Mercurial > octave
annotate libinterp/dldfcn/symrcm.cc @ 23220:092078913d54
maint: Merge stable to default.
author | John W. Eaton <jwe@octave.org> |
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date | Wed, 22 Feb 2017 12:58:07 -0500 |
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6608 | 1 /* |
2 | |
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3 Copyright (C) 2007-2017 Michael Weitzel |
6608 | 4 |
5 This file is part of Octave. | |
6 | |
7 Octave is free software; you can redistribute it and/or modify it | |
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8 under the terms of the GNU General Public License as published by |
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9 the Free Software Foundation; either version 3 of the License, or |
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10 (at your option) any later version. |
6608 | 11 |
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12 Octave is distributed in the hope that it will be useful, but |
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13 WITHOUT ANY WARRANTY; without even the implied warranty of |
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14 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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15 GNU General Public License for more details. |
6608 | 16 |
17 You should have received a copy of the GNU General Public License | |
7016 | 18 along with Octave; see the file COPYING. If not, see |
19 <http://www.gnu.org/licenses/>. | |
6608 | 20 |
21 */ | |
22 | |
23 /* | |
24 An implementation of the Reverse Cuthill-McKee algorithm (symrcm) | |
25 | |
26 The implementation of this algorithm is based in the descriptions found in | |
27 | |
28 @INPROCEEDINGS{, | |
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29 author = {E. Cuthill and J. McKee}, |
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30 title = {Reducing the Bandwidth of Sparse Symmetric Matrices}, |
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31 booktitle = {Proceedings of the 24th ACM National Conference}, |
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32 publisher = {Brandon Press}, |
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33 pages = {157 -- 172}, |
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34 location = {New Jersey}, |
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35 year = {1969} |
6608 | 36 } |
37 | |
38 @BOOK{, | |
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39 author = {Alan George and Joseph W. H. Liu}, |
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40 title = {Computer Solution of Large Sparse Positive Definite Systems}, |
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41 publisher = {Prentice Hall Series in Computational Mathematics}, |
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42 ISBN = {0-13-165274-5}, |
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43 year = {1981} |
6608 | 44 } |
45 | |
46 The algorithm represents a heuristic approach to the NP-complete minimum | |
47 bandwidth problem. | |
48 | |
6610 | 49 Written by Michael Weitzel <michael.weitzel@@uni-siegen.de> |
50 <weitzel@@ldknet.org> | |
6608 | 51 */ |
52 | |
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53 #if defined (HAVE_CONFIG_H) |
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54 # include "config.h" |
6608 | 55 #endif |
56 | |
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57 #include <algorithm> |
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58 |
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59 #include "CSparse.h" |
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60 #include "boolNDArray.h" |
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61 #include "dNDArray.h" |
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62 #include "dSparse.h" |
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63 #include "oct-locbuf.h" |
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64 #include "oct-sparse.h" |
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65 #include "quit.h" |
6608 | 66 |
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67 #include "defun-dld.h" |
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68 #include "errwarn.h" |
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69 #include "ov.h" |
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70 #include "ovl.h" |
6608 | 71 |
72 // A node struct for the Cuthill-McKee algorithm | |
73 struct CMK_Node | |
74 { | |
75 // the node's id (matrix row index) | |
76 octave_idx_type id; | |
77 // the node's degree | |
78 octave_idx_type deg; | |
79 // minimal distance to the root of the spanning tree | |
80 octave_idx_type dist; | |
81 }; | |
82 | |
83 // A simple queue. | |
84 // Queues Q have a fixed maximum size N (rows,cols of the matrix) and are | |
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85 // stored in an array. qh and qt point to queue head and tail. |
6608 | 86 |
87 // Enqueue operation (adds a node "o" at the tail) | |
6959 | 88 |
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89 inline static void |
6959 | 90 Q_enq (CMK_Node *Q, octave_idx_type N, octave_idx_type& qt, const CMK_Node& o) |
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91 { |
6959 | 92 Q[qt] = o; |
93 qt = (qt + 1) % (N + 1); | |
94 } | |
6608 | 95 |
96 // Dequeue operation (removes a node from the head) | |
6959 | 97 |
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98 inline static CMK_Node |
6959 | 99 Q_deq (CMK_Node * Q, octave_idx_type N, octave_idx_type& qh) |
100 { | |
101 CMK_Node r = Q[qh]; | |
102 qh = (qh + 1) % (N + 1); | |
103 return r; | |
104 } | |
6608 | 105 |
106 // Predicate (queue empty) | |
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107 #define Q_empty(Q, N, qh, qt) ((qh) == (qt)) |
6608 | 108 |
109 // A simple, array-based binary heap (used as a priority queue for nodes) | |
110 | |
111 // the left descendant of entry i | |
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112 #define LEFT(i) (((i) << 1) + 1) // = (2*(i)+1) |
6608 | 113 // the right descendant of entry i |
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114 #define RIGHT(i) (((i) << 1) + 2) // = (2*(i)+2) |
6608 | 115 // the parent of entry i |
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116 #define PARENT(i) (((i) - 1) >> 1) // = floor(((i)-1)/2) |
6608 | 117 |
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118 // Builds a min-heap (the root contains the smallest element). A is an array |
6608 | 119 // with the graph's nodes, i is a starting position, size is the length of A. |
6959 | 120 |
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121 static void |
6959 | 122 H_heapify_min (CMK_Node *A, octave_idx_type i, octave_idx_type size) |
123 { | |
124 octave_idx_type j = i; | |
125 for (;;) | |
126 { | |
127 octave_idx_type l = LEFT(j); | |
128 octave_idx_type r = RIGHT(j); | |
129 | |
130 octave_idx_type smallest; | |
131 if (l < size && A[l].deg < A[j].deg) | |
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132 smallest = l; |
6959 | 133 else |
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134 smallest = j; |
6959 | 135 |
136 if (r < size && A[r].deg < A[smallest].deg) | |
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137 smallest = r; |
6959 | 138 |
139 if (smallest != j) | |
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140 { |
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141 std::swap (A[j], A[smallest]); |
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142 j = smallest; |
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143 } |
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144 else |
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145 break; |
6959 | 146 } |
147 } | |
6608 | 148 |
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149 // Heap operation insert. Running time is O(log(n)) |
6959 | 150 |
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151 static void |
6959 | 152 H_insert (CMK_Node *H, octave_idx_type& h, const CMK_Node& o) |
153 { | |
154 octave_idx_type i = h++; | |
155 | |
156 H[i] = o; | |
157 | |
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158 if (i == 0) |
6959 | 159 return; |
160 do | |
161 { | |
162 octave_idx_type p = PARENT(i); | |
163 if (H[i].deg < H[p].deg) | |
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164 { |
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165 std::swap (H[i], H[p]); |
6959 | 166 |
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167 i = p; |
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168 } |
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169 else |
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170 break; |
6959 | 171 } |
172 while (i > 0); | |
173 } | |
6608 | 174 |
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175 // Heap operation remove-min. Removes the smalles element in O(1) and |
6608 | 176 // reorganizes the heap optionally in O(log(n)) |
6959 | 177 |
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178 inline static CMK_Node |
6959 | 179 H_remove_min (CMK_Node *H, octave_idx_type& h, int reorg/*=1*/) |
180 { | |
181 CMK_Node r = H[0]; | |
182 H[0] = H[--h]; | |
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183 if (reorg) |
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184 H_heapify_min (H, 0, h); |
6959 | 185 return r; |
186 } | |
6608 | 187 |
188 // Predicate (heap empty) | |
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189 #define H_empty(H, h) ((h) == 0) |
6608 | 190 |
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191 // Helper function for the Cuthill-McKee algorithm. Tries to determine a |
6608 | 192 // pseudo-peripheral node of the graph as starting node. |
6959 | 193 |
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194 static octave_idx_type |
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195 find_starting_node (octave_idx_type N, const octave_idx_type *ridx, |
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196 const octave_idx_type *cidx, const octave_idx_type *ridx2, |
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197 const octave_idx_type *cidx2, octave_idx_type *D, |
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198 octave_idx_type start) |
6959 | 199 { |
200 CMK_Node w; | |
201 | |
202 OCTAVE_LOCAL_BUFFER (CMK_Node, Q, N+1); | |
203 boolNDArray btmp (dim_vector (1, N), false); | |
204 bool *visit = btmp.fortran_vec (); | |
205 | |
206 octave_idx_type qh = 0; | |
207 octave_idx_type qt = 0; | |
208 CMK_Node x; | |
209 x.id = start; | |
210 x.deg = D[start]; | |
211 x.dist = 0; | |
212 Q_enq (Q, N, qt, x); | |
213 visit[start] = true; | |
214 | |
215 // distance level | |
216 octave_idx_type level = 0; | |
217 // current largest "eccentricity" | |
218 octave_idx_type max_dist = 0; | |
219 | |
220 for (;;) | |
221 { | |
222 while (! Q_empty (Q, N, qh, qt)) | |
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223 { |
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224 CMK_Node v = Q_deq (Q, N, qh); |
6959 | 225 |
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226 if (v.dist > x.dist || (v.id != x.id && v.deg > x.deg)) |
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227 x = v; |
6959 | 228 |
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229 octave_idx_type i = v.id; |
6959 | 230 |
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231 // add all unvisited neighbors to the queue |
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232 octave_idx_type j1 = cidx[i]; |
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233 octave_idx_type j2 = cidx2[i]; |
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234 while (j1 < cidx[i+1] || j2 < cidx2[i+1]) |
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235 { |
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236 octave_quit (); |
6959 | 237 |
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238 if (j1 == cidx[i+1]) |
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239 { |
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240 octave_idx_type r2 = ridx2[j2++]; |
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241 if (! visit[r2]) |
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242 { |
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243 // the distance of node j is dist(i)+1 |
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244 w.id = r2; |
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245 w.deg = D[r2]; |
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246 w.dist = v.dist+1; |
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247 Q_enq (Q, N, qt, w); |
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248 visit[r2] = true; |
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250 if (w.dist > level) |
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251 level = w.dist; |
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252 } |
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253 } |
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254 else if (j2 == cidx2[i+1]) |
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255 { |
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256 octave_idx_type r1 = ridx[j1++]; |
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257 if (! visit[r1]) |
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258 { |
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259 // the distance of node j is dist(i)+1 |
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260 w.id = r1; |
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261 w.deg = D[r1]; |
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262 w.dist = v.dist+1; |
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263 Q_enq (Q, N, qt, w); |
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264 visit[r1] = true; |
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266 if (w.dist > level) |
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267 level = w.dist; |
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268 } |
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269 } |
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270 else |
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271 { |
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272 octave_idx_type r1 = ridx[j1]; |
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273 octave_idx_type r2 = ridx2[j2]; |
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274 if (r1 <= r2) |
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275 { |
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276 if (! visit[r1]) |
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277 { |
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278 w.id = r1; |
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279 w.deg = D[r1]; |
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280 w.dist = v.dist+1; |
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281 Q_enq (Q, N, qt, w); |
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282 visit[r1] = true; |
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284 if (w.dist > level) |
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285 level = w.dist; |
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286 } |
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287 j1++; |
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288 if (r1 == r2) |
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289 j2++; |
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290 } |
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291 else |
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292 { |
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293 if (! visit[r2]) |
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294 { |
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295 w.id = r2; |
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296 w.deg = D[r2]; |
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297 w.dist = v.dist+1; |
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298 Q_enq (Q, N, qt, w); |
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299 visit[r2] = true; |
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301 if (w.dist > level) |
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302 level = w.dist; |
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303 } |
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304 j2++; |
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305 } |
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306 } |
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307 } |
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308 } // finish of BFS |
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310 if (max_dist < x.dist) | |
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311 { |
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312 max_dist = x.dist; |
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314 for (octave_idx_type i = 0; i < N; i++) |
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315 visit[i] = false; |
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317 visit[x.id] = true; |
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318 x.dist = 0; |
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319 qt = qh = 0; |
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320 Q_enq (Q, N, qt, x); |
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321 } |
6959 | 322 else |
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323 break; |
6959 | 324 } |
325 return x.id; | |
326 } | |
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328 // Calculates the node's degrees. This means counting the nonzero elements |
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329 // in the symmetric matrix' rows. This works for non-symmetric matrices |
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332 static octave_idx_type |
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333 calc_degrees (octave_idx_type N, const octave_idx_type *ridx, |
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334 const octave_idx_type *cidx, octave_idx_type *D) |
6959 | 335 { |
336 octave_idx_type max_deg = 0; | |
337 | |
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338 for (octave_idx_type i = 0; i < N; i++) |
6959 | 339 D[i] = 0; |
340 | |
341 for (octave_idx_type j = 0; j < N; j++) | |
342 { | |
343 for (octave_idx_type i = cidx[j]; i < cidx[j+1]; i++) | |
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344 { |
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345 octave_quit (); |
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346 |
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347 octave_idx_type k = ridx[i]; |
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348 // there is a nonzero element (k,j) |
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349 D[k]++; |
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350 if (D[k] > max_deg) |
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351 max_deg = D[k]; |
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352 // if there is no element (j,k) there is one in |
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353 // the symmetric matrix: |
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354 if (k != j) |
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355 { |
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356 bool found = false; |
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357 for (octave_idx_type l = cidx[k]; l < cidx[k + 1]; l++) |
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358 { |
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359 octave_quit (); |
6959 | 360 |
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361 if (ridx[l] == j) |
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362 { |
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363 found = true; |
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364 break; |
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365 } |
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366 else if (ridx[l] > j) |
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367 break; |
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368 } |
6959 | 369 |
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370 if (! found) |
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371 { |
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372 // A(j,k) == 0 |
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373 D[j]++; |
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374 if (D[j] > max_deg) |
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375 max_deg = D[j]; |
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376 } |
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377 } |
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378 } |
6959 | 379 } |
380 return max_deg; | |
381 } | |
6608 | 382 |
383 // Transpose of the structure of a square sparse matrix | |
6959 | 384 |
6608 | 385 static void |
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386 transpose (octave_idx_type N, const octave_idx_type *ridx, |
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387 const octave_idx_type *cidx, octave_idx_type *ridx2, |
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388 octave_idx_type *cidx2) |
6959 | 389 { |
390 octave_idx_type nz = cidx[N]; | |
391 | |
392 OCTAVE_LOCAL_BUFFER (octave_idx_type, w, N + 1); | |
393 for (octave_idx_type i = 0; i < N; i++) | |
394 w[i] = 0; | |
395 for (octave_idx_type i = 0; i < nz; i++) | |
396 w[ridx[i]]++; | |
397 nz = 0; | |
398 for (octave_idx_type i = 0; i < N; i++) | |
399 { | |
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400 octave_quit (); |
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401 |
6959 | 402 cidx2[i] = nz; |
403 nz += w[i]; | |
404 w[i] = cidx2[i]; | |
405 } | |
406 cidx2[N] = nz; | |
407 w[N] = nz; | |
408 | |
409 for (octave_idx_type j = 0; j < N; j++) | |
410 for (octave_idx_type k = cidx[j]; k < cidx[j + 1]; k++) | |
411 { | |
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412 octave_quit (); |
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413 |
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414 octave_idx_type q = w[ridx[k]]++; |
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415 ridx2[q] = j; |
6959 | 416 } |
417 } | |
6608 | 418 |
419 // An implementation of the Cuthill-McKee algorithm. | |
420 DEFUN_DLD (symrcm, args, , | |
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421 doc: /* -*- texinfo -*- |
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422 @deftypefn {} {@var{p} =} symrcm (@var{S}) |
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423 Return the symmetric reverse @nospell{Cuthill-McKee} permutation of @var{S}. |
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424 |
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425 @var{p} is a permutation vector such that |
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426 @code{@var{S}(@var{p}, @var{p})} tends to have its diagonal elements closer |
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427 to the diagonal than @var{S}. This is a good preordering for LU or |
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428 Cholesky@tie{}factorization of matrices that come from ``long, skinny'' |
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429 problems. It works for both symmetric and asymmetric @var{S}. |
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430 |
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431 The algorithm represents a heuristic approach to the NP-complete bandwidth |
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432 minimization problem. The implementation is based in the descriptions found |
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433 in |
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434 |
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435 @nospell{E. Cuthill, J. McKee}. @cite{Reducing the Bandwidth of Sparse |
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436 Symmetric Matrices}. Proceedings of the 24th ACM National Conference, |
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437 157--172 1969, Brandon Press, New Jersey. |
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438 |
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439 @nospell{A. George, J.W.H. Liu}. @cite{Computer Solution of Large Sparse |
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440 Positive Definite Systems}, Prentice Hall Series in Computational |
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441 Mathematics, ISBN 0-13-165274-5, 1981. |
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442 |
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443 @seealso{colperm, colamd, symamd} |
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444 @end deftypefn */) |
6608 | 445 { |
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446 if (args.length () != 1) |
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447 print_usage (); |
6608 | 448 |
6959 | 449 octave_value arg = args(0); |
6608 | 450 |
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451 // the parameter of the matrix is converted into a sparse matrix |
6608 | 452 //(if necessary) |
453 octave_idx_type *cidx; | |
454 octave_idx_type *ridx; | |
455 SparseMatrix Ar; | |
456 SparseComplexMatrix Ac; | |
457 | |
458 if (arg.is_real_type ()) | |
459 { | |
6959 | 460 Ar = arg.sparse_matrix_value (); |
6608 | 461 // Note cidx/ridx are const, so use xridx and xcidx... |
462 cidx = Ar.xcidx (); | |
463 ridx = Ar.xridx (); | |
464 } | |
465 else | |
466 { | |
6959 | 467 Ac = arg.sparse_complex_matrix_value (); |
6608 | 468 cidx = Ac.xcidx (); |
469 ridx = Ac.xridx (); | |
470 } | |
471 | |
472 octave_idx_type nr = arg.rows (); | |
473 octave_idx_type nc = arg.columns (); | |
474 | |
475 if (nr != nc) | |
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476 err_square_matrix_required ("symrcm", "S"); |
6608 | 477 |
478 if (nr == 0 && nc == 0) | |
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479 return ovl (NDArray (dim_vector (1, 0))); |
6959 | 480 |
6608 | 481 // sizes of the heaps |
482 octave_idx_type s = 0; | |
6959 | 483 |
6608 | 484 // head- and tail-indices for the queue |
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485 octave_idx_type qt = 0; |
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486 octave_idx_type qh = 0; |
6608 | 487 CMK_Node v, w; |
488 // dimension of the matrix | |
6959 | 489 octave_idx_type N = nr; |
490 | |
6608 | 491 OCTAVE_LOCAL_BUFFER (octave_idx_type, cidx2, N + 1); |
492 OCTAVE_LOCAL_BUFFER (octave_idx_type, ridx2, cidx[N]); | |
493 transpose (N, ridx, cidx, ridx2, cidx2); | |
494 | |
495 // the permutation vector | |
496 NDArray P (dim_vector (1, N)); | |
497 | |
498 // compute the node degrees | |
6959 | 499 OCTAVE_LOCAL_BUFFER (octave_idx_type, D, N); |
500 octave_idx_type max_deg = calc_degrees (N, ridx, cidx, D); | |
6608 | 501 |
502 // if none of the nodes has a degree > 0 (a matrix of zeros) | |
503 // the return value corresponds to the identity permutation | |
504 if (max_deg == 0) | |
505 { | |
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506 for (octave_idx_type i = 0; i < N; i++) |
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507 P(i) = i; |
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508 |
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509 return ovl (P); |
6608 | 510 } |
511 | |
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512 // a heap for the a node's neighbors. The number of neighbors is |
6608 | 513 // limited by the maximum degree max_deg: |
6959 | 514 OCTAVE_LOCAL_BUFFER (CMK_Node, S, max_deg); |
6608 | 515 |
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516 // a queue for the BFS. The array is always one element larger than |
6608 | 517 // the number of entries that are stored. |
6959 | 518 OCTAVE_LOCAL_BUFFER (CMK_Node, Q, N+1); |
6608 | 519 |
520 // a counter (for building the permutation) | |
6959 | 521 octave_idx_type c = -1; |
6608 | 522 |
6959 | 523 // upper bound for the bandwidth (=quality of solution) |
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524 // initialize the bandwidth of the graph with 0. B contains the |
6608 | 525 // the maximum of the theoretical lower limits of the subgraphs |
526 // bandwidths. | |
6959 | 527 octave_idx_type B = 0; |
6608 | 528 |
529 // mark all nodes as unvisited; with the exception of the nodes | |
530 // that have degree==0 and build a CC of the graph. | |
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531 |
6608 | 532 boolNDArray btmp (dim_vector (1, N), false); |
533 bool *visit = btmp.fortran_vec (); | |
534 | |
535 do | |
536 { | |
537 // locate an unvisited starting node of the graph | |
6959 | 538 octave_idx_type i; |
6608 | 539 for (i = 0; i < N; i++) |
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540 if (! visit[i]) |
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541 break; |
6608 | 542 |
543 // locate a probably better starting node | |
544 v.id = find_starting_node (N, ridx, cidx, ridx2, cidx2, D, i); | |
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545 |
6608 | 546 // mark the node as visited and enqueue it (a starting node |
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547 // for the BFS). Since the node will be a root of a spanning |
6608 | 548 // tree, its dist is 0. |
549 v.deg = D[v.id]; | |
550 v.dist = 0; | |
551 visit[v.id] = true; | |
6959 | 552 Q_enq (Q, N, qt, v); |
6608 | 553 |
6959 | 554 // lower bound for the bandwidth of a subgraph |
6608 | 555 // keep a "level" in the spanning tree (= min. distance to the |
556 // root) for determining the bandwidth of the computed | |
557 // permutation P | |
6959 | 558 octave_idx_type Bsub = 0; |
6608 | 559 // min. dist. to the root is 0 |
6959 | 560 octave_idx_type level = 0; |
6608 | 561 // the root is the first/only node on level 0 |
6959 | 562 octave_idx_type level_N = 1; |
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563 |
6677 | 564 while (! Q_empty (Q, N, qh, qt)) |
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565 { |
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566 v = Q_deq (Q, N, qh); |
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567 i = v.id; |
6608 | 568 |
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569 c++; |
6608 | 570 |
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571 // for computing the inverse permutation P where |
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572 // A(inv(P),inv(P)) or P'*A*P is banded |
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573 // P(i) = c; |
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574 |
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575 // for computing permutation P where |
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576 // A(P(i),P(j)) or P*A*P' is banded |
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577 P(c) = i; |
6608 | 578 |
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579 // put all unvisited neighbors j of node i on the heap |
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580 s = 0; |
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581 octave_idx_type j1 = cidx[i]; |
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582 octave_idx_type j2 = cidx2[i]; |
6608 | 583 |
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584 octave_quit (); |
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585 |
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586 while (j1 < cidx[i+1] || j2 < cidx2[i+1]) |
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587 { |
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588 octave_quit (); |
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589 |
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590 if (j1 == cidx[i+1]) |
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591 { |
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592 octave_idx_type r2 = ridx2[j2++]; |
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593 if (! visit[r2]) |
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594 { |
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595 // the distance of node j is dist(i)+1 |
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596 w.id = r2; |
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597 w.deg = D[r2]; |
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598 w.dist = v.dist+1; |
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599 H_insert (S, s, w); |
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600 visit[r2] = true; |
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601 } |
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602 } |
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603 else if (j2 == cidx2[i+1]) |
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604 { |
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605 octave_idx_type r1 = ridx[j1++]; |
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606 if (! visit[r1]) |
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607 { |
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608 w.id = r1; |
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609 w.deg = D[r1]; |
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610 w.dist = v.dist+1; |
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611 H_insert (S, s, w); |
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612 visit[r1] = true; |
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613 } |
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614 } |
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615 else |
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616 { |
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617 octave_idx_type r1 = ridx[j1]; |
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618 octave_idx_type r2 = ridx2[j2]; |
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619 if (r1 <= r2) |
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620 { |
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621 if (! visit[r1]) |
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622 { |
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623 w.id = r1; |
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624 w.deg = D[r1]; |
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625 w.dist = v.dist+1; |
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626 H_insert (S, s, w); |
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627 visit[r1] = true; |
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628 } |
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629 j1++; |
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630 if (r1 == r2) |
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631 j2++; |
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632 } |
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633 else |
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634 { |
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635 if (! visit[r2]) |
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636 { |
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637 w.id = r2; |
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638 w.deg = D[r2]; |
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639 w.dist = v.dist+1; |
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640 H_insert (S, s, w); |
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641 visit[r2] = true; |
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642 } |
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643 j2++; |
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644 } |
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645 } |
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646 } |
6608 | 647 |
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648 // add the neighbors to the queue (sorted by node degree) |
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649 while (! H_empty (S, s)) |
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650 { |
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651 octave_quit (); |
6608 | 652 |
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653 // locate a neighbor of i with minimal degree in O(log(N)) |
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654 v = H_remove_min (S, s, 1); |
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655 |
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656 // entered the BFS a new level? |
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657 if (v.dist > level) |
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658 { |
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659 // adjustment of bandwith: |
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660 // "[...] the minimum bandwidth that |
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661 // can be obtained [...] is the |
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662 // maximum number of nodes per level" |
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663 if (Bsub < level_N) |
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664 Bsub = level_N; |
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665 |
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666 level = v.dist; |
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667 // v is the first node on the new level |
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668 level_N = 1; |
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669 } |
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670 else |
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671 { |
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672 // there is no new level but another node on |
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673 // this level: |
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674 level_N++; |
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675 } |
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676 |
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677 // enqueue v in O(1) |
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678 Q_enq (Q, N, qt, v); |
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679 } |
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680 |
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681 // synchronize the bandwidth with level_N once again: |
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682 if (Bsub < level_N) |
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683 Bsub = level_N; |
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684 } |
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685 // finish of BFS. If there are still unvisited nodes in the graph |
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686 // then it is split into CCs. The computed bandwidth is the maximum |
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687 // of all subgraphs. Update: |
6608 | 688 if (Bsub > B) |
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689 B = Bsub; |
6608 | 690 } |
691 // are there any nodes left? | |
692 while (c+1 < N); | |
693 | |
694 // compute the reverse-ordering | |
695 s = N / 2 - 1; | |
6959 | 696 for (octave_idx_type i = 0, j = N - 1; i <= s; i++, j--) |
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697 std::swap (P.elem (i), P.elem (j)); |
6608 | 698 |
699 // increment all indices, since Octave is not C | |
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700 return ovl (P+1); |
6608 | 701 } |