/* glpini01.c */

/***********************************************************************

*  This code is part of GLPK (GNU Linear Programming Kit).

*

*  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,

*  2009, 2010 Andrew Makhorin, Department for Applied Informatics,

*  Moscow Aviation Institute, Moscow, Russia. All rights reserved.

*  E-mail: <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 "glpapi.h"

/*----------------------------------------------------------------------

-- triang - find maximal triangular part of a rectangular matrix.

--

-- *Synopsis*

--

-- int triang(int m, int n,

--    void *info, int (*mat)(void *info, int k, int ndx[]),

--    int rn[], int cn[]);

--

-- *Description*

--

-- For a given rectangular (sparse) matrix A with m rows and n columns

-- the routine triang tries to find such permutation matrices P and Q

-- that the first rows and columns of the matrix B = P*A*Q form a lower

-- triangular submatrix of as greatest size as possible:

--

--                   1                       n

--                1  * . . . . . . x x x x x x

--                   * * . . . . . x x x x x x

--                   * * * . . . . x x x x x x

--                   * * * * . . . x x x x x x

--    B = P*A*Q =    * * * * * . . x x x x x x

--                   * * * * * * . x x x x x x

--                   * * * * * * * x x x x x x

--                   x x x x x x x x x x x x x

--                   x x x x x x x x x x x x x

--                m  x x x x x x x x x x x x x

--

-- where: '*' - elements of the lower triangular part, '.' - structural

-- zeros, 'x' - other (either non-zero or zero) elements.

--

-- The parameter info is a transit pointer passed to the formal routine

-- mat (see below).

--

-- The formal routine mat specifies the given matrix A in both row- and

-- column-wise formats. In order to obtain an i-th row of the matrix A

-- the routine triang calls the routine mat with the parameter k = +i,

-- 1 <= i <= m. In response the routine mat should store column indices

-- of (non-zero) elements of the i-th row to the locations ndx[1], ...,

-- ndx[len], where len is number of non-zeros in the i-th row returned

-- on exit. Analogously, in order to obtain a j-th column of the matrix

-- A, the routine mat is called with the parameter k = -j, 1 <= j <= n,

-- and should return pattern of the j-th column in the same way as for

-- row patterns. Note that the routine mat may be called more than once

-- for the same rows and columns.

--

-- On exit the routine computes two resultant arrays rn and cn, which

-- define the permutation matrices P and Q, respectively. The array rn

-- should have at least 1+m locations, where rn[i] = i' (1 <= i <= m)

-- means that i-th row of the original matrix A corresponds to i'-th row

-- of the matrix B = P*A*Q. Similarly, the array cn should have at least

-- 1+n locations, where cn[j] = j' (1 <= j <= n) means that j-th column

-- of the matrix A corresponds to j'-th column of the matrix B.

--

-- *Returns*

--

-- The routine triang returns the size of the lower tringular part of

-- the matrix B = P*A*Q (see the figure above).

--

-- *Complexity*

--

-- The time complexity of the routine triang is O(nnz), where nnz is

-- number of non-zeros in the given matrix A.

--

-- *Algorithm*

--

-- The routine triang starts from the matrix B = P*Q*A, where P and Q

-- are unity matrices, so initially B = A.

--

-- Before the next iteration B = (B1 | B2 | B3), where B1 is partially

-- built a lower triangular submatrix, B2 is the active submatrix, and

-- B3 is a submatrix that contains rejected columns. Thus, the current

-- matrix B looks like follows (initially k1 = 1 and k2 = n):

--

--       1         k1         k2         n

--    1  x . . . . . . . . . . . . . # # #

--       x x . . . . . . . . . . . . # # #

--       x x x . . . . . . . . . . # # # #

--       x x x x . . . . . . . . . # # # #

--       x x x x x . . . . . . . # # # # #

--    k1 x x x x x * * * * * * * # # # # #

--       x x x x x * * * * * * * # # # # #

--       x x x x x * * * * * * * # # # # #

--       x x x x x * * * * * * * # # # # #

--    m  x x x x x * * * * * * * # # # # #

--       <--B1---> <----B2-----> <---B3-->

--

-- On each iteartion the routine looks for a singleton row, i.e. some

-- row that has the only non-zero in the active submatrix B2. If such

-- row exists and the corresponding non-zero is b[i,j], where (by the

-- definition) k1 <= i <= m and k1 <= j <= k2, the routine permutes

-- k1-th and i-th rows and k1-th and j-th columns of the matrix B (in

-- order to place the element in the position b[k1,k1]), removes the

-- k1-th column from the active submatrix B2, and adds this column to

-- the submatrix B1. If no row singletons exist, but B2 is not empty

-- yet, the routine chooses a j-th column, which has maximal number of

-- non-zeros among other columns of B2, removes this column from B2 and

-- adds it to the submatrix B3 in the hope that new row singletons will

-- appear in the active submatrix. */

static int triang(int m, int n,

      void *info, int (*mat)(void *info, int k, int ndx[]),

      int rn[], int cn[])

{     int *ndx; /* int ndx[1+max(m,n)]; */

      /* this array is used for querying row and column patterns of the

         given matrix A (the third parameter to the routine mat) */

      int *rs_len; /* int rs_len[1+m]; */

      /* rs_len[0] is not used;

         rs_len[i], 1 <= i <= m, is number of non-zeros in the i-th row

         of the matrix A, which (non-zeros) belong to the current active

         submatrix */

      int *rs_head; /* int rs_head[1+n]; */

      /* rs_head[len], 0 <= len <= n, is the number i of the first row

         of the matrix A, for which rs_len[i] = len */

      int *rs_prev; /* int rs_prev[1+m]; */

      /* rs_prev[0] is not used;

         rs_prev[i], 1 <= i <= m, is a number i' of the previous row of

         the matrix A, for which rs_len[i] = rs_len[i'] (zero marks the

         end of this linked list) */

      int *rs_next; /* int rs_next[1+m]; */

      /* rs_next[0] is not used;

         rs_next[i], 1 <= i <= m, is a number i' of the next row of the

         matrix A, for which rs_len[i] = rs_len[i'] (zero marks the end

         this linked list) */

      int cs_head;

      /* is a number j of the first column of the matrix A, which has

         maximal number of non-zeros among other columns */

      int *cs_prev; /* cs_prev[1+n]; */

      /* cs_prev[0] is not used;

         cs_prev[j], 1 <= j <= n, is a number of the previous column of

         the matrix A with the same or greater number of non-zeros than

         in the j-th column (zero marks the end of this linked list) */

      int *cs_next; /* cs_next[1+n]; */

      /* cs_next[0] is not used;

         cs_next[j], 1 <= j <= n, is a number of the next column of

         the matrix A with the same or lesser number of non-zeros than

         in the j-th column (zero marks the end of this linked list) */

      int i, j, ii, jj, k1, k2, len, t, size = 0;

      int *head, *rn_inv, *cn_inv;

      if (!(m > 0 && n > 0))

         xerror("triang: m = %d; n = %d; invalid dimension\n", m, n);

      /* allocate working arrays */

      ndx = xcalloc(1+(m >= n ? m : n), sizeof(int));

      rs_len = xcalloc(1+m, sizeof(int));

      rs_head = xcalloc(1+n, sizeof(int));

      rs_prev = xcalloc(1+m, sizeof(int));

      rs_next = xcalloc(1+m, sizeof(int));

      cs_prev = xcalloc(1+n, sizeof(int));

      cs_next = xcalloc(1+n, sizeof(int));

      /* build linked lists of columns of the matrix A with the same

         number of non-zeros */

      head = rs_len; /* currently rs_len is used as working array */

      for (len = 0; len <= m; len ++) head[len] = 0;

      for (j = 1; j <= n; j++)

      {  /* obtain length of the j-th column */

         len = mat(info, -j, ndx);

         xassert(0 <= len && len <= m);

         /* include the j-th column in the corresponding linked list */

         cs_prev[j] = head[len];

         head[len] = j;

      }

      /* merge all linked lists of columns in one linked list, where

         columns are ordered by descending of their lengths */

      cs_head = 0;

      for (len = 0; len <= m; len++)

      {  for (j = head[len]; j != 0; j = cs_prev[j])

         {  cs_next[j] = cs_head;

            cs_head = j;

         }

      }

      jj = 0;

      for (j = cs_head; j != 0; j = cs_next[j])

      {  cs_prev[j] = jj;

         jj = j;

      }

      /* build initial doubly linked lists of rows of the matrix A with

         the same number of non-zeros */

      for (len = 0; len <= n; len++) rs_head[len] = 0;

      for (i = 1; i <= m; i++)

      {  /* obtain length of the i-th row */

         rs_len[i] = len = mat(info, +i, ndx);

         xassert(0 <= len && len <= n);

         /* include the i-th row in the correspondng linked list */

         rs_prev[i] = 0;

         rs_next[i] = rs_head[len];

         if (rs_next[i] != 0) rs_prev[rs_next[i]] = i;

         rs_head[len] = i;

      }

      /* initially all rows and columns of the matrix A are active */

      for (i = 1; i <= m; i++) rn[i] = 0;

      for (j = 1; j <= n; j++) cn[j] = 0;

      /* set initial bounds of the active submatrix */

      k1 = 1, k2 = n;

      /* main loop starts here */

      while (k1 <= k2)

      {  i = rs_head[1];

         if (i != 0)

         {  /* the i-th row of the matrix A is a row singleton, since

               it has the only non-zero in the active submatrix */

            xassert(rs_len[i] == 1);

            /* determine the number j of an active column of the matrix

               A, in which this non-zero is placed */

            j = 0;

            t = mat(info, +i, ndx);

            xassert(0 <= t && t <= n);

            for (t = t; t >= 1; t--)

            {  jj = ndx[t];

               xassert(1 <= jj && jj <= n);

               if (cn[jj] == 0)

               {  xassert(j == 0);

                  j = jj;

               }

            }

            xassert(j != 0);

            /* the singleton is a[i,j]; move a[i,j] to the position

               b[k1,k1] of the matrix B */

            rn[i] = cn[j] = k1;

            /* shift the left bound of the active submatrix */

            k1++;

            /* increase the size of the lower triangular part */

            size++;

         }

         else

         {  /* the current active submatrix has no row singletons */

            /* remove an active column with maximal number of non-zeros

               from the active submatrix */

            j = cs_head;

            xassert(j != 0);

            cn[j] = k2;

            /* shift the right bound of the active submatrix */

            k2--;

         }

         /* the j-th column of the matrix A has been removed from the

            active submatrix */

         /* remove the j-th column from the linked list */

         if (cs_prev[j] == 0)

            cs_head = cs_next[j];

         else

            cs_next[cs_prev[j]] = cs_next[j];

         if (cs_next[j] == 0)

            /* nop */;

         else

            cs_prev[cs_next[j]] = cs_prev[j];

         /* go through non-zeros of the j-th columns and update active

            lengths of the corresponding rows */

         t = mat(info, -j, ndx);

         xassert(0 <= t && t <= m);

         for (t = t; t >= 1; t--)

         {  i = ndx[t];

            xassert(1 <= i && i <= m);

            /* the non-zero a[i,j] has left the active submatrix */

            len = rs_len[i];

            xassert(len >= 1);

            /* remove the i-th row from the linked list of rows with

               active length len */

            if (rs_prev[i] == 0)

               rs_head[len] = rs_next[i];

            else

               rs_next[rs_prev[i]] = rs_next[i];

            if (rs_next[i] == 0)

               /* nop */;

            else

               rs_prev[rs_next[i]] = rs_prev[i];

            /* decrease the active length of the i-th row */

            rs_len[i] = --len;

            /* return the i-th row to the corresponding linked list */

            rs_prev[i] = 0;

            rs_next[i] = rs_head[len];

            if (rs_next[i] != 0) rs_prev[rs_next[i]] = i;

            rs_head[len] = i;

         }

      }

      /* other rows of the matrix A, which are still active, correspond

         to rows k1, ..., m of the matrix B (in arbitrary order) */

      for (i = 1; i <= m; i++) if (rn[i] == 0) rn[i] = k1++;

      /* but for columns this is not needed, because now the submatrix

         B2 has no columns */

      for (j = 1; j <= n; j++) xassert(cn[j] != 0);

      /* perform some optional checks */

      /* make sure that rn is a permutation of {1, ..., m} and cn is a

         permutation of {1, ..., n} */

      rn_inv = rs_len; /* used as working array */

      for (ii = 1; ii <= m; ii++) rn_inv[ii] = 0;

      for (i = 1; i <= m; i++)

      {  ii = rn[i];

         xassert(1 <= ii && ii <= m);

         xassert(rn_inv[ii] == 0);

         rn_inv[ii] = i;

      }

      cn_inv = rs_head; /* used as working array */

      for (jj = 1; jj <= n; jj++) cn_inv[jj] = 0;

      for (j = 1; j <= n; j++)

      {  jj = cn[j];

         xassert(1 <= jj && jj <= n);

         xassert(cn_inv[jj] == 0);

         cn_inv[jj] = j;

      }

      /* make sure that the matrix B = P*A*Q really has the form, which

         was declared */

      for (ii = 1; ii <= size; ii++)

      {  int diag = 0;

         i = rn_inv[ii];

         t = mat(info, +i, ndx);

         xassert(0 <= t && t <= n);

         for (t = t; t >= 1; t--)

         {  j = ndx[t];

            xassert(1 <= j && j <= n);

            jj = cn[j];

            if (jj <= size) xassert(jj <= ii);

            if (jj == ii)

            {  xassert(!diag);

               diag = 1;

            }

         }

         xassert(diag);

      }

      /* free working arrays */

      xfree(ndx);

      xfree(rs_len);

      xfree(rs_head);

      xfree(rs_prev);

      xfree(rs_next);

      xfree(cs_prev);

      xfree(cs_next);

      /* return to the calling program */

      return size;

}

/*----------------------------------------------------------------------

-- adv_basis - construct advanced initial LP basis.

--

-- *Synopsis*

--

-- #include "glpini.h"

-- void adv_basis(glp_prob *lp);

--

-- *Description*

--

-- The routine adv_basis constructs an advanced initial basis for an LP

-- problem object, which the parameter lp points to.

--

-- In order to build the initial basis the routine does the following:

--

-- 1) includes in the basis all non-fixed auxiliary variables;

--

-- 2) includes in the basis as many as possible non-fixed structural

--    variables preserving triangular form of the basis matrix;

--

-- 3) includes in the basis appropriate (fixed) auxiliary variables

--    in order to complete the basis.

--

-- As a result the initial basis has minimum of fixed variables and the

-- corresponding basis matrix is triangular. */

static int mat(void *info, int k, int ndx[])

{     /* this auxiliary routine returns the pattern of a given row or

         a given column of the augmented constraint matrix A~ = (I|-A),

         in which columns of fixed variables are implicitly cleared */

      LPX *lp = info;

      int m = lpx_get_num_rows(lp);

      int n = lpx_get_num_cols(lp);

      int typx, i, j, lll, len = 0;

      if (k > 0)

      {  /* the pattern of the i-th row is required */

         i = +k;

         xassert(1 <= i && i <= m);

#if 0 /* 22/XII-2003 */

         /* if the auxiliary variable x[i] is non-fixed, include its

            element (placed in the i-th column) in the pattern */

         lpx_get_row_bnds(lp, i, &typx, NULL, NULL);

         if (typx != LPX_FX) ndx[++len] = i;

         /* include in the pattern elements placed in columns, which

            correspond to non-fixed structural varables */

         i_beg = aa_ptr[i];

         i_end = i_beg + aa_len[i] - 1;

         for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++)

         {  j = m + sv_ndx[i_ptr];

            lpx_get_col_bnds(lp, j-m, &typx, NULL, NULL);

            if (typx != LPX_FX) ndx[++len] = j;

         }

#else

         lll = lpx_get_mat_row(lp, i, ndx, NULL);

         for (k = 1; k <= lll; k++)

         {  lpx_get_col_bnds(lp, ndx[k], &typx, NULL, NULL);

            if (typx != LPX_FX) ndx[++len] = m + ndx[k];

         }

         lpx_get_row_bnds(lp, i, &typx, NULL, NULL);

         if (typx != LPX_FX) ndx[++len] = i;

#endif

      }

      else

      {  /* the pattern of the j-th column is required */

         j = -k;

         xassert(1 <= j && j <= m+n);

         /* if the (auxiliary or structural) variable x[j] is fixed,

            the pattern of its column is empty */

         if (j <= m)

            lpx_get_row_bnds(lp, j, &typx, NULL, NULL);

         else

            lpx_get_col_bnds(lp, j-m, &typx, NULL, NULL);

         if (typx != LPX_FX)

         {  if (j <= m)

            {  /* x[j] is non-fixed auxiliary variable */

               ndx[++len] = j;

            }

            else

            {  /* x[j] is non-fixed structural variables */

#if 0 /* 22/XII-2003 */

               j_beg = aa_ptr[j];

               j_end = j_beg + aa_len[j] - 1;

               for (j_ptr = j_beg; j_ptr <= j_end; j_ptr++)

                  ndx[++len] = sv_ndx[j_ptr];

#else

               len = lpx_get_mat_col(lp, j-m, ndx, NULL);

#endif

            }

         }

      }

      /* return the length of the row/column pattern */

      return len;

}

static void adv_basis(glp_prob *lp)

{     int m = lpx_get_num_rows(lp);

      int n = lpx_get_num_cols(lp);

      int i, j, jj, k, size;

      int *rn, *cn, *rn_inv, *cn_inv;

      int typx, *tagx = xcalloc(1+m+n, sizeof(int));

      double lb, ub;

      xprintf("Constructing initial basis...\n");

#if 0 /* 13/V-2009 */

      if (m == 0)

         xerror("glp_adv_basis: problem has no rows\n");

      if (n == 0)

         xerror("glp_adv_basis: problem has no columns\n");

#else

      if (m == 0 || n == 0)

      {  glp_std_basis(lp);

         return;

      }

#endif

      /* use the routine triang (see above) to find maximal triangular

         part of the augmented constraint matrix A~ = (I|-A); in order

         to prevent columns of fixed variables to be included in the

         triangular part, such columns are implictly removed from the

         matrix A~ by the routine adv_mat */

      rn = xcalloc(1+m, sizeof(int));

      cn = xcalloc(1+m+n, sizeof(int));

      size = triang(m, m+n, lp, mat, rn, cn);

      if (lpx_get_int_parm(lp, LPX_K_MSGLEV) >= 3)

         xprintf("Size of triangular part = %d\n", size);

      /* the first size rows and columns of the matrix P*A~*Q (where

         P and Q are permutation matrices defined by the arrays rn and

         cn) form a lower triangular matrix; build the arrays (rn_inv

         and cn_inv), which define the matrices inv(P) and inv(Q) */

      rn_inv = xcalloc(1+m, sizeof(int));

      cn_inv = xcalloc(1+m+n, sizeof(int));

      for (i = 1; i <= m; i++) rn_inv[rn[i]] = i;

      for (j = 1; j <= m+n; j++) cn_inv[cn[j]] = j;

      /* include the columns of the matrix A~, which correspond to the

         first size columns of the matrix P*A~*Q, in the basis */

      for (k = 1; k <= m+n; k++) tagx[k] = -1;

      for (jj = 1; jj <= size; jj++)

      {  j = cn_inv[jj];

         /* the j-th column of A~ is the jj-th column of P*A~*Q */

         tagx[j] = LPX_BS;

      }

      /* if size < m, we need to add appropriate columns of auxiliary

         variables to the basis */

      for (jj = size + 1; jj <= m; jj++)

      {  /* the jj-th column of P*A~*Q should be replaced by the column

            of the auxiliary variable, for which the only unity element

            is placed in the position [jj,jj] */

         i = rn_inv[jj];

         /* the jj-th row of P*A~*Q is the i-th row of A~, but in the

            i-th row of A~ the unity element belongs to the i-th column

            of A~; therefore the disired column corresponds to the i-th

            auxiliary variable (note that this column doesn't belong to

            the triangular part found by the routine triang) */

         xassert(1 <= i && i <= m);

         xassert(cn[i] > size);

         tagx[i] = LPX_BS;

      }

      /* free working arrays */

      xfree(rn);

      xfree(cn);

      xfree(rn_inv);

      xfree(cn_inv);

      /* build tags of non-basic variables */

      for (k = 1; k <= m+n; k++)

      {  if (tagx[k] != LPX_BS)

         {  if (k <= m)

               lpx_get_row_bnds(lp, k, &typx, &lb, &ub);

            else

               lpx_get_col_bnds(lp, k-m, &typx, &lb, &ub);

            switch (typx)

            {  case LPX_FR:

                  tagx[k] = LPX_NF; break;

               case LPX_LO:

                  tagx[k] = LPX_NL; break;

               case LPX_UP:

                  tagx[k] = LPX_NU; break;

               case LPX_DB:

                  tagx[k] =

                     (fabs(lb) <= fabs(ub) ? LPX_NL : LPX_NU);

                  break;

               case LPX_FX:

                  tagx[k] = LPX_NS; break;

               default:

                  xassert(typx != typx);

            }

         }

      }

      for (k = 1; k <= m+n; k++)

      {  if (k <= m)

            lpx_set_row_stat(lp, k, tagx[k]);

         else

            lpx_set_col_stat(lp, k-m, tagx[k]);

      }

      xfree(tagx);

      return;

}

/***********************************************************************

*  NAME

*

*  glp_adv_basis - construct advanced initial LP basis

*

*  SYNOPSIS

*

*  void glp_adv_basis(glp_prob *lp, int flags);

*

*  DESCRIPTION

*

*  The routine glp_adv_basis constructs an advanced initial basis for

*  the specified problem object.

*

*  The parameter flags is reserved for use in the future and must be

*  specified as zero. */

void glp_adv_basis(glp_prob *lp, int flags)

{     if (flags != 0)

         xerror("glp_adv_basis: flags = %d; invalid flags\n", flags);

      if (lp->m == 0 || lp->n == 0)

         glp_std_basis(lp);

      else

         adv_basis(lp);

      return;

}

/* eof */