 column 9
d =
0 Inf 10 50 30 60
```