All the possible path between two points without repetition

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I'm looking to get the value of all pairwise data between two points by an iteration. The iteration contains direct and indirect paths. For example if there are 3 points, and having all the pairwise scores between points being available:
a_11 = 0; a_12 = 0.2; a_13 = 0.6; a_32 = -0.3; % a_23 = -a_32 (same for all other scores)
Now if the value for example, the path between 1-2 is considered, we would have (direct + indirect):
a_12_iter1 = a_12 + (a_13 + a_32)% initial paiwsie scores between all points are known, a_12_iter1 means the first iteration
%considering the initial pairwise scores.
a_12_iter1 = 0.2 + (0.6 - 0.3) = 0.5
No path is chosen twice.
Another example is having 4 points (square):
a_12_iter1 = a_12 + (a_14 + a_43 + a_32 + a_13);
a_13_iter1 = a_13 + (a_12 + a_23 + a_14 + a_43 + a_24); %same procedure is used for all other values (a_14, a_24, ...)
The data gets large fast, another instant is a pentagon (5 points):
a_13_iter1 = a_13 + (a_12 + a_23 + a_15 + a_54 + a_43 + a_43 + a_25 + a_53 + a_14)% no path is chosen twice
I am looking for a code that can perform this operation if my original data is large (100 points). The code for 3 points is given as:
Does anyone know how I can write this code? Thank you!

Accepted Answer

Abolfazl Chaman Motlagh
Abolfazl Chaman Motlagh on 26 Feb 2022
You can solve this problem with your data as a matrix, or as a graph. the matrix make it easier i guess:
a traditional method with loop is kinnda this:
Connections = [0 0.2 0.6; -0.2 0 0.3; -0.6 -0.3 0]; % for 3 point problem
Path = zeros(3);
for i=1:3
for j=1:3
Path(i,j) = Connections(i,j); % First Term
for k=setdiff((1:3),[i,j])
Path(i,j) = Path(i,j) + Connections(i,k) + Connections(k,j);
Path = 3×3
0 0.5000 1.1000 -0.5000 0 0.7000 -1.1000 -0.7000 0
you can see it is what you wanted.
of course inner loop can be vectorize to code below:
for i=1:3
for j=1:3
Path(i,j) = Connections(i,j) + sum(Connections(i,k)) + sum(Connections(k,j));
Path can also be initialize from Connections itself instead of zeros.
this can be your function for whatever velue and number of points you have.
function Path = Path_value_cal(Connections)
n = size(Connections);
Path = Connections;
for i=1:n
for j=1:n
Path(i,j) = Path(i,j) + sum(Connections(i,k)) + sum(Connections(k,j));

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