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/* glpqmd.c (quotient minimum degree algorithm) */
/***********************************************************************
* This code is part of GLPK (GNU Linear Programming Kit).
*
* THIS CODE IS THE RESULT OF TRANSLATION OF THE FORTRAN SUBROUTINES
* GENQMD, QMDRCH, QMDQT, QMDUPD, AND QMDMRG FROM THE BOOK:
*
* ALAN GEORGE, JOSEPH W-H LIU. COMPUTER SOLUTION OF LARGE SPARSE
* POSITIVE DEFINITE SYSTEMS. PRENTICE-HALL, 1981.
*
* THE TRANSLATION HAS BEEN DONE WITH THE PERMISSION OF THE AUTHORS
* OF THE ORIGINAL FORTRAN SUBROUTINES: ALAN GEORGE AND JOSEPH LIU,
* UNIVERSITY OF WATERLOO, WATERLOO, ONTARIO, CANADA.
*
* The translation was made by Andrew Makhorin <mao@gnu.org>.
*
* GLPK 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 3 of the License, or
* (at your option) any later version.
*
* GLPK 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 should have received a copy of the GNU General Public License
* along with GLPK. If not, see <http://www.gnu.org/licenses/>.
***********************************************************************/
#include "glpqmd.h"
/***********************************************************************
* NAME
*
* genqmd - GENeral Quotient Minimum Degree algorithm
*
* SYNOPSIS
*
* #include "glpqmd.h"
* void genqmd(int *neqns, int xadj[], int adjncy[], int perm[],
* int invp[], int deg[], int marker[], int rchset[], int nbrhd[],
* int qsize[], int qlink[], int *nofsub);
*
* PURPOSE
*
* This routine implements the minimum degree algorithm. It makes use
* of the implicit representation of the elimination graph by quotient
* graphs, and the notion of indistinguishable nodes.
*
* CAUTION
*
* The adjancy vector adjncy will be destroyed.
*
* INPUT PARAMETERS
*
* neqns - number of equations;
* (xadj, adjncy) -
* the adjancy structure.
*
* OUTPUT PARAMETERS
*
* perm - the minimum degree ordering;
* invp - the inverse of perm.
*
* WORKING PARAMETERS
*
* deg - the degree vector. deg[i] is negative means node i has been
* numbered;
* marker - a marker vector, where marker[i] is negative means node i
* has been merged with another nodeand thus can be ignored;
* rchset - vector used for the reachable set;
* nbrhd - vector used for neighborhood set;
* qsize - vector used to store the size of indistinguishable
* supernodes;
* qlink - vector used to store indistinguishable nodes, i, qlink[i],
* qlink[qlink[i]], ... are the members of the supernode
* represented by i.
*
* PROGRAM SUBROUTINES
*
* qmdrch, qmdqt, qmdupd.
***********************************************************************/
void genqmd(int *_neqns, int xadj[], int adjncy[], int perm[],
int invp[], int deg[], int marker[], int rchset[], int nbrhd[],
int qsize[], int qlink[], int *_nofsub)
{ int inode, ip, irch, j, mindeg, ndeg, nhdsze, node, np, num,
nump1, nxnode, rchsze, search, thresh;
# define neqns (*_neqns)
# define nofsub (*_nofsub)
/* Initialize degree vector and other working variables. */
mindeg = neqns;
nofsub = 0;
for (node = 1; node <= neqns; node++)
{ perm[node] = node;
invp[node] = node;
marker[node] = 0;
qsize[node] = 1;
qlink[node] = 0;
ndeg = xadj[node+1] - xadj[node];
deg[node] = ndeg;
if (ndeg < mindeg) mindeg = ndeg;
}
num = 0;
/* Perform threshold search to get a node of min degree.
Variable search point to where search should start. */
s200: search = 1;
thresh = mindeg;
mindeg = neqns;
s300: nump1 = num + 1;
if (nump1 > search) search = nump1;
for (j = search; j <= neqns; j++)
{ node = perm[j];
if (marker[node] >= 0)
{ ndeg = deg[node];
if (ndeg <= thresh) goto s500;
if (ndeg < mindeg) mindeg = ndeg;
}
}
goto s200;
/* Node has minimum degree. Find its reachable sets by calling
qmdrch. */
s500: search = j;
nofsub += deg[node];
marker[node] = 1;
qmdrch(&node, xadj, adjncy, deg, marker, &rchsze, rchset, &nhdsze,
nbrhd);
/* Eliminate all nodes indistinguishable from node. They are given
by node, qlink[node], ... . */
nxnode = node;
s600: num++;
np = invp[nxnode];
ip = perm[num];
perm[np] = ip;
invp[ip] = np;
perm[num] = nxnode;
invp[nxnode] = num;
deg[nxnode] = -1;
nxnode = qlink[nxnode];
if (nxnode > 0) goto s600;
if (rchsze > 0)
{ /* Update the degrees of the nodes in the reachable set and
identify indistinguishable nodes. */
qmdupd(xadj, adjncy, &rchsze, rchset, deg, qsize, qlink,
marker, &rchset[rchsze+1], &nbrhd[nhdsze+1]);
/* Reset marker value of nodes in reach set. Update threshold
value for cyclic search. Also call qmdqt to form new
quotient graph. */
marker[node] = 0;
for (irch = 1; irch <= rchsze; irch++)
{ inode = rchset[irch];
if (marker[inode] >= 0)
{ marker[inode] = 0;
ndeg = deg[inode];
if (ndeg < mindeg) mindeg = ndeg;
if (ndeg <= thresh)
{ mindeg = thresh;
thresh = ndeg;
search = invp[inode];
}
}
}
if (nhdsze > 0)
qmdqt(&node, xadj, adjncy, marker, &rchsze, rchset, nbrhd);
}
if (num < neqns) goto s300;
return;
# undef neqns
# undef nofsub
}
/***********************************************************************
* NAME
*
* qmdrch - Quotient MD ReaCHable set
*
* SYNOPSIS
*
* #include "glpqmd.h"
* void qmdrch(int *root, int xadj[], int adjncy[], int deg[],
* int marker[], int *rchsze, int rchset[], int *nhdsze,
* int nbrhd[]);
*
* PURPOSE
*
* This subroutine determines the reachable set of a node through a
* given subset. The adjancy structure is assumed to be stored in a
* quotient graph format.
*
* INPUT PARAMETERS
*
* root - the given node not in the subset;
* (xadj, adjncy) -
* the adjancy structure pair;
* deg - the degree vector. deg[i] < 0 means the node belongs to the
* given subset.
*
* OUTPUT PARAMETERS
*
* (rchsze, rchset) -
* the reachable set;
* (nhdsze, nbrhd) -
* the neighborhood set.
*
* UPDATED PARAMETERS
*
* marker - the marker vector for reach and nbrhd sets. > 0 means the
* node is in reach set. < 0 means the node has been merged
* with others in the quotient or it is in nbrhd set.
***********************************************************************/
void qmdrch(int *_root, int xadj[], int adjncy[], int deg[],
int marker[], int *_rchsze, int rchset[], int *_nhdsze,
int nbrhd[])
{ int i, istop, istrt, j, jstop, jstrt, nabor, node;
# define root (*_root)
# define rchsze (*_rchsze)
# define nhdsze (*_nhdsze)
/* Loop through the neighbors of root in the quotient graph. */
nhdsze = 0;
rchsze = 0;
istrt = xadj[root];
istop = xadj[root+1] - 1;
if (istop < istrt) return;
for (i = istrt; i <= istop; i++)
{ nabor = adjncy[i];
if (nabor == 0) return;
if (marker[nabor] == 0)
{ if (deg[nabor] >= 0)
{ /* Include nabor into the reachable set. */
rchsze++;
rchset[rchsze] = nabor;
marker[nabor] = 1;
goto s600;
}
/* nabor has been eliminated. Find nodes reachable from
it. */
marker[nabor] = -1;
nhdsze++;
nbrhd[nhdsze] = nabor;
s300: jstrt = xadj[nabor];
jstop = xadj[nabor+1] - 1;
for (j = jstrt; j <= jstop; j++)
{ node = adjncy[j];
nabor = - node;
if (node < 0) goto s300;
if (node == 0) goto s600;
if (marker[node] == 0)
{ rchsze++;
rchset[rchsze] = node;
marker[node] = 1;
}
}
}
s600: ;
}
return;
# undef root
# undef rchsze
# undef nhdsze
}
/***********************************************************************
* NAME
*
* qmdqt - Quotient MD Quotient graph Transformation
*
* SYNOPSIS
*
* #include "glpqmd.h"
* void qmdqt(int *root, int xadj[], int adjncy[], int marker[],
* int *rchsze, int rchset[], int nbrhd[]);
*
* PURPOSE
*
* This subroutine performs the quotient graph transformation after a
* node has been eliminated.
*
* INPUT PARAMETERS
*
* root - the node just eliminated. It becomes the representative of
* the new supernode;
* (xadj, adjncy) -
* the adjancy structure;
* (rchsze, rchset) -
* the reachable set of root in the old quotient graph;
* nbrhd - the neighborhood set which will be merged with root to form
* the new supernode;
* marker - the marker vector.
*
* UPDATED PARAMETERS
*
* adjncy - becomes the adjncy of the quotient graph.
***********************************************************************/
void qmdqt(int *_root, int xadj[], int adjncy[], int marker[],
int *_rchsze, int rchset[], int nbrhd[])
{ int inhd, irch, j, jstop, jstrt, link, nabor, node;
# define root (*_root)
# define rchsze (*_rchsze)
irch = 0;
inhd = 0;
node = root;
s100: jstrt = xadj[node];
jstop = xadj[node+1] - 2;
if (jstop >= jstrt)
{ /* Place reach nodes into the adjacent list of node. */
for (j = jstrt; j <= jstop; j++)
{ irch++;
adjncy[j] = rchset[irch];
if (irch >= rchsze) goto s400;
}
}
/* Link to other space provided by the nbrhd set. */
link = adjncy[jstop+1];
node = - link;
if (link >= 0)
{ inhd++;
node = nbrhd[inhd];
adjncy[jstop+1] = - node;
}
goto s100;
/* All reachable nodes have been saved. End the adjacent list.
Add root to the neighborhood list of each node in the reach
set. */
s400: adjncy[j+1] = 0;
for (irch = 1; irch <= rchsze; irch++)
{ node = rchset[irch];
if (marker[node] >= 0)
{ jstrt = xadj[node];
jstop = xadj[node+1] - 1;
for (j = jstrt; j <= jstop; j++)
{ nabor = adjncy[j];
if (marker[nabor] < 0)
{ adjncy[j] = root;
goto s600;
}
}
}
s600: ;
}
return;
# undef root
# undef rchsze
}
/***********************************************************************
* NAME
*
* qmdupd - Quotient MD UPDate
*
* SYNOPSIS
*
* #include "glpqmd.h"
* void qmdupd(int xadj[], int adjncy[], int *nlist, int list[],
* int deg[], int qsize[], int qlink[], int marker[], int rchset[],
* int nbrhd[]);
*
* PURPOSE
*
* This routine performs degree update for a set of nodes in the minimum
* degree algorithm.
*
* INPUT PARAMETERS
*
* (xadj, adjncy) -
* the adjancy structure;
* (nlist, list) -
* the list of nodes whose degree has to be updated.
*
* UPDATED PARAMETERS
*
* deg - the degree vector;
* qsize - size of indistinguishable supernodes;
* qlink - linked list for indistinguishable nodes;
* marker - used to mark those nodes in reach/nbrhd sets.
*
* WORKING PARAMETERS
*
* rchset - the reachable set;
* nbrhd - the neighborhood set.
*
* PROGRAM SUBROUTINES
*
* qmdmrg.
***********************************************************************/
void qmdupd(int xadj[], int adjncy[], int *_nlist, int list[],
int deg[], int qsize[], int qlink[], int marker[], int rchset[],
int nbrhd[])
{ int deg0, deg1, il, inhd, inode, irch, j, jstop, jstrt, mark,
nabor, nhdsze, node, rchsze;
# define nlist (*_nlist)
/* Find all eliminated supernodes that are adjacent to some nodes
in the given list. Put them into (nhdsze, nbrhd). deg0 contains
the number of nodes in the list. */
if (nlist <= 0) return;
deg0 = 0;
nhdsze = 0;
for (il = 1; il <= nlist; il++)
{ node = list[il];
deg0 += qsize[node];
jstrt = xadj[node];
jstop = xadj[node+1] - 1;
for (j = jstrt; j <= jstop; j++)
{ nabor = adjncy[j];
if (marker[nabor] == 0 && deg[nabor] < 0)
{ marker[nabor] = -1;
nhdsze++;
nbrhd[nhdsze] = nabor;
}
}
}
/* Merge indistinguishable nodes in the list by calling the
subroutine qmdmrg. */
if (nhdsze > 0)
qmdmrg(xadj, adjncy, deg, qsize, qlink, marker, °0, &nhdsze,
nbrhd, rchset, &nbrhd[nhdsze+1]);
/* Find the new degrees of the nodes that have not been merged. */
for (il = 1; il <= nlist; il++)
{ node = list[il];
mark = marker[node];
if (mark == 0 || mark == 1)
{ marker[node] = 2;
qmdrch(&node, xadj, adjncy, deg, marker, &rchsze, rchset,
&nhdsze, nbrhd);
deg1 = deg0;
if (rchsze > 0)
{ for (irch = 1; irch <= rchsze; irch++)
{ inode = rchset[irch];
deg1 += qsize[inode];
marker[inode] = 0;
}
}
deg[node] = deg1 - 1;
if (nhdsze > 0)
{ for (inhd = 1; inhd <= nhdsze; inhd++)
{ inode = nbrhd[inhd];
marker[inode] = 0;
}
}
}
}
return;
# undef nlist
}
/***********************************************************************
* NAME
*
* qmdmrg - Quotient MD MeRGe
*
* SYNOPSIS
*
* #include "qmdmrg.h"
* void qmdmrg(int xadj[], int adjncy[], int deg[], int qsize[],
* int qlink[], int marker[], int *deg0, int *nhdsze, int nbrhd[],
* int rchset[], int ovrlp[]);
*
* PURPOSE
*
* This routine merges indistinguishable nodes in the minimum degree
* ordering algorithm. It also computes the new degrees of these new
* supernodes.
*
* INPUT PARAMETERS
*
* (xadj, adjncy) -
* the adjancy structure;
* deg0 - the number of nodes in the given set;
* (nhdsze, nbrhd) -
* the set of eliminated supernodes adjacent to some nodes in
* the set.
*
* UPDATED PARAMETERS
*
* deg - the degree vector;
* qsize - size of indistinguishable nodes;
* qlink - linked list for indistinguishable nodes;
* marker - the given set is given by those nodes with marker value set
* to 1. Those nodes with degree updated will have marker value
* set to 2.
*
* WORKING PARAMETERS
*
* rchset - the reachable set;
* ovrlp - temp vector to store the intersection of two reachable sets.
***********************************************************************/
void qmdmrg(int xadj[], int adjncy[], int deg[], int qsize[],
int qlink[], int marker[], int *_deg0, int *_nhdsze, int nbrhd[],
int rchset[], int ovrlp[])
{ int deg1, head, inhd, iov, irch, j, jstop, jstrt, link, lnode,
mark, mrgsze, nabor, node, novrlp, rchsze, root;
# define deg0 (*_deg0)
# define nhdsze (*_nhdsze)
/* Initialization. */
if (nhdsze <= 0) return;
for (inhd = 1; inhd <= nhdsze; inhd++)
{ root = nbrhd[inhd];
marker[root] = 0;
}
/* Loop through each eliminated supernode in the set
(nhdsze, nbrhd). */
for (inhd = 1; inhd <= nhdsze; inhd++)
{ root = nbrhd[inhd];
marker[root] = -1;
rchsze = 0;
novrlp = 0;
deg1 = 0;
s200: jstrt = xadj[root];
jstop = xadj[root+1] - 1;
/* Determine the reachable set and its intersection with the
input reachable set. */
for (j = jstrt; j <= jstop; j++)
{ nabor = adjncy[j];
root = - nabor;
if (nabor < 0) goto s200;
if (nabor == 0) break;
mark = marker[nabor];
if (mark == 0)
{ rchsze++;
rchset[rchsze] = nabor;
deg1 += qsize[nabor];
marker[nabor] = 1;
}
else if (mark == 1)
{ novrlp++;
ovrlp[novrlp] = nabor;
marker[nabor] = 2;
}
}
/* From the overlapped set, determine the nodes that can be
merged together. */
head = 0;
mrgsze = 0;
for (iov = 1; iov <= novrlp; iov++)
{ node = ovrlp[iov];
jstrt = xadj[node];
jstop = xadj[node+1] - 1;
for (j = jstrt; j <= jstop; j++)
{ nabor = adjncy[j];
if (marker[nabor] == 0)
{ marker[node] = 1;
goto s1100;
}
}
/* Node belongs to the new merged supernode. Update the
vectors qlink and qsize. */
mrgsze += qsize[node];
marker[node] = -1;
lnode = node;
s900: link = qlink[lnode];
if (link > 0)
{ lnode = link;
goto s900;
}
qlink[lnode] = head;
head = node;
s1100: ;
}
if (head > 0)
{ qsize[head] = mrgsze;
deg[head] = deg0 + deg1 - 1;
marker[head] = 2;
}
/* Reset marker values. */
root = nbrhd[inhd];
marker[root] = 0;
if (rchsze > 0)
{ for (irch = 1; irch <= rchsze; irch++)
{ node = rchset[irch];
marker[node] = 0;
}
}
}
return;
# undef deg0
# undef nhdsze
}
/* eof */
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