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view libinterp/dldfcn/symrcm.cc @ 21301:40de9f8f23a6
Use '#include "config.h"' rather than <config.h>.
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op-str-m.cc, op-str-s.cc, op-str-str.cc, op-struct.cc, op-ui16-ui16.cc,
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dim-vector.cc, fCColVector.cc, fCDiagMatrix.cc, fCMatrix.cc, fCNDArray.cc,
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fRowVector.cc, idx-vector.cc, int16NDArray.cc, int32NDArray.cc,
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lu.cc, oct-convn.cc, oct-fftw.cc, oct-norm.cc, oct-rand.cc, oct-spparms.cc,
qr.cc, qrp.cc, randgamma.c, randmtzig.c, randpoisson.c, schur.cc,
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lo-ieee.cc, lo-regexp.cc, lo-utils.cc, oct-base64.cc, oct-glob.cc,
oct-inttypes.cc, oct-locbuf.cc, oct-mutex.cc, oct-rl-edit.c, oct-rl-hist.c,
oct-shlib.cc, oct-sort.cc, pathsearch.cc, singleton-cleanup.cc, sparse-sort.cc,
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octave-config.in.cc:
Use '#include "config.h"' rather than <config.h>.
author | Rik <rik@octave.org> |
---|---|
date | Thu, 18 Feb 2016 13:34:50 -0800 |
parents | fcac5dbbf9ed |
children | aba2e6293dd8 |
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/* Copyright (C) 2007-2015 Michael Weitzel 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 <http://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> */ #ifdef HAVE_CONFIG_H # include "config.h" #endif #include "ov.h" #include "defun-dld.h" #include "error.h" #include "errwarn.h" #include "utils.h" #include "oct-locbuf.h" #include "ov-re-mat.h" #include "ov-re-sparse.h" #include "ov-cx-sparse.h" #include "oct-sparse.h" // 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 smalles 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_DLD (symrcm, args, , "-*- texinfo -*-\n\ @deftypefn {} {@var{p} =} symrcm (@var{S})\n\ Return the symmetric reverse @nospell{Cuthill-McKee} permutation of @var{S}.\n\ \n\ @var{p} is a permutation vector such that\n\ @code{@var{S}(@var{p}, @var{p})} tends to have its diagonal elements closer\n\ to the diagonal than @var{S}. This is a good preordering for LU or\n\ Cholesky@tie{}factorization of matrices that come from ``long, skinny''\n\ problems. It works for both symmetric and asymmetric @var{S}.\n\ \n\ The algorithm represents a heuristic approach to the NP-complete bandwidth\n\ minimization problem. The implementation is based in the descriptions found\n\ in\n\ \n\ @nospell{E. Cuthill, J. McKee}. @cite{Reducing the Bandwidth of Sparse\n\ Symmetric Matrices}. Proceedings of the 24th ACM National Conference,\n\ 157--172 1969, Brandon Press, New Jersey.\n\ \n\ @nospell{A. George, J.W.H. Liu}. @cite{Computer Solution of Large Sparse\n\ Positive Definite Systems}, Prentice Hall Series in Computational\n\ Mathematics, ISBN 0-13-165274-5, 1981.\n\ \n\ @seealso{colperm, colamd, symamd}\n\ @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.is_real_type ()) { 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 bandwith: // "[...] 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); }