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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function varargout = euclideanMST(points)
%EUCLIDEANMST Build euclidean minimal spanning tree of a set of points.
%
% EDGES = euclideanMST(POINTS)
% POINTS is a [NxP] array, N being the number of points and P being the
% dimension.
% Result EDGES is a [Mx2] array, containing indices of each vertex for
% each edges.
%
% [EDGES DISTS] = euclideanMST(POINTS)
% Also returns the lengths of edges computed by MST algorithm.
%
% Algorithm first computes Delaunay triangulation of the set of points,
% then computes euclidean length of each edge of triangulation, and
% finally uses prim algorithm to simplify the graph.
%
% Example
% % choose random points in the plane and display their Euclidean MST
% pts = rand(50, 2)*100;
% edges = euclideanMST(pts);
% drawGraph(pts, edges)
%
% See also
% prim_mst, distancePoints, delaunayn
%
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2007-07-27, using Matlab 7.4.0.287 (R2007a)
% Copyright 2007-2023 INRA - BIA PV Nantes - MIAJ Jouy-en-Josas
% dimension
D = size(points, 2);
Df = factorial(D);
% compute all couples of vertices in unit triangle, tetrahedron, or n-dim
% simplex
subs = zeros(Df, 2);
k = 1;
for i = 1:D
for j = i+1:D+1
subs(k, 1) = i;
subs(k, 2) = j;
k = k + 1;
end
end
% compute delaunay triangulation in D dimensions
tri = delaunayn(points);
Nt = size(tri, 1);
% compute all possible edges
edges = zeros(Nt*Df, 2);
for t = 1:Nt
for i = 1:Df
edges((t-1)*Df+i, 1) = tri(t, subs(i, 1));
edges((t-1)*Df+i, 2) = tri(t, subs(i, 2));
end
end
% simplify edges
edges = unique(sort(edges, 2), 'rows');
% compute euclidean length of each edge
val = zeros(size(edges, 1), 1);
for i = 1:size(edges,1)
val(i) = distancePoints(points(edges(i,1), :), points(edges(i,2), :));
end
% compute MST of created graph
[edges2, vals2] = prim_mst(edges, val);
% process output arguments
if nargout == 1
varargout{1} = edges2;
elseif nargout==2
varargout{1} = edges2;
varargout{2} = vals2;
end
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