amesos_cholmod_rowcolcounts.c

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/* ========================================================================== */
/* === Cholesky/cholmod_rowcolcounts ======================================== */
/* ========================================================================== */

/* -----------------------------------------------------------------------------
 * CHOLMOD/Cholesky Module.  Copyright (C) 2005-2006, Timothy A. Davis
 * The CHOLMOD/Cholesky Module is licensed under Version 2.1 of the GNU
 * Lesser General Public License.  See lesser.txt for a text of the license.
 * CHOLMOD is also available under other licenses; contact authors for details.
 * http://www.cise.ufl.edu/research/sparse
 * -------------------------------------------------------------------------- */

/* Compute the row and column counts of the Cholesky factor L of the matrix
 * A or A*A'.  The etree and its postordering must already be computed (see
 * cholmod_etree and cholmod_postorder) and given as inputs to this routine.
 *
 * For the symmetric case (LL'=A), A is accessed by column.  Only the lower
 * triangular part of A is used.  Entries not in this part of the matrix are
 * ignored.  This is the same as storing the upper triangular part of A by
 * rows, with entries in the lower triangular part being ignored.  NOTE: this
 * representation is the TRANSPOSE of the input to cholmod_etree.
 *
 * For the unsymmetric case (LL'=AA'), A is accessed by column.  Equivalently,
 * if A is viewed as a matrix in compressed-row form, this routine computes
 * the row and column counts for L where LL'=A'A.  If the input vector f is
 * present, then F*F' is analyzed instead, where F = A(:,f).
 *
 * The set f is held in fset and fsize.
 *  fset = NULL means ":" in MATLAB. fset is ignored.
 *  fset != NULL means f = fset [0..fset-1].
 *  fset != NULL and fsize = 0 means f is the empty set.
 *  Common->status is set to CHOLMOD_INVALID if fset is invalid.
 *
 * In both cases, the columns of A need not be sorted.
 * A can be packed or unpacked.
 *
 * References:
 * J. Gilbert, E. Ng, B. Peyton, "An efficient algorithm to compute row and
 * column counts for sparse Cholesky factorization", SIAM J. Matrix Analysis &
 * Applic., vol 15, 1994, pp. 1075-1091.
 *
 * J. Gilbert, X. Li, E. Ng, B. Peyton, "Computing row and column counts for
 * sparse QR and LU factorization", BIT, vol 41, 2001, pp. 693-710.
 *
 * workspace:
 *  if symmetric:   Flag (nrow), Iwork (2*nrow)
 *  if unsymmetric: Flag (nrow), Iwork (2*nrow+ncol), Head (nrow+1)
 *
 * Supports any xtype (pattern, real, complex, or zomplex).
 */

#ifndef NCHOLESKY

#include "amesos_cholmod_internal.h"
#include "amesos_cholmod_cholesky.h"

/* ========================================================================== */
/* === initialize_node ====================================================== */
/* ========================================================================== */

static int amesos_initialize_node  /* initial work for kth node in postordered etree */
(
    Int k,    /* at the kth step of the algorithm (and kth node) */
    Int Post [ ], /* Post [k] = i, the kth node in postordered etree */
    Int Parent [ ], /* Parent [i] is the parent of i in the etree */
    Int ColCount [ ], /* ColCount [c] is the current weight of node c */
    Int PrevNbr [ ] /* PrevNbr [u] = k if u was last considered at step k */
)
{
    Int p, parent ;
    /* determine p, the kth node in the postordered etree */
    p = Post [k] ;
    /* adjust the weight if p is not a root of the etree */
    parent = Parent [p] ;
    if (parent != EMPTY)
    {
  ColCount [parent]-- ;
    }
    /* flag node p to exclude self edges (p,p) */
    PrevNbr [p] = k ;
    return (p) ;
}


/* ========================================================================== */
/* === process_edge ========================================================= */
/* ========================================================================== */

/* edge (p,u) is being processed.  p < u is a descendant of its ancestor u in
 * the etree.  node p is the kth node in the postordered etree.  */

static void amesos_process_edge
(
    Int p,    /* process edge (p,u) of the matrix */
    Int u,
    Int k,    /* we are at the kth node in the postordered etree */
    Int First [ ],  /* First [i] = k if the postordering of first
       * descendent of node i is k */
    Int PrevNbr [ ],  /* u was last considered at step k = PrevNbr [u] */
    Int ColCount [ ], /* ColCount [c] is the current weight of node c */
    Int PrevLeaf [ ], /* s = PrevLeaf [u] means that s was the last leaf
       * seen in the subtree rooted at u.  */
    Int RowCount [ ], /* RowCount [i] is # of nonzeros in row i of L,
       * including the diagonal.  Not computed if NULL. */
    Int SetParent [ ],  /* the FIND/UNION data structure, which forms a set
       * of trees.  A root i has i = SetParent [i].  Following
       * a path from i to the root q of the subtree containing
       * i means that q is the SetParent representative of i.
       * All nodes in the tree could have their SetParent
       * equal to the root q; the tree representation is used
       * to save time.  When a path is traced from i to its
       * root q, the path is re-traversed to set the SetParent
       * of the whole path to be the root q. */
    Int Level [ ]  /* Level [i] = length of path from node i to root */
)
{
    Int prevleaf, q, s, sparent ;
    if (First [p] > PrevNbr [u])
    {
  /* p is a leaf of the subtree of u */
  ColCount [p]++ ;
  prevleaf = PrevLeaf [u] ;
  if (prevleaf == EMPTY)
  {
      /* p is the first leaf of subtree of u; RowCount will be incremented
       * by the length of the path in the etree from p up to u. */
      q = u ;
  }
  else
  {
      /* q = FIND (prevleaf): find the root q of the
       * SetParent tree containing prevleaf */
      for (q = prevleaf ; q != SetParent [q] ; q = SetParent [q])
      {
    ;
      }
      /* the root q has been found; re-traverse the path and
       * perform path compression */
      s = prevleaf ;
      for (s = prevleaf ; s != q ; s = sparent)
      {
    sparent = SetParent [s] ;
    SetParent [s] = q ;
      }
      /* adjust the RowCount and ColCount; RowCount will be incremented by
       * the length of the path from p to the SetParent root q, and
       * decrement the ColCount of q by one. */
      ColCount [q]-- ;
  }
  if (RowCount != NULL)
  {
      /* if RowCount is being computed, increment it by the length of
       * the path from p to q */
      RowCount [u] += (Level [p] - Level [q]) ;
  }
  /* p is a leaf of the subtree of u, so mark PrevLeaf [u] to be p */
  PrevLeaf [u] = p ;
    }
    /* flag u has having been processed at step k */
    PrevNbr [u] = k ;
}


/* ========================================================================== */
/* === finalize_node ======================================================== */
/* ========================================================================== */

static void amesos_finalize_node    /* compute UNION (p, Parent [p]) */
(
    Int p,
    Int Parent [ ], /* Parent [p] is the parent of p in the etree */
    Int SetParent [ ] /* see process_edge, above */
)
{
    /* all nodes in the SetParent tree rooted at p now have as their final
     * root the node Parent [p].  This computes UNION (p, Parent [p]) */
    if (Parent [p] != EMPTY)
    {
  SetParent [p] = Parent [p] ;
    }
}


/* ========================================================================== */
/* === cholmod_rowcolcounts ================================================= */
/* ========================================================================== */

int CHOLMOD(rowcolcounts)
(
    /* ---- input ---- */
    cholmod_sparse *A,  /* matrix to analyze */
    Int *fset,    /* subset of 0:(A->ncol)-1 */
    size_t fsize, /* size of fset */
    Int *Parent,  /* size nrow.  Parent [i] = p if p is the parent of i */
    Int *Post,    /* size nrow.  Post [k] = i if i is the kth node in
       * the postordered etree. */
    /* ---- output --- */
    Int *RowCount,  /* size nrow. RowCount [i] = # entries in the ith row of
       * L, including the diagonal. */
    Int *ColCount,  /* size nrow. ColCount [i] = # entries in the ith
       * column of L, including the diagonal. */
    Int *First,   /* size nrow.  First [i] = k is the least postordering
       * of any descendant of i. */
    Int *Level,   /* size nrow.  Level [i] is the length of the path from
       * i to the root, with Level [root] = 0. */
    /* --------------- */
    cholmod_common *Common
)
{
    double fl, ff ;
    Int *Ap, *Ai, *Anz, *PrevNbr, *SetParent, *Head, *PrevLeaf, *Anext, *Ipost,
  *Iwork ;
    Int i, j, r, k, len, s, p, pend, inew, stype, nf, anz, inode, parent,
  nrow, ncol, packed, use_fset, jj ;
    size_t w ;
    int ok = TRUE ;

    /* ---------------------------------------------------------------------- */
    /* check inputs */
    /* ---------------------------------------------------------------------- */

    RETURN_IF_NULL_COMMON (FALSE) ;
    RETURN_IF_NULL (A, FALSE) ;
    RETURN_IF_NULL (Parent, FALSE) ;
    RETURN_IF_NULL (Post, FALSE) ;
    RETURN_IF_NULL (ColCount, FALSE) ;
    RETURN_IF_NULL (First, FALSE) ;
    RETURN_IF_NULL (Level, FALSE) ;
    RETURN_IF_XTYPE_INVALID (A, CHOLMOD_PATTERN, CHOLMOD_ZOMPLEX, FALSE) ;
    stype = A->stype ;
    if (stype > 0)
    {
  /* symmetric with upper triangular part not supported */
  ERROR (CHOLMOD_INVALID, "symmetric upper not supported") ;
  return (FALSE) ;
    }
    Common->status = CHOLMOD_OK ;

    /* ---------------------------------------------------------------------- */
    /* allocate workspace */
    /* ---------------------------------------------------------------------- */

    nrow = A->nrow ;  /* the number of rows of A */
    ncol = A->ncol ;  /* the number of columns of A */

    /* w = 2*nrow + (stype ? 0 : ncol) */
    w = CHOLMOD(mult_size_t) (nrow, 2, &ok) ;
    w = CHOLMOD(add_size_t) (w, (stype ? 0 : ncol), &ok) ;
    if (!ok)
    {
  ERROR (CHOLMOD_TOO_LARGE, "problem too large") ;
  return (FALSE) ;
    }

    CHOLMOD(allocate_work) (nrow, w, 0, Common) ;
    if (Common->status < CHOLMOD_OK)
    {
  return (FALSE) ;
    }

    ASSERT (CHOLMOD(dump_perm) (Post, nrow, nrow, "Post", Common)) ;
    ASSERT (CHOLMOD(dump_parent) (Parent, nrow, "Parent", Common)) ;

    /* ---------------------------------------------------------------------- */
    /* get inputs */
    /* ---------------------------------------------------------------------- */

    Ap = A->p ; /* size ncol+1, column pointers for A */
    Ai = A->i ; /* the row indices of A, of size nz=Ap[ncol+1] */
    Anz = A->nz ;
    packed = A->packed ;
    ASSERT (IMPLIES (!packed, Anz != NULL)) ;

    /* ---------------------------------------------------------------------- */
    /* get workspace */
    /* ---------------------------------------------------------------------- */

    Iwork = Common->Iwork ;
    SetParent = Iwork ;       /* size nrow (i/i/l) */
    PrevNbr   = Iwork + nrow ;      /* size nrow (i/i/l) */
    Anext     = Iwork + 2*((size_t) nrow) ;    /* size ncol (i/i/l) (unsym only) */
    PrevLeaf  = Common->Flag ;      /* size nrow */
    Head      = Common->Head ;      /* size nrow+1 (unsym only)*/

    /* ---------------------------------------------------------------------- */
    /* find the first descendant and level of each node in the tree */
    /* ---------------------------------------------------------------------- */

    /* First [i] = k if the postordering of first descendent of node i is k */
    /* Level [i] = length of path from node i to the root (Level [root] = 0) */

    for (i = 0 ; i < nrow ; i++)
    {
  First [i] = EMPTY ;
    }

    /* postorder traversal of the etree */
    for (k = 0 ; k < nrow ; k++)
    {
  /* node i of the etree is the kth node in the postordered etree */
  i = Post [k] ;

  /* i is a leaf if First [i] is still EMPTY */
  /* ColCount [i] starts at 1 if i is a leaf, zero otherwise */
  ColCount [i] = (First [i] == EMPTY) ? 1 : 0 ;

  /* traverse the path from node i to the root, stopping if we find a
   * node r whose First [r] is already defined. */
  len = 0 ;
  for (r = i ; (r != EMPTY) && (First [r] == EMPTY) ; r = Parent [r])
  {
      First [r] = k ;
      len++ ;
  }
  if (r == EMPTY)
  {
      /* we hit a root node, the level of which is zero */
      len-- ;
  }
  else
  {
      /* we stopped at node r, where Level [r] is already defined */
      len += Level [r] ;
  }
  /* re-traverse the path from node i to r; set the level of each node */
  for (s = i ; s != r ; s = Parent [s])
  {
      Level [s] = len-- ;
  }
    }

    /* ---------------------------------------------------------------------- */
    /* AA' case: sort columns of A according to first postordered row index */
    /* ---------------------------------------------------------------------- */

    fl = 0.0 ;
    if (stype == 0)
    {
  /* [ use PrevNbr [0..nrow-1] as workspace for Ipost */
  Ipost = PrevNbr ;
  /* Ipost [i] = k if i is the kth node in the postordered etree. */
  for (k = 0 ; k < nrow ; k++)
  {
      Ipost [Post [k]] = k ;
  }
  use_fset = (fset != NULL) ;
  if (use_fset)
  {
      nf = fsize ;
      /* clear Anext to check fset */
      for (j = 0 ; j < ncol ; j++)
      {
    Anext [j] = -2 ;
      }
      /* find the first postordered row in each column of A (post,f)
       * and place the column in the corresponding link list */
      for (jj = 0 ; jj < nf ; jj++)
      {
    j = fset [jj] ;
    if (j < 0 || j > ncol || Anext [j] != -2)
    {
        /* out-of-range or duplicate entry in fset */
        ERROR (CHOLMOD_INVALID, "fset invalid") ;
        return (FALSE) ;
    }
    /* flag column j as having been seen */
    Anext [j] = EMPTY ;
      }
      /* fset is now valid */
      ASSERT (CHOLMOD(dump_perm) (fset, nf, ncol, "fset", Common)) ;
  }
  else
  {
      nf = ncol ;
  }
  for (jj = 0 ; jj < nf ; jj++)
  {
      j = (use_fset) ? (fset [jj]) : jj ;
      /* column j is in the fset; find the smallest row (if any) */
      p = Ap [j] ;
      pend = (packed) ? (Ap [j+1]) : (p + Anz [j]) ;
      ff = (double) MAX (0, pend - p) ;
      fl += ff*ff + ff ;
      if (pend > p)
      {
    k = Ipost [Ai [p]] ;
    for ( ; p < pend ; p++)
    {
        inew = Ipost [Ai [p]] ;
        k = MIN (k, inew) ;
    }
    /* place column j in link list k */
    ASSERT (k >= 0 && k < nrow) ;
    Anext [j] = Head [k] ;
    Head [k] = j ;
      }
  }
  /* Ipost no longer needed for inverse postordering ]
   * Head [k] contains a link list of all columns whose first
   * postordered row index is equal to k, for k = 0 to nrow-1. */
    }

    /* ---------------------------------------------------------------------- */
    /* compute the row counts and node weights */
    /* ---------------------------------------------------------------------- */

    if (RowCount != NULL)
    {
  for (i = 0 ; i < nrow ; i++)
  {
      RowCount [i] = 1 ;
  }
    }
    for (i = 0 ; i < nrow ; i++)
    {
  PrevLeaf [i] = EMPTY ;
  PrevNbr [i] = EMPTY ;
  SetParent [i] = i ; /* every node is in its own set, by itself */
    }

    if (stype != 0)
    {

  /* ------------------------------------------------------------------ */
  /* symmetric case: LL' = A */
  /* ------------------------------------------------------------------ */

  /* also determine the number of entries in triu(A) */
  anz = nrow ;
  for (k = 0 ; k < nrow ; k++)
  {
      /* j is the kth node in the postordered etree */
      j = amesos_initialize_node (k, Post, Parent, ColCount, PrevNbr) ;

      /* for all nonzeros A(i,j) below the diagonal, in column j of A */
      p = Ap [j] ;
      pend = (packed) ? (Ap [j+1]) : (p + Anz [j]) ;
      for ( ; p < pend ; p++)
      {
    i = Ai [p] ;
    if (i > j)
    {
        /* j is a descendant of i in etree(A) */
        anz++ ;
        amesos_process_edge (j, i, k, First, PrevNbr, ColCount,
          PrevLeaf, RowCount, SetParent, Level) ;
    }
      }
      /* update SetParent: UNION (j, Parent [j]) */
      amesos_finalize_node (j, Parent, SetParent) ;
  }
  Common->anz = anz ;
    }
    else
    {

  /* ------------------------------------------------------------------ */
  /* unsymmetric case: LL' = AA' */
  /* ------------------------------------------------------------------ */

  for (k = 0 ; k < nrow ; k++)
  {
      /* inode is the kth node in the postordered etree */
      inode = amesos_initialize_node (k, Post, Parent, ColCount, PrevNbr) ;

      /* for all cols j whose first postordered row is k: */
      for (j = Head [k] ; j != EMPTY ; j = Anext [j])
      {
    /* k is the first postordered row in column j of A */
    /* for all rows i in column j: */
    p = Ap [j] ;
    pend = (packed) ? (Ap [j+1]) : (p + Anz [j]) ;
    for ( ; p < pend ; p++)
    {
        i = Ai [p] ;
        /* has i already been considered at this step k */
        if (PrevNbr [i] < k)
        {
      /* inode is a descendant of i in etree(AA') */
      /* process edge (inode,i) and set PrevNbr[i] to k */
      amesos_process_edge (inode, i, k, First, PrevNbr, ColCount,
        PrevLeaf, RowCount, SetParent, Level) ;
        }
    }
      }
      /* clear link list k */
      Head [k] = EMPTY ;
      /* update SetParent: UNION (inode, Parent [inode]) */
      amesos_finalize_node (inode, Parent, SetParent) ;
  }
    }

    /* ---------------------------------------------------------------------- */
    /* finish computing the column counts */
    /* ---------------------------------------------------------------------- */

    for (j = 0 ; j < nrow ; j++)
    {
  parent = Parent [j] ;
  if (parent != EMPTY)
  {
      /* add the ColCount of j to its parent */
      ColCount [parent] += ColCount [j] ;
  }
    }

    /* ---------------------------------------------------------------------- */
    /* clear workspace */
    /* ---------------------------------------------------------------------- */

    Common->mark = EMPTY ;
    CHOLMOD(clear_flag) (Common) ;
    ASSERT (CHOLMOD(dump_work) (TRUE, TRUE, 0, Common)) ;

    /* ---------------------------------------------------------------------- */
    /* flop count and nnz(L) for subsequent LL' numerical factorization */
    /* ---------------------------------------------------------------------- */

    /* use double to avoid integer overflow.  lnz cannot be NaN. */
    Common->aatfl = fl ;
    Common->lnz = 0. ;
    fl = 0 ;
    for (j = 0 ; j < nrow ; j++)
    {
  ff = (double) (ColCount [j]) ;
  Common->lnz += ff ;
  fl += ff*ff ;
    }

    Common->fl = fl ;
    PRINT1 (("rowcol fl %g lnz %g\n", Common->fl, Common->lnz)) ;

    return (TRUE) ;
}
#endif