$title Minimum Spanning Tree (MST,SEQ=94)

$onText
The problem is to find the minimum spanning tree in a network.
The network of the gamslib problem (sroute) is used as an example.
The algorithm is started at all nodes in order to demonstrate that
the algorithm can start from any node.

Hillier, F S, and Lieberman G J, Introduction to Operations Research.
Holden-Day, San Francisco, 1967.

Keywords: minimum spanning tree, graph theory
$offText

Set i 'cities' / boston    , chicago   , dallas
                 kansas-cty, losangeles, memphis
                 portland  , salt-lake , wash-dc /;

Alias (i,ip,ipp);

Parameter uarc(i,ip) 'arcs and cost'
                     / boston  .chicago      58, kansas-cty .memphis    27
                       boston  .wash-dc      25, kansas-cty .salt-lake  66
                       chicago .kansas-cty   29, kansas-cty .wash-dc    62
                       chicago .memphis      32, losangeles .portland   58
                       chicago .portland    130, losangeles .salt-lake  43
                       chicago .salt-lake    85, memphis    .wash-dc    53
                       dallas  .kansas-cty   29, portland   .salt-lake  48
                       dallas  .losangeles   85, dallas     .memphis    28
                       dallas  .salt-lake    75                            /;

Set
   a(i)   'assigned nodes'
   u(i)   'unassigned nodes'
   nan(i) 'newly assigned node'
   s      'sequence' / 1*10 /;

Parameter
   mst(i,ip,ipp)  'minimum spanning trees starting from all nodes'
   fold(i,ip,ipp) 'folded minimum spanning trees'
   darc(i,ip)     'directed arcs'
   dmin           'min distance';

darc(i,ip) = max(uarc(i,ip),uarc(ip,i));

option  darc:0;
display darc;

loop(ip,
   nan(ip) = yes;
   a(i)    = nan(i);
   u(i)    = not a(i);
   loop(s$card(nan),
      nan(nan) = no;
      dmin     = smin((a,u)$darc(a,u), darc(a,u));
      loop((a,u)$(darc(a,u) and dmin = darc(a,u)),
         mst(a,u,ip) = ord(s);
         dmin = 0;
      );
      nan(u) = sum(a$(mst(a,u,ip) = ord(s)), yes);
      u(nan) =  no;
      a(nan) = yes;
   );
);

option  mst:0:2:1;
display mst;

fold(i,ip,ipp)$(ord(i) < ord(ip)) =  mst(i,ip,ipp) - mst(ip,i,ipp);

option  fold:0:2:1;
display fold;