symrcm.cc

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/*
An implementation of the Reverse Cuthill-McKee algorithm (symrcm)
 
The implementation of this algorithm is based in the descriptions found in
 
@INPROCEEDINGS{,
        author = {E. Cuthill and J. McKee},
        title = {Reducing the Bandwidth of Sparse Symmetric Matrices},
        booktitle = {Proceedings of the 24th ACM National Conference},
        publisher = {Brandon Press},
        pages = {157 -- 172},
        location = {New Jersey},
        year = {1969}
}
 
@BOOK{,
        author = {Alan George and Joseph W. H. Liu},
        title = {Computer Solution of Large Sparse Positive Definite Systems},
        publisher = {Prentice Hall Series in Computational Mathematics},
        ISBN = {0-13-165274-5},
        year = {1981}
}
 
The algorithm represents a heuristic approach to the NP-complete minimum
bandwidth problem.
 
Written by Michael Weitzel <michael.weitzel@@uni-siegen.de>
                           <weitzel@@ldknet.org>
*/
 
#if defined (HAVE_CONFIG_H)
#  include "config.h"
#endif
 
#include <algorithm>
 
#include "CSparse.h"
#include "boolNDArray.h"
#include "dNDArray.h"
#include "dSparse.h"
#include "oct-locbuf.h"
#include "oct-sparse.h"
#include "quit.h"
 
#include "defun.h"
#include "errwarn.h"
#include "ov.h"
#include "ovl.h"
 
OCTAVE_BEGIN_NAMESPACE(octave)
 
// A node struct for the Cuthill-McKee algorithm
struct CMK_Node
{
  // the node's id (matrix row index)
  octave_idx_type id;
  // the node's degree
  octave_idx_type deg;
  // minimal distance to the root of the spanning tree
  octave_idx_type dist;
};
 
// A simple queue.
// Queues Q have a fixed maximum size N (rows,cols of the matrix) and are
// stored in an array.  qh and qt point to queue head and tail.
 
// Enqueue operation (adds a node "o" at the tail)
 
inline static void
Q_enq (CMK_Node *Q, octave_idx_type N, octave_idx_type& qt, const CMK_Node& o)
{
  Q[qt] = o;
  qt = (qt + 1) % (N + 1);
}
 
// Dequeue operation (removes a node from the head)
 
inline static CMK_Node
Q_deq (CMK_Node *Q, octave_idx_type N, octave_idx_type& qh)
{
  CMK_Node r = Q[qh];
  qh = (qh + 1) % (N + 1);
  return r;
}
 
// Predicate (queue empty)
#define Q_empty(Q, N, qh, qt)   ((qh) == (qt))
 
// A simple, array-based binary heap (used as a priority queue for nodes)
 
// the left descendant of entry i
#define LEFT(i)         (((i) << 1) + 1)        // = (2*(i)+1)
                                                // the right descendant of entry i
#define RIGHT(i)        (((i) << 1) + 2)        // = (2*(i)+2)
                                                // the parent of entry i
#define PARENT(i)       (((i) - 1) >> 1)        // = floor(((i)-1)/2)
 
// Builds a min-heap (the root contains the smallest element).  A is an array
// with the graph's nodes, i is a starting position, size is the length of A.
 
static void
H_heapify_min (CMK_Node *A, octave_idx_type i, octave_idx_type size)
{
  octave_idx_type j = i;
  for (;;)
    {
      octave_idx_type l = LEFT(j);
      octave_idx_type r = RIGHT(j);
 
      octave_idx_type smallest;
      if (l < size && A[l].deg < A[j].deg)
        smallest = l;
      else
        smallest = j;
 
      if (r < size && A[r].deg < A[smallest].deg)
        smallest = r;
 
      if (smallest != j)
        {
          std::swap (A[j], A[smallest]);
          j = smallest;
        }
      else
        break;
    }
}
 
// Heap operation insert.  Running time is O(log(n))
 
static void
H_insert (CMK_Node *H, octave_idx_type& h, const CMK_Node& o)
{
  octave_idx_type i = h++;
 
  H[i] = o;
 
  if (i == 0)
    return;
  do
    {
      octave_idx_type p = PARENT(i);
      if (H[i].deg < H[p].deg)
        {
          std::swap (H[i], H[p]);
 
          i = p;
        }
      else
        break;
    }
  while (i > 0);
}
 
// Heap operation remove-min.  Removes the smallest element in O(1) and
// reorganizes the heap optionally in O(log(n))
 
inline static CMK_Node
H_remove_min (CMK_Node *H, octave_idx_type& h, int reorg/*=1*/)
{
  CMK_Node r = H[0];
  H[0] = H[--h];
  if (reorg)
    H_heapify_min (H, 0, h);
  return r;
}
 
// Predicate (heap empty)
#define H_empty(H, h)   ((h) == 0)
 
// Helper function for the Cuthill-McKee algorithm.  Tries to determine a
// pseudo-peripheral node of the graph as starting node.
 
static octave_idx_type
find_starting_node (octave_idx_type N, const octave_idx_type *ridx,
                    const octave_idx_type *cidx, const octave_idx_type *ridx2,
                    const octave_idx_type *cidx2, octave_idx_type *D,
                    octave_idx_type start)
{
  CMK_Node w;
 
  OCTAVE_LOCAL_BUFFER (CMK_Node, Q, N+1);
  boolNDArray btmp (dim_vector (1, N), false);
  bool *visit = btmp.rwdata ();
 
  octave_idx_type qh = 0;
  octave_idx_type qt = 0;
  CMK_Node x;
  x.id = start;
  x.deg = D[start];
  x.dist = 0;
  Q_enq (Q, N, qt, x);
  visit[start] = true;
 
  // distance level
  octave_idx_type level = 0;
  // current largest "eccentricity"
  octave_idx_type max_dist = 0;
 
  for (;;)
    {
      while (! Q_empty (Q, N, qh, qt))
        {
          CMK_Node v = Q_deq (Q, N, qh);
 
          if (v.dist > x.dist || (v.id != x.id && v.deg > x.deg))
            x = v;
 
          octave_idx_type i = v.id;
 
          // add all unvisited neighbors to the queue
          octave_idx_type j1 = cidx[i];
          octave_idx_type j2 = cidx2[i];
          while (j1 < cidx[i+1] || j2 < cidx2[i+1])
            {
              if (j1 == cidx[i+1])
                {
                  octave_idx_type r2 = ridx2[j2++];
                  if (! visit[r2])
                    {
                      // the distance of node j is dist(i)+1
                      w.id = r2;
                      w.deg = D[r2];
                      w.dist = v.dist+1;
                      Q_enq (Q, N, qt, w);
                      visit[r2] = true;
 
                      if (w.dist > level)
                        level = w.dist;
                    }
                }
              else if (j2 == cidx2[i+1])
                {
                  octave_idx_type r1 = ridx[j1++];
                  if (! visit[r1])
                    {
                      // the distance of node j is dist(i)+1
                      w.id = r1;
                      w.deg = D[r1];
                      w.dist = v.dist+1;
                      Q_enq (Q, N, qt, w);
                      visit[r1] = true;
 
                      if (w.dist > level)
                        level = w.dist;
                    }
                }
              else
                {
                  octave_idx_type r1 = ridx[j1];
                  octave_idx_type r2 = ridx2[j2];
                  if (r1 <= r2)
                    {
                      if (! visit[r1])
                        {
                          w.id = r1;
                          w.deg = D[r1];
                          w.dist = v.dist+1;
                          Q_enq (Q, N, qt, w);
                          visit[r1] = true;
 
                          if (w.dist > level)
                            level = w.dist;
                        }
                      j1++;
                      if (r1 == r2)
                        j2++;
                    }
                  else
                    {
                      if (! visit[r2])
                        {
                          w.id = r2;
                          w.deg = D[r2];
                          w.dist = v.dist+1;
                          Q_enq (Q, N, qt, w);
                          visit[r2] = true;
 
                          if (w.dist > level)
                            level = w.dist;
                        }
                      j2++;
                    }
                }
            }
        } // finish of BFS
 
      if (max_dist < x.dist)
        {
          max_dist = x.dist;
 
          for (octave_idx_type i = 0; i < N; i++)
            visit[i] = false;
 
          visit[x.id] = true;
          x.dist = 0;
          qt = qh = 0;
          Q_enq (Q, N, qt, x);
        }
      else
        break;
    }
  return x.id;
}
 
// Calculates the node's degrees.  This means counting the nonzero elements
// in the symmetric matrix' rows.  This works for non-symmetric matrices
// as well.
 
static octave_idx_type
calc_degrees (octave_idx_type N, octave_idx_type *cidx, octave_idx_type *ridx,
              octave_idx_type *cidx2, octave_idx_type *ridx2, octave_idx_type *D)
{
  octave_idx_type max_deg = 0;
  for (octave_idx_type i = 0; i < N; i++)
    D[i] = 0;
 
  for (octave_idx_type j = 0; j < N; j++)
    for (octave_idx_type i = cidx[j]; i < cidx[j+1]; i++)
      D[ridx[i]]++;
 
  for (octave_idx_type j = 0; j < N; j++)
    for (octave_idx_type i = cidx2[j]; i < cidx2[j+1]; i++)
      D[ridx2[i]]++;
 
  for (octave_idx_type i = 0; i < N; i++)
    if (D[i] > max_deg)
      max_deg = D[i];
 
  return max_deg;
}
 
// Transpose of the structure of a square sparse matrix
 
static void
transpose (octave_idx_type N, const octave_idx_type *ridx,
           const octave_idx_type *cidx, octave_idx_type *ridx2,
           octave_idx_type *cidx2)
{
  octave_idx_type nz = cidx[N];
 
  OCTAVE_LOCAL_BUFFER (octave_idx_type, w, N + 1);
  for (octave_idx_type i = 0; i < N; i++)
    w[i] = 0;
  for (octave_idx_type i = 0; i < nz; i++)
    w[ridx[i]]++;
  nz = 0;
  for (octave_idx_type i = 0; i < N; i++)
    {
      cidx2[i] = nz;
      nz += w[i];
      w[i] = cidx2[i];
    }
  cidx2[N] = nz;
  w[N] = nz;
 
  for (octave_idx_type j = 0; j < N; j++)
    for (octave_idx_type k = cidx[j]; k < cidx[j + 1]; k++)
      {
        octave_idx_type q = w[ridx[k]]++;
        ridx2[q] = j;
      }
}
 
// An implementation of the Cuthill-McKee algorithm.
DEFUN (symrcm, args, ,
       doc: /* -*- texinfo -*-
@deftypefn {} {@var{p} =} symrcm (@var{S})
Return the symmetric reverse @nospell{Cuthill-McKee} permutation of @var{S}.
 
@var{p} is a permutation vector such that
@code{@var{S}(@var{p}, @var{p})} tends to have its diagonal elements closer
to the diagonal than @var{S}.  This is a good preordering for LU or
Cholesky@tie{}factorization of matrices that come from ``long, skinny''
problems.  It works for both symmetric and asymmetric @var{S}.
 
The algorithm represents a heuristic approach to the NP-complete bandwidth
minimization problem.  The implementation is based in the descriptions found
in
 
@nospell{E. Cuthill, J. McKee}.
@cite{Reducing the Bandwidth of Sparse Symmetric Matrices}.
Proceedings of the 24th @nospell{ACM} National Conference,
157--172 1969, Brandon Press, New Jersey.
 
@nospell{A. George, J.W.H. Liu}.  @cite{Computer Solution of Large Sparse
Positive Definite Systems}, Prentice Hall Series in Computational
Mathematics, ISBN 0-13-165274-5, 1981.
 
@seealso{colperm, colamd, symamd}
@end deftypefn */)
{
  if (args.length () != 1)
    print_usage ();
 
  octave_value arg = args(0);
 
  octave_idx_type nr = arg.rows ();
  octave_idx_type nc = arg.columns ();
 
  if (nr != nc)
    err_square_matrix_required ("symrcm", "S");
 
  if (nr == 0 && nc == 0)
    return ovl (NDArray (dim_vector (1, 0)));
 
  // dimension of the matrix
  octave_idx_type N = nr;
 
  // the parameter of the matrix is converted into a sparse matrix
  //(if necessary)
  SparseMatrix Ar;
 
  octave_quit ();
 
  if (arg.isreal ())
    {
      Ar = arg.sparse_matrix_value ();
    }
  else
    {
      SparseComplexMatrix Ac = arg.sparse_complex_matrix_value ();
      Ar = max (max (real (Ac), -real (Ac)), max (imag (Ac), -imag (Ac)));
    }
 
  octave_quit ();
 
  // Note cidx/ridx are const, so use xridx and xcidx...
  octave_idx_type *cidx = Ar.xcidx ();
  octave_idx_type *ridx = Ar.xridx ();
 
  // transpose
  OCTAVE_LOCAL_BUFFER (octave_idx_type, cidx2, N + 1);
  OCTAVE_LOCAL_BUFFER (octave_idx_type, ridx2, cidx[N]);
  transpose (N, ridx, cidx, ridx2, cidx2);
 
  octave_quit ();
 
  // vertex degrees
  OCTAVE_LOCAL_BUFFER (octave_idx_type, D, N);
  octave_idx_type max_deg = calc_degrees (N, cidx, ridx, cidx2, ridx2, D);
 
  octave_quit ();
 
  // the permutation vector
  NDArray P (dim_vector (1, N));
 
  // if none of the nodes has a degree > 0 (a matrix of zeros)
  // the return value corresponds to the identity permutation
  if (max_deg == 0)
    {
      for (octave_idx_type i = 0; i < N; i++)
        P(i) = i+1;  // +1 to convert from base-0 to base-1
 
      return ovl (P);
    }
 
  // At this point, all early returns have completed.
  // Proceed to BFS.
 
  // sizes of the heaps
  octave_idx_type s = 0;
 
  // head- and tail-indices for the queue
  octave_idx_type qt = 0;
  octave_idx_type qh = 0;
  CMK_Node v, w;
 
  // a heap for the a node's neighbors.  The number of neighbors is
  // limited by the maximum degree max_deg:
  OCTAVE_LOCAL_BUFFER (CMK_Node, S, max_deg);
 
  // a queue for the BFS.  The array is always one element larger than
  // the number of entries that are stored.
  OCTAVE_LOCAL_BUFFER (CMK_Node, Q, N+1);
 
  // a counter (for building the permutation)
  octave_idx_type c = -1;
 
  // upper bound for the bandwidth (=quality of solution)
  // initialize the bandwidth of the graph with 0.  B contains the
  // the maximum of the theoretical lower limits of the subgraphs
  // bandwidths.
  octave_idx_type B = 0;
 
  // mark all nodes as unvisited; with the exception of the nodes
  // that have degree==0 and build a CC of the graph.
 
  boolNDArray btmp (dim_vector (1, N), false);
  bool *visit = btmp.rwdata ();
 
  octave_quit ();
 
  do
    {
      // locate an unvisited starting node of the graph
      octave_idx_type i;
      for (i = 0; i < N; i++)
        if (! visit[i])
          break;
 
      // locate a probably better starting node
      v.id = find_starting_node (N, ridx, cidx, ridx2, cidx2, D, i);
 
      // mark the node as visited and enqueue it (a starting node
      // for the BFS).  Since the node will be a root of a spanning
      // tree, its dist is 0.
      v.deg = D[v.id];
      v.dist = 0;
      visit[v.id] = true;
      Q_enq (Q, N, qt, v);
 
      // lower bound for the bandwidth of a subgraph
      // keep a "level" in the spanning tree (= min. distance to the
      // root) for determining the bandwidth of the computed
      // permutation P
      octave_idx_type Bsub = 0;
      // min. dist. to the root is 0
      octave_idx_type level = 0;
      // the root is the first/only node on level 0
      octave_idx_type level_N = 1;
 
      while (! Q_empty (Q, N, qh, qt))
        {
          v = Q_deq (Q, N, qh);
          i = v.id;
 
          c++;
 
          // for computing the inverse permutation P where
          // A(inv(P),inv(P)) or P'*A*P is banded
          //         P(i) = c;
 
          // for computing permutation P where
          // A(P(i),P(j)) or P*A*P' is banded
          P(c) = i;
 
          // put all unvisited neighbors j of node i on the heap
          s = 0;
          octave_idx_type j1 = cidx[i];
          octave_idx_type j2 = cidx2[i];
 
          while (j1 < cidx[i+1] || j2 < cidx2[i+1])
            {
              if (j1 == cidx[i+1])
                {
                  octave_idx_type r2 = ridx2[j2++];
                  if (! visit[r2])
                    {
                      // the distance of node j is dist(i)+1
                      w.id = r2;
                      w.deg = D[r2];
                      w.dist = v.dist+1;
                      H_insert (S, s, w);
                      visit[r2] = true;
                    }
                }
              else if (j2 == cidx2[i+1])
                {
                  octave_idx_type r1 = ridx[j1++];
                  if (! visit[r1])
                    {
                      w.id = r1;
                      w.deg = D[r1];
                      w.dist = v.dist+1;
                      H_insert (S, s, w);
                      visit[r1] = true;
                    }
                }
              else
                {
                  octave_idx_type r1 = ridx[j1];
                  octave_idx_type r2 = ridx2[j2];
                  if (r1 <= r2)
                    {
                      if (! visit[r1])
                        {
                          w.id = r1;
                          w.deg = D[r1];
                          w.dist = v.dist+1;
                          H_insert (S, s, w);
                          visit[r1] = true;
                        }
                      j1++;
                      if (r1 == r2)
                        j2++;
                    }
                  else
                    {
                      if (! visit[r2])
                        {
                          w.id = r2;
                          w.deg = D[r2];
                          w.dist = v.dist+1;
                          H_insert (S, s, w);
                          visit[r2] = true;
                        }
                      j2++;
                    }
                }
            }
 
          // add the neighbors to the queue (sorted by node degree)
          while (! H_empty (S, s))
            {
              // locate a neighbor of i with minimal degree in O(log(N))
              v = H_remove_min (S, s, 1);
 
              // entered the BFS a new level?
              if (v.dist > level)
                {
                  // adjustment of bandwidth:
                  // "[...] the minimum bandwidth that
                  // can be obtained [...] is the
                  //  maximum number of nodes per level"
                  if (Bsub < level_N)
                    Bsub = level_N;
 
                  level = v.dist;
                  // v is the first node on the new level
                  level_N = 1;
                }
              else
                {
                  // there is no new level but another node on
                  // this level:
                  level_N++;
                }
 
              // enqueue v in O(1)
              Q_enq (Q, N, qt, v);
            }
 
          // synchronize the bandwidth with level_N once again:
          if (Bsub < level_N)
            Bsub = level_N;
        }
      // finish of BFS.  If there are still unvisited nodes in the graph
      // then it is split into CCs.  The computed bandwidth is the maximum
      // of all subgraphs.  Update:
      if (Bsub > B)
        B = Bsub;
    }
  // are there any nodes left?
  while (c+1 < N);
 
  // compute the reverse-ordering
  s = N / 2 - 1;
  for (octave_idx_type i = 0, j = N - 1; i <= s; i++, j--)
    std::swap (P.elem (i), P.elem (j));
 
  // increment all indices, since Octave is not C
  return ovl (P+1);
}
 
/*
 
  basic functionality test, with icosahedron:
%!test <*64718>
%! adj = [ 0 1 1 1 1 1 0 0 0 0 0 0;
%!         1 0 1 0 0 1 1 0 0 0 1 0;
%!         1 1 0 1 0 0 1 1 0 0 0 0;
%!         1 0 1 0 1 0 0 1 1 0 0 0;
%!         1 0 0 1 0 1 0 0 1 1 0 0;
%!         1 1 0 0 1 0 0 0 0 1 1 0;
%!         0 1 1 0 0 0 0 1 0 0 1 1;
%!         0 0 1 1 0 0 1 0 1 0 0 1;
%!         0 0 0 1 1 0 0 1 0 1 0 1;
%!         0 0 0 0 1 1 0 0 1 0 1 1;
%!         0 1 0 0 0 1 1 0 0 1 0 1;
%!         0 0 0 0 0 0 1 1 1 1 1 0 ];
%! p = symrcm (adj);
%! assert (p, [12 8 9 10 11 7 3 4 5 6 2 1]);
%! assert (bandwidth (adj), 9);
%! assert (bandwidth (adj(p, p)), 6);
 
  handle zero-matrix properly:
%!test <*64718>
%! adj = false (5);
%! p = symrcm (adj);
%! assert (p, 1:5);
 
*/
 
OCTAVE_END_NAMESPACE(octave)