Mercurial > octave
view libinterp/corefcn/symrcm.cc @ 31605:e88a07dec498 stable
maint: Use macros to begin/end C++ namespaces.
* oct-conf-post-public.in.h: Define two macros (OCTAVE_BEGIN_NAMESPACE,
OCTAVE_END_NAMESPACE) that can be used to start/end a namespace.
* mk-opts.pl, build-env.h, build-env.in.cc, __betainc__.cc, __contourc__.cc,
__dsearchn__.cc, __eigs__.cc, __expint__.cc, __ftp__.cc, __gammainc__.cc,
__ichol__.cc, __ilu__.cc, __isprimelarge__.cc, __lin_interpn__.cc,
__magick_read__.cc, __pchip_deriv__.cc, __qp__.cc, amd.cc, auto-shlib.cc,
auto-shlib.h, balance.cc, base-text-renderer.cc, base-text-renderer.h,
besselj.cc, bitfcns.cc, bsxfun.cc, c-file-ptr-stream.cc, c-file-ptr-stream.h,
call-stack.cc, call-stack.h, ccolamd.cc, cellfun.cc, chol.cc, colamd.cc,
colloc.cc, conv2.cc, daspk.cc, dasrt.cc, dassl.cc, data.cc, data.h, debug.cc,
defaults.cc, defaults.h, defun-int.h, defun.cc, det.cc, dirfns.cc, display.cc,
display.h, dlmread.cc, dmperm.cc, dot.cc, dynamic-ld.cc, dynamic-ld.h, eig.cc,
ellipj.cc, environment.cc, environment.h, error.cc, error.h, errwarn.h,
event-manager.cc, event-manager.h, event-queue.cc, event-queue.h, fcn-info.cc,
fcn-info.h, fft.cc, fft2.cc, fftn.cc, file-io.cc, filter.cc, find.cc,
ft-text-renderer.cc, ft-text-renderer.h, gcd.cc, getgrent.cc, getpwent.cc,
getrusage.cc, givens.cc, gl-render.cc, gl-render.h, gl2ps-print.cc,
gl2ps-print.h, graphics-toolkit.cc, graphics-toolkit.h, graphics.cc,
graphics.in.h, gsvd.cc, gtk-manager.cc, gtk-manager.h, hash.cc, help.cc,
help.h, hess.cc, hex2num.cc, hook-fcn.cc, hook-fcn.h, input.cc, input.h,
interpreter-private.cc, interpreter-private.h, interpreter.cc, interpreter.h,
inv.cc, jsondecode.cc, jsonencode.cc, kron.cc, latex-text-renderer.cc,
latex-text-renderer.h, load-path.cc, load-path.h, load-save.cc, load-save.h,
lookup.cc, ls-ascii-helper.cc, ls-ascii-helper.h, ls-oct-text.cc, ls-utils.cc,
ls-utils.h, lsode.cc, lu.cc, mappers.cc, matrix_type.cc, max.cc, mex-private.h,
mex.cc, mgorth.cc, nproc.cc, oct-fstrm.cc, oct-fstrm.h, oct-hdf5-types.cc,
oct-hdf5-types.h, oct-hist.cc, oct-hist.h, oct-iostrm.cc, oct-iostrm.h,
oct-opengl.h, oct-prcstrm.cc, oct-prcstrm.h, oct-procbuf.cc, oct-procbuf.h,
oct-process.cc, oct-process.h, oct-stdstrm.h, oct-stream.cc, oct-stream.h,
oct-strstrm.cc, oct-strstrm.h, oct-tex-lexer.in.ll, oct-tex-parser.yy,
ordqz.cc, ordschur.cc, pager.cc, pager.h, pinv.cc, pow2.cc, pr-flt-fmt.cc,
pr-output.cc, procstream.cc, procstream.h, psi.cc, qr.cc, quad.cc, quadcc.cc,
qz.cc, rand.cc, rcond.cc, regexp.cc, schur.cc, settings.cc, settings.h,
sighandlers.cc, sighandlers.h, sparse-xdiv.cc, sparse-xdiv.h, sparse-xpow.cc,
sparse-xpow.h, sparse.cc, spparms.cc, sqrtm.cc, stack-frame.cc, stack-frame.h,
stream-euler.cc, strfind.cc, strfns.cc, sub2ind.cc, svd.cc, sylvester.cc,
symbfact.cc, syminfo.cc, syminfo.h, symrcm.cc, symrec.cc, symrec.h,
symscope.cc, symscope.h, symtab.cc, symtab.h, syscalls.cc, sysdep.cc, sysdep.h,
text-engine.cc, text-engine.h, text-renderer.cc, text-renderer.h, time.cc,
toplev.cc, tril.cc, tsearch.cc, typecast.cc, url-handle-manager.cc,
url-handle-manager.h, urlwrite.cc, utils.cc, utils.h, variables.cc,
variables.h, xdiv.cc, xdiv.h, xnorm.cc, xnorm.h, xpow.cc, xpow.h,
__delaunayn__.cc, __fltk_uigetfile__.cc, __glpk__.cc, __init_fltk__.cc,
__init_gnuplot__.cc, __ode15__.cc, __voronoi__.cc, audiodevinfo.cc,
audioread.cc, convhulln.cc, fftw.cc, gzip.cc, mk-build-env-features.sh,
mk-builtins.pl, cdef-class.cc, cdef-class.h, cdef-fwd.h, cdef-manager.cc,
cdef-manager.h, cdef-method.cc, cdef-method.h, cdef-object.cc, cdef-object.h,
cdef-package.cc, cdef-package.h, cdef-property.cc, cdef-property.h,
cdef-utils.cc, cdef-utils.h, ov-base.cc, ov-base.h, ov-bool-mat.cc,
ov-builtin.h, ov-cell.cc, ov-class.cc, ov-class.h, ov-classdef.cc,
ov-classdef.h, ov-complex.cc, ov-fcn-handle.cc, ov-fcn-handle.h, ov-fcn.h,
ov-java.cc, ov-java.h, ov-mex-fcn.h, ov-null-mat.cc, ov-oncleanup.cc,
ov-struct.cc, ov-typeinfo.cc, ov-typeinfo.h, ov-usr-fcn.cc, ov-usr-fcn.h,
ov.cc, ov.h, octave.cc, octave.h, mk-ops.sh, op-b-b.cc, op-b-bm.cc,
op-b-sbm.cc, op-bm-b.cc, op-bm-bm.cc, op-bm-sbm.cc, op-cdm-cdm.cc, op-cell.cc,
op-chm.cc, op-class.cc, op-cm-cm.cc, op-cm-cs.cc, op-cm-m.cc, op-cm-s.cc,
op-cm-scm.cc, op-cm-sm.cc, op-cs-cm.cc, op-cs-cs.cc, op-cs-m.cc, op-cs-s.cc,
op-cs-scm.cc, op-cs-sm.cc, op-dm-dm.cc, op-dm-scm.cc, op-dm-sm.cc,
op-dm-template.cc, op-dms-template.cc, op-fcdm-fcdm.cc, op-fcm-fcm.cc,
op-fcm-fcs.cc, op-fcm-fm.cc, op-fcm-fs.cc, op-fcn.cc, op-fcs-fcm.cc,
op-fcs-fcs.cc, op-fcs-fm.cc, op-fcs-fs.cc, op-fdm-fdm.cc, op-fm-fcm.cc,
op-fm-fcs.cc, op-fm-fm.cc, op-fm-fs.cc, op-fs-fcm.cc, op-fs-fcs.cc,
op-fs-fm.cc, op-fs-fs.cc, op-i16-i16.cc, op-i32-i32.cc, op-i64-i64.cc,
op-i8-i8.cc, op-int-concat.cc, op-m-cm.cc, op-m-cs.cc, op-m-m.cc, op-m-s.cc,
op-m-scm.cc, op-m-sm.cc, op-mi.cc, op-pm-pm.cc, op-pm-scm.cc, op-pm-sm.cc,
op-pm-template.cc, op-range.cc, op-s-cm.cc, op-s-cs.cc, op-s-m.cc, op-s-s.cc,
op-s-scm.cc, op-s-sm.cc, op-sbm-b.cc, op-sbm-bm.cc, op-sbm-sbm.cc,
op-scm-cm.cc, op-scm-cs.cc, op-scm-m.cc, op-scm-s.cc, op-scm-scm.cc,
op-scm-sm.cc, op-sm-cm.cc, op-sm-cs.cc, op-sm-m.cc, op-sm-s.cc, op-sm-scm.cc,
op-sm-sm.cc, op-str-m.cc, op-str-s.cc, op-str-str.cc, op-struct.cc,
op-ui16-ui16.cc, op-ui32-ui32.cc, op-ui64-ui64.cc, op-ui8-ui8.cc, ops.h,
anon-fcn-validator.cc, anon-fcn-validator.h, bp-table.cc, bp-table.h,
comment-list.cc, comment-list.h, filepos.h, lex.h, lex.ll, oct-lvalue.cc,
oct-lvalue.h, oct-parse.yy, parse.h, profiler.cc, profiler.h,
pt-anon-scopes.cc, pt-anon-scopes.h, pt-arg-list.cc, pt-arg-list.h,
pt-args-block.cc, pt-args-block.h, pt-array-list.cc, pt-array-list.h,
pt-assign.cc, pt-assign.h, pt-binop.cc, pt-binop.h, pt-bp.cc, pt-bp.h,
pt-cbinop.cc, pt-cbinop.h, pt-cell.cc, pt-cell.h, pt-check.cc, pt-check.h,
pt-classdef.cc, pt-classdef.h, pt-cmd.h, pt-colon.cc, pt-colon.h, pt-const.cc,
pt-const.h, pt-decl.cc, pt-decl.h, pt-eval.cc, pt-eval.h, pt-except.cc,
pt-except.h, pt-exp.cc, pt-exp.h, pt-fcn-handle.cc, pt-fcn-handle.h, pt-id.cc,
pt-id.h, pt-idx.cc, pt-idx.h, pt-jump.h, pt-loop.cc, pt-loop.h, pt-mat.cc,
pt-mat.h, pt-misc.cc, pt-misc.h, pt-pr-code.cc, pt-pr-code.h, pt-select.cc,
pt-select.h, pt-spmd.cc, pt-spmd.h, pt-stmt.cc, pt-stmt.h, pt-tm-const.cc,
pt-tm-const.h, pt-unop.cc, pt-unop.h, pt-vm-eval.cc, pt-walk.cc, pt-walk.h,
pt.cc, pt.h, token.cc, token.h, Range.cc, Range.h, idx-vector.cc, idx-vector.h,
range-fwd.h, CollocWt.cc, CollocWt.h, aepbalance.cc, aepbalance.h, chol.cc,
chol.h, gepbalance.cc, gepbalance.h, gsvd.cc, gsvd.h, hess.cc, hess.h,
lo-mappers.cc, lo-mappers.h, lo-specfun.cc, lo-specfun.h, lu.cc, lu.h,
oct-convn.cc, oct-convn.h, oct-fftw.cc, oct-fftw.h, oct-norm.cc, oct-norm.h,
oct-rand.cc, oct-rand.h, oct-spparms.cc, oct-spparms.h, qr.cc, qr.h, qrp.cc,
qrp.h, randgamma.cc, randgamma.h, randmtzig.cc, randmtzig.h, randpoisson.cc,
randpoisson.h, schur.cc, schur.h, sparse-chol.cc, sparse-chol.h, sparse-lu.cc,
sparse-lu.h, sparse-qr.cc, sparse-qr.h, svd.cc, svd.h, child-list.cc,
child-list.h, dir-ops.cc, dir-ops.h, file-ops.cc, file-ops.h, file-stat.cc,
file-stat.h, lo-sysdep.cc, lo-sysdep.h, lo-sysinfo.cc, lo-sysinfo.h,
mach-info.cc, mach-info.h, oct-env.cc, oct-env.h, oct-group.cc, oct-group.h,
oct-password.cc, oct-password.h, oct-syscalls.cc, oct-syscalls.h, oct-time.cc,
oct-time.h, oct-uname.cc, oct-uname.h, action-container.cc, action-container.h,
base-list.h, cmd-edit.cc, cmd-edit.h, cmd-hist.cc, cmd-hist.h, f77-fcn.h,
file-info.cc, file-info.h, lo-array-errwarn.cc, lo-array-errwarn.h, lo-hash.cc,
lo-hash.h, lo-ieee.h, lo-regexp.cc, lo-regexp.h, lo-utils.cc, lo-utils.h,
oct-base64.cc, oct-base64.h, oct-glob.cc, oct-glob.h, oct-inttypes.h,
oct-mutex.cc, oct-mutex.h, oct-refcount.h, oct-shlib.cc, oct-shlib.h,
oct-sparse.cc, oct-sparse.h, oct-string.h, octave-preserve-stream-state.h,
pathsearch.cc, pathsearch.h, quit.cc, quit.h, unwind-prot.cc, unwind-prot.h,
url-transfer.cc, url-transfer.h : Use new macros to begin/end C++ namespaces.
author | Rik <rik@octave.org> |
---|---|
date | Thu, 01 Dec 2022 14:23:45 -0800 |
parents | 796f54d4ddbf |
children | aac27ad79be6 |
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//////////////////////////////////////////////////////////////////////// // // Copyright (C) 2007-2022 The Octave Project Developers // // See the file COPYRIGHT.md in the top-level directory of this // distribution or <https://octave.org/copyright/>. // // This file is part of Octave. // // Octave 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. // // Octave 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 Octave; see the file COPYING. If not, see // <https://www.gnu.org/licenses/>. // //////////////////////////////////////////////////////////////////////// /* 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.fortran_vec (); 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]) { octave_quit (); 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, const octave_idx_type *ridx, const octave_idx_type *cidx, 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++) { octave_quit (); octave_idx_type k = ridx[i]; // there is a nonzero element (k,j) D[k]++; if (D[k] > max_deg) max_deg = D[k]; // if there is no element (j,k) there is one in // the symmetric matrix: if (k != j) { bool found = false; for (octave_idx_type l = cidx[k]; l < cidx[k + 1]; l++) { octave_quit (); if (ridx[l] == j) { found = true; break; } else if (ridx[l] > j) break; } if (! found) { // A(j,k) == 0 D[j]++; if (D[j] > max_deg) max_deg = D[j]; } } } } 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++) { octave_quit (); 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_quit (); 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); // the parameter of the matrix is converted into a sparse matrix //(if necessary) octave_idx_type *cidx; octave_idx_type *ridx; SparseMatrix Ar; SparseComplexMatrix Ac; if (arg.isreal ()) { Ar = arg.sparse_matrix_value (); // Note cidx/ridx are const, so use xridx and xcidx... cidx = Ar.xcidx (); ridx = Ar.xridx (); } else { Ac = arg.sparse_complex_matrix_value (); cidx = Ac.xcidx (); ridx = Ac.xridx (); } 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))); // 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; // dimension of the matrix octave_idx_type N = nr; OCTAVE_LOCAL_BUFFER (octave_idx_type, cidx2, N + 1); OCTAVE_LOCAL_BUFFER (octave_idx_type, ridx2, cidx[N]); transpose (N, ridx, cidx, ridx2, cidx2); // the permutation vector NDArray P (dim_vector (1, N)); // compute the node degrees OCTAVE_LOCAL_BUFFER (octave_idx_type, D, N); octave_idx_type max_deg = calc_degrees (N, ridx, cidx, D); // 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; return ovl (P); } // 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.fortran_vec (); 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]; octave_quit (); while (j1 < cidx[i+1] || j2 < cidx2[i+1]) { octave_quit (); 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)) { octave_quit (); // 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); } OCTAVE_END_NAMESPACE(octave)