Who owes who money optimization

Question

Say you have n people, each who owe each other money. In general it should be possible to reduce the amount of transactions that need to take place. i.e. if X owes Y £4 and

Answer 1

I have a solution to the problem written in matlab. It is based on a matrix of who owes who what. The number in the (i,j) means that person j owes person i the number. E.g.

B owes A 2 and A owes B 1

of course in this case it is trivial that B should just give A 1

This becomes more complex with more entries. However, with the algorithm i wrote i can guarantee that no more than N-1 transactions occurs where N is the number of persones 2 in this case.

Here is the code i wrote.

function out = whooweswho(matrix)
%input sanitation
if ~isposintscalar(matrix)
    [N M] = size(matrix);
    if N ~= M
        error('Matrix must be square');
    end

    for i=1:N
        if matrix(N,N) ~= 0
            error('Matrix must have zero on diagonals');
        end
    end
else
    %construction of example matrix if input is a positive scalar
    disp('scalar input: Showing example of NxN matrix randomly filled');
    N = matrix;
    matrix = round(10*N*rand(N,N)).*(ones(N,N)-eye(N))
end

%construction of vector containing each persons balance
net = zeros(N,1);
for i=1:N
  net(i) = sum(matrix(i,:))-sum(matrix(:,i));
end

%zero matrix, so it can be used for the result
matrix = zeros(size(matrix));

%sum(net) == 0 always as no money dissappears. So if min(net) == 0 it
%implies that all balances are zero and we are done.
while min(net) ~= 0
  %find the poorest and the richest.
  [rec_amount reciever] = max(net);
  [give_amount giver] = min(net);
  %balance so at least one of them gets zero balance.
  amount =min(abs(rec_amount),abs(give_amount));
  net(reciever) = net(reciever) - amount;
  net(giver) = net(giver) + amount;
  %store result in matrix.
  matrix(reciever,giver) = amount;
end
%output equivalent matrix of input just reduced.
out = matrix;

end