Find lenght of shortest path from nodal incidence matrix

Hello, I want to find the lenght of the shortest path between two nodes out of the given nodal-terminal-incidence-matrix (nti). In the nti the number of rows equals the number of nodes and the number of columns equals the number of terminals. Every connection of two nodes is represented by a single path with two terminals. The first on the first node, second on second node. So for instance for a 4-node graph (see picture),

the nti is

    KKT=[1 0 0 0 1 0 0 0;
         0 1 0 0 0 0 1 0;
         0 0 1 0 0 1 0 0;
         0 0 0 1 0 0 0 1];

How can i calculate the adjacency-matrix from this and how can i find now the length of shortest path?

[Guillaume]

• As far as I can tell that KKT matrix tells you nothing about connectivity. It only tells you which node belongs to which terminal.
• Do the terminals actually matter. You could certainly store the adjacency matrix for the nodes and completely forget about the terminals.
• Shortest path between what and what?

Answer 1

[Marc Gebhardt]

I have already calculated the adjacence and incidence matrix with this code, where the vectors fN and tN are from-node and to-node information:

fN1=find(nti(:,:)==1);
fN=fN1(1:2:size(fN1,1)-1);
[fN,~]=ind2sub(size(nti),fN);
tN=fN1(2:2:size(fN1,1));
[tN,~]=ind2sub(size(nti),tN);

G=graph(fN,tN);
inc=incidence(G);
adj=adjacency(G);

Now I have to find the lenght of each shortest path between two nodes.

Answer 2

[Rostislav Teryaev]

It seems not like an incidence matrix. Incidence matrix represents nodes connections. There are can be two approaches:

1. Terminals as graph nodes

IncTerminals = [1 1 0 0 1 0 0 0
1 1 0 0 0 0 1 0
0 0 1 1 0 1 0 0
0 0 1 1 0 0 0 1
1 0 0 0 1 1 0 0
0 0 1 0 1 1 0 0
0 1 0 0 0 0 1 1
0 0 0 1 0 0 1 1]

2. Nodes as graph nodes

IncNodes = [1 1 1 0
1 1 0 1
1 0 1 1
0 1 1 1]

But I can not remember wether it possible or not to convert incinence matrix to adjacency matrix or build a graph object with it.

But it is possible to build any kind of matrix or graph object if you would set edges as a pairs of nodes (check the doc for graph function). Again two ways:

1. Terminals as graph nodes

s = [1 1 2 3 3 4 5 7]; % Vector of starting nodes
t = [2 5 7 4 6 8 6 8]; % Vector of target nodes

2. Nodes as graph nodes

s = [1 1 2 3]; % Vector of starting nodes
t = [2 3 4 4]; % Vector of target nodes

Now you can create a graph object by

G = graph(s,t);

plotting it by

plot(G)

gives

1. Terminals

2. Nodes

Then you can extract from graph object any matrix you want

Adj = adjacency(G);

or

Inc = incidence(G);

And the shortest path also

P = shortestpath(G,1,3)

[Marc Gebhardt]

Yeah, but now I got it, that with

distances(G)

you can get the wanted values.