function [p,s]=v_minspane(x)
%V_MINSPANE calculate minimum spanning tree using euclidean distance [p,s]=X
%
% Inputs:  x(n,d)    d-dimensional data points, one per row
%
% Outputs: p(n-1,1)  indices of the parent of each node within the tree
%                      (data point n is the root)
%          s(n-1,1)  list of the edges in ascending order of euclidean
%                      distance. Thus the shortest edge goes from node
%                      s(1) to node p(s(1))
%
% The minimum spanning tree (or shortest spanning tree ) defines a set of
% n-1 links that interconnect n points (or nodes) with the minimum total
% length. We represent these links in the form of a tree with node n as
% the root. Each node (except node n) has a unique parent node that is
% given in the output vector p; it is possible for several nodes to share
% the same parent.
% It can be useful to know which of the links are the longest, so the output
% argument lists them in ascending order. s could be calculated directly by
% calculaing the length, l, of each link as follows:
%       l=sqrt(sum((x(1:n-1,:) - x(p,:)).^2),2);
%       [v,s]=sort(l);

%      Copyright (C) Mike Brookes 2000-2009
%      Version: $Id: v_minspane.m 10865 2018-09-21 17:22:45Z dmb $
%
%   VOICEBOX is a MATLAB toolbox for speech processing.
%   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   This program is free software; you can redistribute it and/or modify
%   it under the terms of the GNU General Public License as published by
%   the Free Software Foundation; either version 2 of the License, or
%   (at your option) any later version.
%
%   This program is distributed in the hope that it will be useful,
%   but WITHOUT ANY WARRANTY; without even the implied warranty of
%   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%   GNU General Public License for more details.
%
%   You can obtain a copy of the GNU General Public License from
%   http://www.gnu.org/copyleft/gpl.html or by writing to
%   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[np,nd]=size(x);

% first do delauny tessalation to find feasible edges

t=delaunayn(x); % nd+1 vertices per polytope
nt=size(t,1);
cn=v_choosenk(nd+1,2);
nf=size(cn,1);
ee=zeros(nt,nf,2);     % space for all the edges
ee(:,:,1)=t(:,cn(:,1));
ee(:,:,2)=t(:,cn(:,2));
ee=reshape(ee,[],2);
mk=ee(:,1)>ee(:,2);
ee(mk,:)=ee(mk,2:-1:1); % make all edges in ascending order
% ees=sparse(ee(:,1),ee(:,2),1);      % remove duplicates
[er,ec]=find(sparse(ee(:,1),ee(:,2),1));      % remove duplicates
ee=[er,ec];
ne=size(ee,1);

% now apply Kruskal shortest spanning tree algorithm

sz=sum((x(ee(:,1),:)-x(ee(:,2),:)).^2,2);     % length^2 of each edge
[vz,mz]=sort(sz);
ee=ee(mz,:);        % sort edges into ascenging length order
ts=ones(ne,1);      % size of each component
tp=zeros(np,1);     % root node links
ei=zeros(np-1,1);     % index of shortest spanning tree edges
k=0;
for i=1:ne
    i1=ee(i,1);
    j=tp(i1);
    while j         % find root node for ee(i,1)
        i1=j;
        j=tp(i1);
    end
    i2=ee(i,2);
    j=tp(i2);
    while j         % find root node for ee(i,2)
        i2=j;
        j=tp(i2);
    end
    if i1~=i2       % if they are different, merge them
        k=k+1;
        ei(k)=i;    % add to the shortest spanning tree
        if ts(i1)>ts(i2)
            tp(i2)=i1;          % make i2 a sub tree of i1
            ts(i1)=ts(i1)+ts(i2);
        else
            tp(i1)=i2;          % make i1 a sub tree of i2
            ts(i2)=ts(i1)+ts(i2);
        end
    end
end
ee=ee(ei,:);        % refine the edges to include only those in the minimum spanning tree

% now arrange as a tree with point np as the head

eet=sparse(ee(:,1),ee(:,2),1:np-1,np,np);
p=zeros(np-1,1);              % points to parent node
s=zeros(np-1,1);              % sorted index
chn=np;                                 % start with the root node of the tree
[rf,cf]=find(eet(:,chn));               % find any nodes that connect to it
[rg,cg]=find(eet(chn,:));
while ~isempty(rf) || ~isempty(rg)
    p(rf)=chn(cf);                      % set the parents
    rcf=rf(:)+np*(chn(cf(:))-1);
    s(eet(rcf))=rf(:);
    eet(rcf)=0;                         % delete the edges linking them to their parents
    p(cg)=chn(rg);
    rcg=chn(rg(:))+np*(cg(:)-1);
    s(eet(rcg))=cg(:);
    eet(rcg)=0;
    chn=[rf(:); cg(:)];                 % now search for their children
    [rf,cf]=find(eet(:,chn));
    [rg,cg]=find(eet(chn,:));
end

