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template<typename Scalar , typename Index >
void internal::minimum_degree_ordering ( SparseMatrix< Scalar, ColMajor, Index > &  C,
PermutationMatrix< Dynamic > &  perm 
)

Approximate minimum degree ordering algorithm.

Returns:
the permutation P reducing the fill-in of the input matrix C The input matrix C must be a selfadjoint compressed column major SparseMatrix object. Both the upper and lower parts have to be stored, but the diagonal entries are optional. On exit the values of C are destroyed

Definition at line 106 of file Amd.h.

References SparseMatrix< _Scalar, _Options, _Index >::cols(), PlainObjectBase< Derived >::conservativeResize(), PlainObjectBase< Derived >::data(), PermutationMatrix< SizeAtCompileTime, MaxSizeAtCompileTime, IndexType >::indices(), SparseMatrix< _Scalar, _Options, _Index >::nonZeros(), PermutationBase< Derived >::resize(), SparseMatrix< _Scalar, _Options, _Index >::resizeNonZeros(), and sqrt().

Referenced by SimplicialCholesky< _MatrixType, _UpLo >::analyzePattern().

{
  typedef SparseMatrix<Scalar,ColMajor,Index> CCS;
  
  int 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;
  unsigned int h;
  
  Index n = C.cols();
  dense = std::max<Index> (16, 10 * sqrt ((double) n));   /* find dense threshold */
  dense = std::min<Index> (n-2, dense);
  
  Index cnz = C.nonZeros();
  perm.resize(n+1);
  t = cnz + cnz/5 + 2*n;                 /* add elbow room to C */
  C.resizeNonZeros(t);
  
  Index* W       = new Index[8*(n+1)]; /* get workspace */
  Index* len     = W;
  Index* nv      = W +   (n+1);
  Index* next    = W + 2*(n+1);
  Index* head    = W + 3*(n+1);
  Index* elen    = W + 4*(n+1);
  Index* degree  = W + 5*(n+1);
  Index* w       = W + 6*(n+1);
  Index* hhead   = W + 7*(n+1);
  Index* last    = perm.indices().data();                              /* use P as workspace for last */
  
  /* --- Initialize quotient graph ---------------------------------------- */
  Index* Cp = C._outerIndexPtr();
  Index* 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 = cs_wclear (0, 0, w, n);         /* clear w */
  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 */
  
  /* --- Initialize degree lists ------------------------------------------ */
  for(i = 0; i < n; i++)
  {
    d = degree[i];
    if(d == 0)                         /* 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)                 /* node i is dense */
    {
      nv[i] = 0;                    /* absorb i into element n */
      elen[i] = -1;                 /* node i is dead */
      nel++;
      Cp[i] = CS_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;
    }
  }
  
  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] = CS_FLIP (j);    /* first entry is now CS_FLIP(j) */
        }
      }
      for(q = 0, p = 0; p < cnz; ) /* scan all of memory */
      {
        if((j = CS_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] = CS_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 = cs_wclear (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] = CS_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] = CS_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<Index> (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<Index>(lemax, dk);
    mark = cs_wclear (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] = CS_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<Index> (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<Index> (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] = CS_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 = cs_tdfs (i, k, head, next, perm.indices().data(), w);
  }
  
  perm.indices().conservativeResize(n);

  delete[] W;
}

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