text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2010 Gael Guennebaud <gael.guennebaud@inria.fr>
  
  /*
  
  NOTE: this routine has been adapted from the CSparse library:
  
  Copyright (c) 2006, Timothy A. Davis.
  http://www.cise.ufl.edu/research/sparse/CSparse
  
  CSparse is free software; you can redistribute it and/or
  modify it under the terms of the GNU Lesser General Public
  License as published by the Free Software Foundation; either
  version 2.1 of the License, or (at your option) any later version.
  
  CSparse 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
  Lesser General Public License for more details.
  
  You should have received a copy of the GNU Lesser General Public
  License along with this Module; if not, write to the Free Software
  Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
  
  */
  
  #include "../Core/util/NonMPL2.h"
  
  #ifndef EIGEN_SPARSE_AMD_H
  #define EIGEN_SPARSE_AMD_H
  
  namespace Eigen { 
  
  namespace internal {
    
  template<typename T> inline T amd_flip(const T& i) { return -i-2; }
  template<typename T> inline T amd_unflip(const T& i) { return i<0 ? amd_flip(i) : i; }
  template<typename T0, typename T1> inline bool amd_marked(const T0* w, const T1& j) { return w[j]<0; }
  template<typename T0, typename T1> inline void amd_mark(const T0* w, const T1& j) { return w[j] = amd_flip(w[j]); }
  
  /* clear w */
  template<typename StorageIndex>
  static StorageIndex cs_wclear (StorageIndex mark, StorageIndex lemax, StorageIndex *w, StorageIndex n)
  {
    StorageIndex k;
    if(mark < 2 || (mark + lemax < 0))
    {
      for(k = 0; k < n; k++)
        if(w[k] != 0)
          w[k] = 1;
      mark = 2;
    }
    return (mark);     /* at this point, w[0..n-1] < mark holds */
  }
  
  /* depth-first search and postorder of a tree rooted at node j */
  template<typename StorageIndex>
  StorageIndex cs_tdfs(StorageIndex j, StorageIndex k, StorageIndex *head, const StorageIndex *next, StorageIndex *post, StorageIndex *stack)
  {
    StorageIndex i, p, top = 0;
    if(!head || !next || !post || !stack) return (-1);    /* check inputs */
    stack[0] = j;                 /* place j on the stack */
    while (top >= 0)                /* while (stack is not empty) */
    {
      p = stack[top];           /* p = top of stack */
      i = head[p];              /* i = youngest child of p */
      if(i == -1)
      {
        top--;                 /* p has no unordered children left */
        post[k++] = p;        /* node p is the kth postordered node */
      }
      else
      {
        head[p] = next[i];   /* remove i from children of p */
        stack[++top] = i;     /* start dfs on child node i */
      }
    }
    return k;
  }
  
  
  /** \internal
    * \ingroup OrderingMethods_Module 
    * Approximate minimum degree ordering algorithm.
    *
    * \param[in] C the input selfadjoint matrix stored in compressed column major format.
    * \param[out] perm the permutation P reducing the fill-in of the input matrix \a C
    *
    * Note that the input matrix \a C must be complete, that is both the upper and lower parts have to be stored, as well as the diagonal entries.
    * On exit the values of C are destroyed */
  template<typename Scalar, typename StorageIndex>
  void minimum_degree_ordering(SparseMatrix<Scalar,ColMajor,StorageIndex>& C, PermutationMatrix<Dynamic,Dynamic,StorageIndex>& perm)
  {
    using std::sqrt;
    
    StorageIndex d, dk, dext, lemax = 0, e, elenk, eln, i, j, k, k1,
                  k2, k3, jlast, ln, dense, nzmax, mindeg = 0, nvi, nvj, nvk, mark, wnvi,
                  ok, nel = 0, p, p1, p2, p3, p4, pj, pk, pk1, pk2, pn, q, t, h;
    
    StorageIndex n = StorageIndex(C.cols());
    dense = std::max<StorageIndex> (16, StorageIndex(10 * sqrt(double(n))));   /* find dense threshold */
    dense = (std::min)(n-2, dense);
    
    StorageIndex cnz = StorageIndex(C.nonZeros());
    perm.resize(n+1);
    t = cnz + cnz/5 + 2*n;                 /* add elbow room to C */
    C.resizeNonZeros(t);
    
    // get workspace
    ei_declare_aligned_stack_constructed_variable(StorageIndex,W,8*(n+1),0);
    StorageIndex* len     = W;
    StorageIndex* nv      = W +   (n+1);
    StorageIndex* next    = W + 2*(n+1);
    StorageIndex* head    = W + 3*(n+1);
    StorageIndex* elen    = W + 4*(n+1);
    StorageIndex* degree  = W + 5*(n+1);
    StorageIndex* w       = W + 6*(n+1);
    StorageIndex* hhead   = W + 7*(n+1);
    StorageIndex* last    = perm.indices().data();                              /* use P as workspace for last */
    
    /* --- Initialize quotient graph ---------------------------------------- */
    StorageIndex* Cp = C.outerIndexPtr();
    StorageIndex* Ci = C.innerIndexPtr();
    for(k = 0; k < n; k++)
      len[k] = Cp[k+1] - Cp[k];
    len[n] = 0;
    nzmax = t;
    
    for(i = 0; i <= n; i++)
    {
      head[i]   = -1;                     // degree list i is empty
      last[i]   = -1;
      next[i]   = -1;
      hhead[i]  = -1;                     // hash list i is empty 
      nv[i]     = 1;                      // node i is just one node
      w[i]      = 1;                      // node i is alive
      elen[i]   = 0;                      // Ek of node i is empty
      degree[i] = len[i];                 // degree of node i
    }
    mark = internal::cs_wclear<StorageIndex>(0, 0, w, n);         /* clear w */
    
    /* --- Initialize degree lists ------------------------------------------ */
    for(i = 0; i < n; i++)
    {
      bool has_diag = false;
      for(p = Cp[i]; p<Cp[i+1]; ++p)
        if(Ci[p]==i)
        {
          has_diag = true;
          break;
        }
     
      d = degree[i];
      if(d == 1 && has_diag)           /* node i is empty */
      {
        elen[i] = -2;                 /* element i is dead */
        nel++;
        Cp[i] = -1;                   /* i is a root of assembly tree */
        w[i] = 0;
      }
      else if(d > dense || !has_diag)  /* node i is dense or has no structural diagonal element */
      {
        nv[i] = 0;                    /* absorb i into element n */
        elen[i] = -1;                 /* node i is dead */
        nel++;
        Cp[i] = amd_flip (n);
        nv[n]++;
      }
      else
      {
        if(head[d] != -1) last[head[d]] = i;
        next[i] = head[d];           /* put node i in degree list d */
        head[d] = i;
      }
    }
    
    elen[n] = -2;                         /* n is a dead element */
    Cp[n] = -1;                           /* n is a root of assembly tree */
    w[n] = 0;                             /* n is a dead element */
    
    while (nel < n)                         /* while (selecting pivots) do */
    {
      /* --- Select node of minimum approximate degree -------------------- */
      for(k = -1; mindeg < n && (k = head[mindeg]) == -1; mindeg++) {}
      if(next[k] != -1) last[next[k]] = -1;
      head[mindeg] = next[k];          /* remove k from degree list */
      elenk = elen[k];                  /* elenk = |Ek| */
      nvk = nv[k];                      /* # of nodes k represents */
      nel += nvk;                        /* nv[k] nodes of A eliminated */
      
      /* --- Garbage collection ------------------------------------------- */
      if(elenk > 0 && cnz + mindeg >= nzmax)
      {
        for(j = 0; j < n; j++)
        {
          if((p = Cp[j]) >= 0)      /* j is a live node or element */
          {
            Cp[j] = Ci[p];          /* save first entry of object */
            Ci[p] = amd_flip (j);    /* first entry is now amd_flip(j) */
          }
        }
        for(q = 0, p = 0; p < cnz; ) /* scan all of memory */
        {
          if((j = amd_flip (Ci[p++])) >= 0)  /* found object j */
          {
            Ci[q] = Cp[j];       /* restore first entry of object */
            Cp[j] = q++;          /* new pointer to object j */
            for(k3 = 0; k3 < len[j]-1; k3++) Ci[q++] = Ci[p++];
          }
        }
        cnz = q;                       /* Ci[cnz...nzmax-1] now free */
      }
      
      /* --- Construct new element ---------------------------------------- */
      dk = 0;
      nv[k] = -nvk;                     /* flag k as in Lk */
      p = Cp[k];
      pk1 = (elenk == 0) ? p : cnz;      /* do in place if elen[k] == 0 */
      pk2 = pk1;
      for(k1 = 1; k1 <= elenk + 1; k1++)
      {
        if(k1 > elenk)
        {
          e = k;                     /* search the nodes in k */
          pj = p;                    /* list of nodes starts at Ci[pj]*/
          ln = len[k] - elenk;      /* length of list of nodes in k */
        }
        else
        {
          e = Ci[p++];              /* search the nodes in e */
          pj = Cp[e];
          ln = len[e];              /* length of list of nodes in e */
        }
        for(k2 = 1; k2 <= ln; k2++)
        {
          i = Ci[pj++];
          if((nvi = nv[i]) <= 0) continue; /* node i dead, or seen */
          dk += nvi;                 /* degree[Lk] += size of node i */
          nv[i] = -nvi;             /* negate nv[i] to denote i in Lk*/
          Ci[pk2++] = i;            /* place i in Lk */
          if(next[i] != -1) last[next[i]] = last[i];
          if(last[i] != -1)         /* remove i from degree list */
          {
            next[last[i]] = next[i];
          }
          else
          {
            head[degree[i]] = next[i];
          }
        }
        if(e != k)
        {
          Cp[e] = amd_flip (k);      /* absorb e into k */
          w[e] = 0;                 /* e is now a dead element */
        }
      }
      if(elenk != 0) cnz = pk2;         /* Ci[cnz...nzmax] is free */
      degree[k] = dk;                   /* external degree of k - |Lk\i| */
      Cp[k] = pk1;                      /* element k is in Ci[pk1..pk2-1] */
      len[k] = pk2 - pk1;
      elen[k] = -2;                     /* k is now an element */
      
      /* --- Find set differences ----------------------------------------- */
      mark = internal::cs_wclear<StorageIndex>(mark, lemax, w, n);  /* clear w if necessary */
      for(pk = pk1; pk < pk2; pk++)    /* scan 1: find |Le\Lk| */
      {
        i = Ci[pk];
        if((eln = elen[i]) <= 0) continue;/* skip if elen[i] empty */
        nvi = -nv[i];                      /* nv[i] was negated */
        wnvi = mark - nvi;
        for(p = Cp[i]; p <= Cp[i] + eln - 1; p++)  /* scan Ei */
        {
          e = Ci[p];
          if(w[e] >= mark)
          {
            w[e] -= nvi;          /* decrement |Le\Lk| */
          }
          else if(w[e] != 0)        /* ensure e is a live element */
          {
            w[e] = degree[e] + wnvi; /* 1st time e seen in scan 1 */
          }
        }
      }
      
      /* --- Degree update ------------------------------------------------ */
      for(pk = pk1; pk < pk2; pk++)    /* scan2: degree update */
      {
        i = Ci[pk];                   /* consider node i in Lk */
        p1 = Cp[i];
        p2 = p1 + elen[i] - 1;
        pn = p1;
        for(h = 0, d = 0, p = p1; p <= p2; p++)    /* scan Ei */
        {
          e = Ci[p];
          if(w[e] != 0)             /* e is an unabsorbed element */
          {
            dext = w[e] - mark;   /* dext = |Le\Lk| */
            if(dext > 0)
            {
              d += dext;         /* sum up the set differences */
              Ci[pn++] = e;     /* keep e in Ei */
              h += e;            /* compute the hash of node i */
            }
            else
            {
              Cp[e] = amd_flip (k);  /* aggressive absorb. e->k */
              w[e] = 0;             /* e is a dead element */
            }
          }
        }
        elen[i] = pn - p1 + 1;        /* elen[i] = |Ei| */
        p3 = pn;
        p4 = p1 + len[i];
        for(p = p2 + 1; p < p4; p++) /* prune edges in Ai */
        {
          j = Ci[p];
          if((nvj = nv[j]) <= 0) continue; /* node j dead or in Lk */
          d += nvj;                  /* degree(i) += |j| */
          Ci[pn++] = j;             /* place j in node list of i */
          h += j;                    /* compute hash for node i */
        }
        if(d == 0)                     /* check for mass elimination */
        {
          Cp[i] = amd_flip (k);      /* absorb i into k */
          nvi = -nv[i];
          dk -= nvi;                 /* |Lk| -= |i| */
          nvk += nvi;                /* |k| += nv[i] */
          nel += nvi;
          nv[i] = 0;
          elen[i] = -1;             /* node i is dead */
        }
        else
        {
          degree[i] = std::min<StorageIndex> (degree[i], d);   /* update degree(i) */
          Ci[pn] = Ci[p3];         /* move first node to end */
          Ci[p3] = Ci[p1];         /* move 1st el. to end of Ei */
          Ci[p1] = k;               /* add k as 1st element in of Ei */
          len[i] = pn - p1 + 1;     /* new len of adj. list of node i */
          h %= n;                    /* finalize hash of i */
          next[i] = hhead[h];      /* place i in hash bucket */
          hhead[h] = i;
          last[i] = h;      /* save hash of i in last[i] */
        }
      }                                   /* scan2 is done */
      degree[k] = dk;                   /* finalize |Lk| */
      lemax = std::max<StorageIndex>(lemax, dk);
      mark = internal::cs_wclear<StorageIndex>(mark+lemax, lemax, w, n);    /* clear w */
      
      /* --- Supernode detection ------------------------------------------ */
      for(pk = pk1; pk < pk2; pk++)
      {
        i = Ci[pk];
        if(nv[i] >= 0) continue;         /* skip if i is dead */
        h = last[i];                      /* scan hash bucket of node i */
        i = hhead[h];
        hhead[h] = -1;                    /* hash bucket will be empty */
        for(; i != -1 && next[i] != -1; i = next[i], mark++)
        {
          ln = len[i];
          eln = elen[i];
          for(p = Cp[i]+1; p <= Cp[i] + ln-1; p++) w[Ci[p]] = mark;
          jlast = i;
          for(j = next[i]; j != -1; ) /* compare i with all j */
          {
            ok = (len[j] == ln) && (elen[j] == eln);
            for(p = Cp[j] + 1; ok && p <= Cp[j] + ln - 1; p++)
            {
              if(w[Ci[p]] != mark) ok = 0;    /* compare i and j*/
            }
            if(ok)                     /* i and j are identical */
            {
              Cp[j] = amd_flip (i);  /* absorb j into i */
              nv[i] += nv[j];
              nv[j] = 0;
              elen[j] = -1;         /* node j is dead */
              j = next[j];          /* delete j from hash bucket */
              next[jlast] = j;
            }
            else
            {
              jlast = j;             /* j and i are different */
              j = next[j];
            }
          }
        }
      }
      
      /* --- Finalize new element------------------------------------------ */
      for(p = pk1, pk = pk1; pk < pk2; pk++)   /* finalize Lk */
      {
        i = Ci[pk];
        if((nvi = -nv[i]) <= 0) continue;/* skip if i is dead */
        nv[i] = nvi;                      /* restore nv[i] */
        d = degree[i] + dk - nvi;         /* compute external degree(i) */
        d = std::min<StorageIndex> (d, n - nel - nvi);
        if(head[d] != -1) last[head[d]] = i;
        next[i] = head[d];               /* put i back in degree list */
        last[i] = -1;
        head[d] = i;
        mindeg = std::min<StorageIndex> (mindeg, d);       /* find new minimum degree */
        degree[i] = d;
        Ci[p++] = i;                      /* place i in Lk */
      }
      nv[k] = nvk;                      /* # nodes absorbed into k */
      if((len[k] = p-pk1) == 0)         /* length of adj list of element k*/
      {
        Cp[k] = -1;                   /* k is a root of the tree */
        w[k] = 0;                     /* k is now a dead element */
      }
      if(elenk != 0) cnz = p;           /* free unused space in Lk */
    }
    
    /* --- Postordering ----------------------------------------------------- */
    for(i = 0; i < n; i++) Cp[i] = amd_flip (Cp[i]);/* fix assembly tree */
    for(j = 0; j <= n; j++) head[j] = -1;
    for(j = n; j >= 0; j--)              /* place unordered nodes in lists */
    {
      if(nv[j] > 0) continue;          /* skip if j is an element */
      next[j] = head[Cp[j]];          /* place j in list of its parent */
      head[Cp[j]] = j;
    }
    for(e = n; e >= 0; e--)              /* place elements in lists */
    {
      if(nv[e] <= 0) continue;         /* skip unless e is an element */
      if(Cp[e] != -1)
      {
        next[e] = head[Cp[e]];      /* place e in list of its parent */
        head[Cp[e]] = e;
      }
    }
    for(k = 0, i = 0; i <= n; i++)       /* postorder the assembly tree */
    {
      if(Cp[i] == -1) k = internal::cs_tdfs<StorageIndex>(i, k, head, next, perm.indices().data(), w);
    }
    
    perm.indices().conservativeResize(n);
  }
  
  } // namespace internal
  
  } // end namespace Eigen
  
  #endif // EIGEN_SPARSE_AMD_H
  