aboutsummaryrefslogtreecommitdiffstats
path: root/test/monniaux/glpk-4.65/src/colamd/colamd.c
blob: 86ddd6b74f63516627bc2ef9375da4339ebb277d (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
/* ========================================================================== */
/* === colamd/symamd - a sparse matrix column ordering algorithm ============ */
/* ========================================================================== */

/* COLAMD / SYMAMD

    colamd:  an approximate minimum degree column ordering algorithm,
        for LU factorization of symmetric or unsymmetric matrices,
        QR factorization, least squares, interior point methods for
        linear programming problems, and other related problems.

    symamd:  an approximate minimum degree ordering algorithm for Cholesky
        factorization of symmetric matrices.

    Purpose:

        Colamd computes a permutation Q such that the Cholesky factorization of
        (AQ)'(AQ) has less fill-in and requires fewer floating point operations
        than A'A.  This also provides a good ordering for sparse partial
        pivoting methods, P(AQ) = LU, where Q is computed prior to numerical
        factorization, and P is computed during numerical factorization via
        conventional partial pivoting with row interchanges.  Colamd is the
        column ordering method used in SuperLU, part of the ScaLAPACK library.
        It is also available as built-in function in MATLAB Version 6,
        available from MathWorks, Inc. (http://www.mathworks.com).  This
        routine can be used in place of colmmd in MATLAB.

        Symamd computes a permutation P of a symmetric matrix A such that the
        Cholesky factorization of PAP' has less fill-in and requires fewer
        floating point operations than A.  Symamd constructs a matrix M such
        that M'M has the same nonzero pattern of A, and then orders the columns
        of M using colmmd.  The column ordering of M is then returned as the
        row and column ordering P of A.

    Authors:

        The authors of the code itself are Stefan I. Larimore and Timothy A.
        Davis (davis at cise.ufl.edu), University of Florida.  The algorithm was
        developed in collaboration with John Gilbert, Xerox PARC, and Esmond
        Ng, Oak Ridge National Laboratory.

    Acknowledgements:

        This work was supported by the National Science Foundation, under
        grants DMS-9504974 and DMS-9803599.

    Copyright and License:

        Copyright (c) 1998-2007, Timothy A. Davis, All Rights Reserved.
        COLAMD is also available under alternate licenses, contact T. Davis
        for details.

        This library is free software; you can redistribute it and/or
        modify it under the terms of the GNU Lesser General Public
        License as published by the Free Software Foundation; either
        version 2.1 of the License, or (at your option) any later version.

        This library 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
        Lesser General Public License for more details.

        You should have received a copy of the GNU Lesser General Public
        License along with this library; if not, write to the Free Software
        Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301
        USA

        Permission is hereby granted to use or copy this program under the
        terms of the GNU LGPL, provided that the Copyright, this License,
        and the Availability of the original version is retained on all copies.
        User documentation of any code that uses this code or any modified
        version of this code must cite the Copyright, this License, the
        Availability note, and "Used by permission." Permission to modify
        the code and to distribute modified code is granted, provided the
        Copyright, this License, and the Availability note are retained,
        and a notice that the code was modified is included.

    Availability:

        The colamd/symamd library is available at

            http://www.cise.ufl.edu/research/sparse/colamd/

        This is the http://www.cise.ufl.edu/research/sparse/colamd/colamd.c
        file.  It requires the colamd.h file.  It is required by the colamdmex.c
        and symamdmex.c files, for the MATLAB interface to colamd and symamd.
        Appears as ACM Algorithm 836.

    See the ChangeLog file for changes since Version 1.0.

    References:

        T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, An approximate column
        minimum degree ordering algorithm, ACM Transactions on Mathematical
        Software, vol. 30, no. 3., pp. 353-376, 2004.

        T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, Algorithm 836: COLAMD,
        an approximate column minimum degree ordering algorithm, ACM
        Transactions on Mathematical Software, vol. 30, no. 3., pp. 377-380,
        2004.

*/

/* ========================================================================== */
/* === Description of user-callable routines ================================ */
/* ========================================================================== */

/* COLAMD includes both int and UF_long versions of all its routines.  The
 * description below is for the int version.  For UF_long, all int arguments
 * become UF_long.  UF_long is normally defined as long, except for WIN64.

    ----------------------------------------------------------------------------
    colamd_recommended:
    ----------------------------------------------------------------------------

        C syntax:

            #include "colamd.h"
            size_t colamd_recommended (int nnz, int n_row, int n_col) ;
            size_t colamd_l_recommended (UF_long nnz, UF_long n_row,
                UF_long n_col) ;

        Purpose:

            Returns recommended value of Alen for use by colamd.  Returns 0
            if any input argument is negative.  The use of this routine
            is optional.  Not needed for symamd, which dynamically allocates
            its own memory.

            Note that in v2.4 and earlier, these routines returned int or long.
            They now return a value of type size_t.

        Arguments (all input arguments):

            int nnz ;           Number of nonzeros in the matrix A.  This must
                                be the same value as p [n_col] in the call to
                                colamd - otherwise you will get a wrong value
                                of the recommended memory to use.

            int n_row ;         Number of rows in the matrix A.

            int n_col ;         Number of columns in the matrix A.

    ----------------------------------------------------------------------------
    colamd_set_defaults:
    ----------------------------------------------------------------------------

        C syntax:

            #include "colamd.h"
            colamd_set_defaults (double knobs [COLAMD_KNOBS]) ;
            colamd_l_set_defaults (double knobs [COLAMD_KNOBS]) ;

        Purpose:

            Sets the default parameters.  The use of this routine is optional.

        Arguments:

            double knobs [COLAMD_KNOBS] ;       Output only.

                NOTE: the meaning of the dense row/col knobs has changed in v2.4

                knobs [0] and knobs [1] control dense row and col detection:

                Colamd: rows with more than
                max (16, knobs [COLAMD_DENSE_ROW] * sqrt (n_col))
                entries are removed prior to ordering.  Columns with more than
                max (16, knobs [COLAMD_DENSE_COL] * sqrt (MIN (n_row,n_col)))
                entries are removed prior to
                ordering, and placed last in the output column ordering.

                Symamd: uses only knobs [COLAMD_DENSE_ROW], which is knobs [0].
                Rows and columns with more than
                max (16, knobs [COLAMD_DENSE_ROW] * sqrt (n))
                entries are removed prior to ordering, and placed last in the
                output ordering.

                COLAMD_DENSE_ROW and COLAMD_DENSE_COL are defined as 0 and 1,
                respectively, in colamd.h.  Default values of these two knobs
                are both 10.  Currently, only knobs [0] and knobs [1] are
                used, but future versions may use more knobs.  If so, they will
                be properly set to their defaults by the future version of
                colamd_set_defaults, so that the code that calls colamd will
                not need to change, assuming that you either use
                colamd_set_defaults, or pass a (double *) NULL pointer as the
                knobs array to colamd or symamd.

            knobs [2]: aggressive absorption

                knobs [COLAMD_AGGRESSIVE] controls whether or not to do
                aggressive absorption during the ordering.  Default is TRUE.


    ----------------------------------------------------------------------------
    colamd:
    ----------------------------------------------------------------------------

        C syntax:

            #include "colamd.h"
            int colamd (int n_row, int n_col, int Alen, int *A, int *p,
                double knobs [COLAMD_KNOBS], int stats [COLAMD_STATS]) ;
            UF_long colamd_l (UF_long n_row, UF_long n_col, UF_long Alen,
                UF_long *A, UF_long *p, double knobs [COLAMD_KNOBS],
                UF_long stats [COLAMD_STATS]) ;

        Purpose:

            Computes a column ordering (Q) of A such that P(AQ)=LU or
            (AQ)'AQ=LL' have less fill-in and require fewer floating point
            operations than factorizing the unpermuted matrix A or A'A,
            respectively.

        Returns:

            TRUE (1) if successful, FALSE (0) otherwise.

        Arguments:

            int n_row ;         Input argument.

                Number of rows in the matrix A.
                Restriction:  n_row >= 0.
                Colamd returns FALSE if n_row is negative.

            int n_col ;         Input argument.

                Number of columns in the matrix A.
                Restriction:  n_col >= 0.
                Colamd returns FALSE if n_col is negative.

            int Alen ;          Input argument.

                Restriction (see note):
                Alen >= 2*nnz + 6*(n_col+1) + 4*(n_row+1) + n_col
                Colamd returns FALSE if these conditions are not met.

                Note:  this restriction makes an modest assumption regarding
                the size of the two typedef's structures in colamd.h.
                We do, however, guarantee that

                        Alen >= colamd_recommended (nnz, n_row, n_col)

                will be sufficient.  Note: the macro version does not check
                for integer overflow, and thus is not recommended.  Use
                the colamd_recommended routine instead.

            int A [Alen] ;      Input argument, undefined on output.

                A is an integer array of size Alen.  Alen must be at least as
                large as the bare minimum value given above, but this is very
                low, and can result in excessive run time.  For best
                performance, we recommend that Alen be greater than or equal to
                colamd_recommended (nnz, n_row, n_col), which adds
                nnz/5 to the bare minimum value given above.

                On input, the row indices of the entries in column c of the
                matrix are held in A [(p [c]) ... (p [c+1]-1)].  The row indices
                in a given column c need not be in ascending order, and
                duplicate row indices may be be present.  However, colamd will
                work a little faster if both of these conditions are met
                (Colamd puts the matrix into this format, if it finds that the
                the conditions are not met).

                The matrix is 0-based.  That is, rows are in the range 0 to
                n_row-1, and columns are in the range 0 to n_col-1.  Colamd
                returns FALSE if any row index is out of range.

                The contents of A are modified during ordering, and are
                undefined on output.

            int p [n_col+1] ;   Both input and output argument.

                p is an integer array of size n_col+1.  On input, it holds the
                "pointers" for the column form of the matrix A.  Column c of
                the matrix A is held in A [(p [c]) ... (p [c+1]-1)].  The first
                entry, p [0], must be zero, and p [c] <= p [c+1] must hold
                for all c in the range 0 to n_col-1.  The value p [n_col] is
                thus the total number of entries in the pattern of the matrix A.
                Colamd returns FALSE if these conditions are not met.

                On output, if colamd returns TRUE, the array p holds the column
                permutation (Q, for P(AQ)=LU or (AQ)'(AQ)=LL'), where p [0] is
                the first column index in the new ordering, and p [n_col-1] is
                the last.  That is, p [k] = j means that column j of A is the
                kth pivot column, in AQ, where k is in the range 0 to n_col-1
                (p [0] = j means that column j of A is the first column in AQ).

                If colamd returns FALSE, then no permutation is returned, and
                p is undefined on output.

            double knobs [COLAMD_KNOBS] ;       Input argument.

                See colamd_set_defaults for a description.

            int stats [COLAMD_STATS] ;          Output argument.

                Statistics on the ordering, and error status.
                See colamd.h for related definitions.
                Colamd returns FALSE if stats is not present.

                stats [0]:  number of dense or empty rows ignored.

                stats [1]:  number of dense or empty columns ignored (and
                                ordered last in the output permutation p)
                                Note that a row can become "empty" if it
                                contains only "dense" and/or "empty" columns,
                                and similarly a column can become "empty" if it
                                only contains "dense" and/or "empty" rows.

                stats [2]:  number of garbage collections performed.
                                This can be excessively high if Alen is close
                                to the minimum required value.

                stats [3]:  status code.  < 0 is an error code.
                            > 1 is a warning or notice.

                        0       OK.  Each column of the input matrix contained
                                row indices in increasing order, with no
                                duplicates.

                        1       OK, but columns of input matrix were jumbled
                                (unsorted columns or duplicate entries).  Colamd
                                had to do some extra work to sort the matrix
                                first and remove duplicate entries, but it
                                still was able to return a valid permutation
                                (return value of colamd was TRUE).

                                        stats [4]: highest numbered column that
                                                is unsorted or has duplicate
                                                entries.
                                        stats [5]: last seen duplicate or
                                                unsorted row index.
                                        stats [6]: number of duplicate or
                                                unsorted row indices.

                        -1      A is a null pointer

                        -2      p is a null pointer

                        -3      n_row is negative

                                        stats [4]: n_row

                        -4      n_col is negative

                                        stats [4]: n_col

                        -5      number of nonzeros in matrix is negative

                                        stats [4]: number of nonzeros, p [n_col]

                        -6      p [0] is nonzero

                                        stats [4]: p [0]

                        -7      A is too small

                                        stats [4]: required size
                                        stats [5]: actual size (Alen)

                        -8      a column has a negative number of entries

                                        stats [4]: column with < 0 entries
                                        stats [5]: number of entries in col

                        -9      a row index is out of bounds

                                        stats [4]: column with bad row index
                                        stats [5]: bad row index
                                        stats [6]: n_row, # of rows of matrx

                        -10     (unused; see symamd.c)

                        -999    (unused; see symamd.c)

                Future versions may return more statistics in the stats array.

        Example:

            See http://www.cise.ufl.edu/research/sparse/colamd/example.c
            for a complete example.

            To order the columns of a 5-by-4 matrix with 11 nonzero entries in
            the following nonzero pattern

                x 0 x 0
                x 0 x x
                0 x x 0
                0 0 x x
                x x 0 0

            with default knobs and no output statistics, do the following:

                #include "colamd.h"
                #define ALEN 100
                int A [ALEN] = {0, 1, 4, 2, 4, 0, 1, 2, 3, 1, 3} ;
                int p [ ] = {0, 3, 5, 9, 11} ;
                int stats [COLAMD_STATS] ;
                colamd (5, 4, ALEN, A, p, (double *) NULL, stats) ;

            The permutation is returned in the array p, and A is destroyed.

    ----------------------------------------------------------------------------
    symamd:
    ----------------------------------------------------------------------------

        C syntax:

            #include "colamd.h"
            int symamd (int n, int *A, int *p, int *perm,
                double knobs [COLAMD_KNOBS], int stats [COLAMD_STATS],
                void (*allocate) (size_t, size_t), void (*release) (void *)) ;
            UF_long symamd_l (UF_long n, UF_long *A, UF_long *p, UF_long *perm,
                double knobs [COLAMD_KNOBS], UF_long stats [COLAMD_STATS],
                void (*allocate) (size_t, size_t), void (*release) (void *)) ;

        Purpose:

            The symamd routine computes an ordering P of a symmetric sparse
            matrix A such that the Cholesky factorization PAP' = LL' remains
            sparse.  It is based on a column ordering of a matrix M constructed
            so that the nonzero pattern of M'M is the same as A.  The matrix A
            is assumed to be symmetric; only the strictly lower triangular part
            is accessed.  You must pass your selected memory allocator (usually
            calloc/free or mxCalloc/mxFree) to symamd, for it to allocate
            memory for the temporary matrix M.

        Returns:

            TRUE (1) if successful, FALSE (0) otherwise.

        Arguments:

            int n ;             Input argument.

                Number of rows and columns in the symmetrix matrix A.
                Restriction:  n >= 0.
                Symamd returns FALSE if n is negative.

            int A [nnz] ;       Input argument.

                A is an integer array of size nnz, where nnz = p [n].

                The row indices of the entries in column c of the matrix are
                held in A [(p [c]) ... (p [c+1]-1)].  The row indices in a
                given column c need not be in ascending order, and duplicate
                row indices may be present.  However, symamd will run faster
                if the columns are in sorted order with no duplicate entries.

                The matrix is 0-based.  That is, rows are in the range 0 to
                n-1, and columns are in the range 0 to n-1.  Symamd
                returns FALSE if any row index is out of range.

                The contents of A are not modified.

            int p [n+1] ;       Input argument.

                p is an integer array of size n+1.  On input, it holds the
                "pointers" for the column form of the matrix A.  Column c of
                the matrix A is held in A [(p [c]) ... (p [c+1]-1)].  The first
                entry, p [0], must be zero, and p [c] <= p [c+1] must hold
                for all c in the range 0 to n-1.  The value p [n] is
                thus the total number of entries in the pattern of the matrix A.
                Symamd returns FALSE if these conditions are not met.

                The contents of p are not modified.

            int perm [n+1] ;    Output argument.

                On output, if symamd returns TRUE, the array perm holds the
                permutation P, where perm [0] is the first index in the new
                ordering, and perm [n-1] is the last.  That is, perm [k] = j
                means that row and column j of A is the kth column in PAP',
                where k is in the range 0 to n-1 (perm [0] = j means
                that row and column j of A are the first row and column in
                PAP').  The array is used as a workspace during the ordering,
                which is why it must be of length n+1, not just n.

            double knobs [COLAMD_KNOBS] ;       Input argument.

                See colamd_set_defaults for a description.

            int stats [COLAMD_STATS] ;          Output argument.

                Statistics on the ordering, and error status.
                See colamd.h for related definitions.
                Symamd returns FALSE if stats is not present.

                stats [0]:  number of dense or empty row and columns ignored
                                (and ordered last in the output permutation
                                perm).  Note that a row/column can become
                                "empty" if it contains only "dense" and/or
                                "empty" columns/rows.

                stats [1]:  (same as stats [0])

                stats [2]:  number of garbage collections performed.

                stats [3]:  status code.  < 0 is an error code.
                            > 1 is a warning or notice.

                        0       OK.  Each column of the input matrix contained
                                row indices in increasing order, with no
                                duplicates.

                        1       OK, but columns of input matrix were jumbled
                                (unsorted columns or duplicate entries).  Symamd
                                had to do some extra work to sort the matrix
                                first and remove duplicate entries, but it
                                still was able to return a valid permutation
                                (return value of symamd was TRUE).

                                        stats [4]: highest numbered column that
                                                is unsorted or has duplicate
                                                entries.
                                        stats [5]: last seen duplicate or
                                                unsorted row index.
                                        stats [6]: number of duplicate or
                                                unsorted row indices.

                        -1      A is a null pointer

                        -2      p is a null pointer

                        -3      (unused, see colamd.c)

                        -4      n is negative

                                        stats [4]: n

                        -5      number of nonzeros in matrix is negative

                                        stats [4]: # of nonzeros (p [n]).

                        -6      p [0] is nonzero

                                        stats [4]: p [0]

                        -7      (unused)

                        -8      a column has a negative number of entries

                                        stats [4]: column with < 0 entries
                                        stats [5]: number of entries in col

                        -9      a row index is out of bounds

                                        stats [4]: column with bad row index
                                        stats [5]: bad row index
                                        stats [6]: n_row, # of rows of matrx

                        -10     out of memory (unable to allocate temporary
                                workspace for M or count arrays using the
                                "allocate" routine passed into symamd).

                Future versions may return more statistics in the stats array.

            void * (*allocate) (size_t, size_t)

                A pointer to a function providing memory allocation.  The
                allocated memory must be returned initialized to zero.  For a
                C application, this argument should normally be a pointer to
                calloc.  For a MATLAB mexFunction, the routine mxCalloc is
                passed instead.

            void (*release) (size_t, size_t)

                A pointer to a function that frees memory allocated by the
                memory allocation routine above.  For a C application, this
                argument should normally be a pointer to free.  For a MATLAB
                mexFunction, the routine mxFree is passed instead.


    ----------------------------------------------------------------------------
    colamd_report:
    ----------------------------------------------------------------------------

        C syntax:

            #include "colamd.h"
            colamd_report (int stats [COLAMD_STATS]) ;
            colamd_l_report (UF_long stats [COLAMD_STATS]) ;

        Purpose:

            Prints the error status and statistics recorded in the stats
            array on the standard error output (for a standard C routine)
            or on the MATLAB output (for a mexFunction).

        Arguments:

            int stats [COLAMD_STATS] ;  Input only.  Statistics from colamd.


    ----------------------------------------------------------------------------
    symamd_report:
    ----------------------------------------------------------------------------

        C syntax:

            #include "colamd.h"
            symamd_report (int stats [COLAMD_STATS]) ;
            symamd_l_report (UF_long stats [COLAMD_STATS]) ;

        Purpose:

            Prints the error status and statistics recorded in the stats
            array on the standard error output (for a standard C routine)
            or on the MATLAB output (for a mexFunction).

        Arguments:

            int stats [COLAMD_STATS] ;  Input only.  Statistics from symamd.


*/

/* ========================================================================== */
/* === Scaffolding code definitions  ======================================== */
/* ========================================================================== */

/* Ensure that debugging is turned off: */
#ifndef NDEBUG
#define NDEBUG
#endif

/* turn on debugging by uncommenting the following line
 #undef NDEBUG
*/

/*
   Our "scaffolding code" philosophy:  In our opinion, well-written library
   code should keep its "debugging" code, and just normally have it turned off
   by the compiler so as not to interfere with performance.  This serves
   several purposes:

   (1) assertions act as comments to the reader, telling you what the code
        expects at that point.  All assertions will always be true (unless
        there really is a bug, of course).

   (2) leaving in the scaffolding code assists anyone who would like to modify
        the code, or understand the algorithm (by reading the debugging output,
        one can get a glimpse into what the code is doing).

   (3) (gasp!) for actually finding bugs.  This code has been heavily tested
        and "should" be fully functional and bug-free ... but you never know...

    The code will become outrageously slow when debugging is
    enabled.  To control the level of debugging output, set an environment
    variable D to 0 (little), 1 (some), 2, 3, or 4 (lots).  When debugging,
    you should see the following message on the standard output:

        colamd: debug version, D = 1 (THIS WILL BE SLOW!)

    or a similar message for symamd.  If you don't, then debugging has not
    been enabled.

*/

/* ========================================================================== */
/* === Include files ======================================================== */
/* ========================================================================== */

#include "colamd.h"

#if 0 /* by mao */
#include <limits.h>
#include <math.h>

#ifdef MATLAB_MEX_FILE
#include "mex.h"
#include "matrix.h"
#endif /* MATLAB_MEX_FILE */

#if !defined (NPRINT) || !defined (NDEBUG)
#include <stdio.h>
#endif

#ifndef NULL
#define NULL ((void *) 0)
#endif
#endif

/* ========================================================================== */
/* === int or UF_long ======================================================= */
/* ========================================================================== */

#if 0 /* by mao */
/* define UF_long */
#include "UFconfig.h"
#endif

#ifdef DLONG

#define Int UF_long
#define ID  UF_long_id
#define Int_MAX UF_long_max

#define COLAMD_recommended colamd_l_recommended
#define COLAMD_set_defaults colamd_l_set_defaults
#define COLAMD_MAIN colamd_l
#define SYMAMD_MAIN symamd_l
#define COLAMD_report colamd_l_report
#define SYMAMD_report symamd_l_report

#else

#define Int int
#define ID "%d"
#define Int_MAX INT_MAX

#define COLAMD_recommended colamd_recommended
#define COLAMD_set_defaults colamd_set_defaults
#define COLAMD_MAIN colamd
#define SYMAMD_MAIN symamd
#define COLAMD_report colamd_report
#define SYMAMD_report symamd_report

#endif

/* ========================================================================== */
/* === Row and Column structures ============================================ */
/* ========================================================================== */

/* User code that makes use of the colamd/symamd routines need not directly */
/* reference these structures.  They are used only for colamd_recommended. */

typedef struct Colamd_Col_struct
{
    Int start ;         /* index for A of first row in this column, or DEAD */
                        /* if column is dead */
    Int length ;        /* number of rows in this column */
    union
    {
        Int thickness ; /* number of original columns represented by this */
                        /* col, if the column is alive */
        Int parent ;    /* parent in parent tree super-column structure, if */
                        /* the column is dead */
    } shared1 ;
    union
    {
        Int score ;     /* the score used to maintain heap, if col is alive */
        Int order ;     /* pivot ordering of this column, if col is dead */
    } shared2 ;
    union
    {
        Int headhash ;  /* head of a hash bucket, if col is at the head of */
                        /* a degree list */
        Int hash ;      /* hash value, if col is not in a degree list */
        Int prev ;      /* previous column in degree list, if col is in a */
                        /* degree list (but not at the head of a degree list) */
    } shared3 ;
    union
    {
        Int degree_next ;       /* next column, if col is in a degree list */
        Int hash_next ;         /* next column, if col is in a hash list */
    } shared4 ;

} Colamd_Col ;

typedef struct Colamd_Row_struct
{
    Int start ;         /* index for A of first col in this row */
    Int length ;        /* number of principal columns in this row */
    union
    {
        Int degree ;    /* number of principal & non-principal columns in row */
        Int p ;         /* used as a row pointer in init_rows_cols () */
    } shared1 ;
    union
    {
        Int mark ;      /* for computing set differences and marking dead rows*/
        Int first_column ;/* first column in row (used in garbage collection) */
    } shared2 ;

} Colamd_Row ;

/* ========================================================================== */
/* === Definitions ========================================================== */
/* ========================================================================== */

/* Routines are either PUBLIC (user-callable) or PRIVATE (not user-callable) */
#define PUBLIC
#define PRIVATE static

#define DENSE_DEGREE(alpha,n) \
    ((Int) MAX (16.0, (alpha) * sqrt ((double) (n))))

#define MAX(a,b) (((a) > (b)) ? (a) : (b))
#define MIN(a,b) (((a) < (b)) ? (a) : (b))

#define ONES_COMPLEMENT(r) (-(r)-1)

/* -------------------------------------------------------------------------- */
/* Change for version 2.1:  define TRUE and FALSE only if not yet defined */
/* -------------------------------------------------------------------------- */

#ifndef TRUE
#define TRUE (1)
#endif

#ifndef FALSE
#define FALSE (0)
#endif

/* -------------------------------------------------------------------------- */

#define EMPTY   (-1)

/* Row and column status */
#define ALIVE   (0)
#define DEAD    (-1)

/* Column status */
#define DEAD_PRINCIPAL          (-1)
#define DEAD_NON_PRINCIPAL      (-2)

/* Macros for row and column status update and checking. */
#define ROW_IS_DEAD(r)                  ROW_IS_MARKED_DEAD (Row[r].shared2.mark)
#define ROW_IS_MARKED_DEAD(row_mark)    (row_mark < ALIVE)
#define ROW_IS_ALIVE(r)                 (Row [r].shared2.mark >= ALIVE)
#define COL_IS_DEAD(c)                  (Col [c].start < ALIVE)
#define COL_IS_ALIVE(c)                 (Col [c].start >= ALIVE)
#define COL_IS_DEAD_PRINCIPAL(c)        (Col [c].start == DEAD_PRINCIPAL)
#define KILL_ROW(r)                     { Row [r].shared2.mark = DEAD ; }
#define KILL_PRINCIPAL_COL(c)           { Col [c].start = DEAD_PRINCIPAL ; }
#define KILL_NON_PRINCIPAL_COL(c)       { Col [c].start = DEAD_NON_PRINCIPAL ; }

/* ========================================================================== */
/* === Colamd reporting mechanism =========================================== */
/* ========================================================================== */

#if defined (MATLAB_MEX_FILE) || defined (MATHWORKS)
/* In MATLAB, matrices are 1-based to the user, but 0-based internally */
#define INDEX(i) ((i)+1)
#else
/* In C, matrices are 0-based and indices are reported as such in *_report */
#define INDEX(i) (i)
#endif

/* All output goes through the PRINTF macro.  */
#define PRINTF(params) { if (colamd_printf != NULL) (void) colamd_printf params ; }

/* ========================================================================== */
/* === Prototypes of PRIVATE routines ======================================= */
/* ========================================================================== */

PRIVATE Int init_rows_cols
(
    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A [],
    Int p [],
    Int stats [COLAMD_STATS]
) ;

PRIVATE void init_scoring
(
    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A [],
    Int head [],
    double knobs [COLAMD_KNOBS],
    Int *p_n_row2,
    Int *p_n_col2,
    Int *p_max_deg
) ;

PRIVATE Int find_ordering
(
    Int n_row,
    Int n_col,
    Int Alen,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A [],
    Int head [],
    Int n_col2,
    Int max_deg,
    Int pfree,
    Int aggressive
) ;

PRIVATE void order_children
(
    Int n_col,
    Colamd_Col Col [],
    Int p []
) ;

PRIVATE void detect_super_cols
(

#ifndef NDEBUG
    Int n_col,
    Colamd_Row Row [],
#endif /* NDEBUG */

    Colamd_Col Col [],
    Int A [],
    Int head [],
    Int row_start,
    Int row_length
) ;

PRIVATE Int garbage_collection
(
    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A [],
    Int *pfree
) ;

PRIVATE Int clear_mark
(
    Int tag_mark,
    Int max_mark,
    Int n_row,
    Colamd_Row Row []
) ;

PRIVATE void print_report
(
    char *method,
    Int stats [COLAMD_STATS]
) ;

/* ========================================================================== */
/* === Debugging prototypes and definitions ================================= */
/* ========================================================================== */

#ifndef NDEBUG

#if 0 /* by mao */
#include <assert.h>
#endif

/* colamd_debug is the *ONLY* global variable, and is only */
/* present when debugging */

PRIVATE Int colamd_debug = 0 ;  /* debug print level */

#define DEBUG0(params) { PRINTF (params) ; }
#define DEBUG1(params) { if (colamd_debug >= 1) PRINTF (params) ; }
#define DEBUG2(params) { if (colamd_debug >= 2) PRINTF (params) ; }
#define DEBUG3(params) { if (colamd_debug >= 3) PRINTF (params) ; }
#define DEBUG4(params) { if (colamd_debug >= 4) PRINTF (params) ; }

#if 0 /* by mao */
#ifdef MATLAB_MEX_FILE
#define ASSERT(expression) (mxAssert ((expression), ""))
#else
#define ASSERT(expression) (assert (expression))
#endif /* MATLAB_MEX_FILE */
#else
#define ASSERT xassert
#endif

PRIVATE void colamd_get_debug   /* gets the debug print level from getenv */
(
    char *method
) ;

PRIVATE void debug_deg_lists
(
    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int head [],
    Int min_score,
    Int should,
    Int max_deg
) ;

PRIVATE void debug_mark
(
    Int n_row,
    Colamd_Row Row [],
    Int tag_mark,
    Int max_mark
) ;

PRIVATE void debug_matrix
(
    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A []
) ;

PRIVATE void debug_structures
(
    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A [],
    Int n_col2
) ;

#else /* NDEBUG */

/* === No debugging ========================================================= */

#define DEBUG0(params) ;
#define DEBUG1(params) ;
#define DEBUG2(params) ;
#define DEBUG3(params) ;
#define DEBUG4(params) ;

#define ASSERT(expression)

#endif /* NDEBUG */

/* ========================================================================== */
/* === USER-CALLABLE ROUTINES: ============================================== */
/* ========================================================================== */

/* ========================================================================== */
/* === colamd_recommended =================================================== */
/* ========================================================================== */

/*
    The colamd_recommended routine returns the suggested size for Alen.  This
    value has been determined to provide good balance between the number of
    garbage collections and the memory requirements for colamd.  If any
    argument is negative, or if integer overflow occurs, a 0 is returned as an
    error condition.  2*nnz space is required for the row and column
    indices of the matrix. COLAMD_C (n_col) + COLAMD_R (n_row) space is
    required for the Col and Row arrays, respectively, which are internal to
    colamd (roughly 6*n_col + 4*n_row).  An additional n_col space is the
    minimal amount of "elbow room", and nnz/5 more space is recommended for
    run time efficiency.

    Alen is approximately 2.2*nnz + 7*n_col + 4*n_row + 10.

    This function is not needed when using symamd.
*/

/* add two values of type size_t, and check for integer overflow */
static size_t t_add (size_t a, size_t b, int *ok)
{
    (*ok) = (*ok) && ((a + b) >= MAX (a,b)) ;
    return ((*ok) ? (a + b) : 0) ;
}

/* compute a*k where k is a small integer, and check for integer overflow */
static size_t t_mult (size_t a, size_t k, int *ok)
{
    size_t i, s = 0 ;
    for (i = 0 ; i < k ; i++)
    {
        s = t_add (s, a, ok) ;
    }
    return (s) ;
}

/* size of the Col and Row structures */
#define COLAMD_C(n_col,ok) \
    ((t_mult (t_add (n_col, 1, ok), sizeof (Colamd_Col), ok) / sizeof (Int)))

#define COLAMD_R(n_row,ok) \
    ((t_mult (t_add (n_row, 1, ok), sizeof (Colamd_Row), ok) / sizeof (Int)))


PUBLIC size_t COLAMD_recommended        /* returns recommended value of Alen. */
(
    /* === Parameters ======================================================= */

    Int nnz,                    /* number of nonzeros in A */
    Int n_row,                  /* number of rows in A */
    Int n_col                   /* number of columns in A */
)
{
    size_t s, c, r ;
    int ok = TRUE ;
    if (nnz < 0 || n_row < 0 || n_col < 0)
    {
        return (0) ;
    }
    s = t_mult (nnz, 2, &ok) ;      /* 2*nnz */
    c = COLAMD_C (n_col, &ok) ;     /* size of column structures */
    r = COLAMD_R (n_row, &ok) ;     /* size of row structures */
    s = t_add (s, c, &ok) ;
    s = t_add (s, r, &ok) ;
    s = t_add (s, n_col, &ok) ;     /* elbow room */
    s = t_add (s, nnz/5, &ok) ;     /* elbow room */
    ok = ok && (s < Int_MAX) ;
    return (ok ? s : 0) ;
}


/* ========================================================================== */
/* === colamd_set_defaults ================================================== */
/* ========================================================================== */

/*
    The colamd_set_defaults routine sets the default values of the user-
    controllable parameters for colamd and symamd:

        Colamd: rows with more than max (16, knobs [0] * sqrt (n_col))
        entries are removed prior to ordering.  Columns with more than
        max (16, knobs [1] * sqrt (MIN (n_row,n_col))) entries are removed
        prior to ordering, and placed last in the output column ordering.

        Symamd: Rows and columns with more than max (16, knobs [0] * sqrt (n))
        entries are removed prior to ordering, and placed last in the
        output ordering.

        knobs [0]       dense row control

        knobs [1]       dense column control

        knobs [2]       if nonzero, do aggresive absorption

        knobs [3..19]   unused, but future versions might use this

*/

PUBLIC void COLAMD_set_defaults
(
    /* === Parameters ======================================================= */

    double knobs [COLAMD_KNOBS]         /* knob array */
)
{
    /* === Local variables ================================================== */

    Int i ;

    if (!knobs)
    {
        return ;                        /* no knobs to initialize */
    }
    for (i = 0 ; i < COLAMD_KNOBS ; i++)
    {
        knobs [i] = 0 ;
    }
    knobs [COLAMD_DENSE_ROW] = 10 ;
    knobs [COLAMD_DENSE_COL] = 10 ;
    knobs [COLAMD_AGGRESSIVE] = TRUE ;  /* default: do aggressive absorption*/
}


/* ========================================================================== */
/* === symamd =============================================================== */
/* ========================================================================== */

PUBLIC Int SYMAMD_MAIN                  /* return TRUE if OK, FALSE otherwise */
(
    /* === Parameters ======================================================= */

    Int n,                              /* number of rows and columns of A */
    Int A [],                           /* row indices of A */
    Int p [],                           /* column pointers of A */
    Int perm [],                        /* output permutation, size n+1 */
    double knobs [COLAMD_KNOBS],        /* parameters (uses defaults if NULL) */
    Int stats [COLAMD_STATS],           /* output statistics and error codes */
    void * (*allocate) (size_t, size_t),
                                        /* pointer to calloc (ANSI C) or */
                                        /* mxCalloc (for MATLAB mexFunction) */
    void (*release) (void *)
                                        /* pointer to free (ANSI C) or */
                                        /* mxFree (for MATLAB mexFunction) */
)
{
    /* === Local variables ================================================== */

    Int *count ;                /* length of each column of M, and col pointer*/
    Int *mark ;                 /* mark array for finding duplicate entries */
    Int *M ;                    /* row indices of matrix M */
    size_t Mlen ;               /* length of M */
    Int n_row ;                 /* number of rows in M */
    Int nnz ;                   /* number of entries in A */
    Int i ;                     /* row index of A */
    Int j ;                     /* column index of A */
    Int k ;                     /* row index of M */
    Int mnz ;                   /* number of nonzeros in M */
    Int pp ;                    /* index into a column of A */
    Int last_row ;              /* last row seen in the current column */
    Int length ;                /* number of nonzeros in a column */

    double cknobs [COLAMD_KNOBS] ;              /* knobs for colamd */
    double default_knobs [COLAMD_KNOBS] ;       /* default knobs for colamd */

#ifndef NDEBUG
    colamd_get_debug ("symamd") ;
#endif /* NDEBUG */

    /* === Check the input arguments ======================================== */

    if (!stats)
    {
        DEBUG0 (("symamd: stats not present\n")) ;
        return (FALSE) ;
    }
    for (i = 0 ; i < COLAMD_STATS ; i++)
    {
        stats [i] = 0 ;
    }
    stats [COLAMD_STATUS] = COLAMD_OK ;
    stats [COLAMD_INFO1] = -1 ;
    stats [COLAMD_INFO2] = -1 ;

    if (!A)
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_A_not_present ;
        DEBUG0 (("symamd: A not present\n")) ;
        return (FALSE) ;
    }

    if (!p)             /* p is not present */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_p_not_present ;
        DEBUG0 (("symamd: p not present\n")) ;
        return (FALSE) ;
    }

    if (n < 0)          /* n must be >= 0 */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_ncol_negative ;
        stats [COLAMD_INFO1] = n ;
        DEBUG0 (("symamd: n negative %d\n", n)) ;
        return (FALSE) ;
    }

    nnz = p [n] ;
    if (nnz < 0)        /* nnz must be >= 0 */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_nnz_negative ;
        stats [COLAMD_INFO1] = nnz ;
        DEBUG0 (("symamd: number of entries negative %d\n", nnz)) ;
        return (FALSE) ;
    }

    if (p [0] != 0)
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_p0_nonzero ;
        stats [COLAMD_INFO1] = p [0] ;
        DEBUG0 (("symamd: p[0] not zero %d\n", p [0])) ;
        return (FALSE) ;
    }

    /* === If no knobs, set default knobs =================================== */

    if (!knobs)
    {
        COLAMD_set_defaults (default_knobs) ;
        knobs = default_knobs ;
    }

    /* === Allocate count and mark ========================================== */

    count = (Int *) ((*allocate) (n+1, sizeof (Int))) ;
    if (!count)
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_out_of_memory ;
        DEBUG0 (("symamd: allocate count (size %d) failed\n", n+1)) ;
        return (FALSE) ;
    }

    mark = (Int *) ((*allocate) (n+1, sizeof (Int))) ;
    if (!mark)
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_out_of_memory ;
        (*release) ((void *) count) ;
        DEBUG0 (("symamd: allocate mark (size %d) failed\n", n+1)) ;
        return (FALSE) ;
    }

    /* === Compute column counts of M, check if A is valid ================== */

    stats [COLAMD_INFO3] = 0 ;  /* number of duplicate or unsorted row indices*/

    for (i = 0 ; i < n ; i++)
    {
        mark [i] = -1 ;
    }

    for (j = 0 ; j < n ; j++)
    {
        last_row = -1 ;

        length = p [j+1] - p [j] ;
        if (length < 0)
        {
            /* column pointers must be non-decreasing */
            stats [COLAMD_STATUS] = COLAMD_ERROR_col_length_negative ;
            stats [COLAMD_INFO1] = j ;
            stats [COLAMD_INFO2] = length ;
            (*release) ((void *) count) ;
            (*release) ((void *) mark) ;
            DEBUG0 (("symamd: col %d negative length %d\n", j, length)) ;
            return (FALSE) ;
        }

        for (pp = p [j] ; pp < p [j+1] ; pp++)
        {
            i = A [pp] ;
            if (i < 0 || i >= n)
            {
                /* row index i, in column j, is out of bounds */
                stats [COLAMD_STATUS] = COLAMD_ERROR_row_index_out_of_bounds ;
                stats [COLAMD_INFO1] = j ;
                stats [COLAMD_INFO2] = i ;
                stats [COLAMD_INFO3] = n ;
                (*release) ((void *) count) ;
                (*release) ((void *) mark) ;
                DEBUG0 (("symamd: row %d col %d out of bounds\n", i, j)) ;
                return (FALSE) ;
            }

            if (i <= last_row || mark [i] == j)
            {
                /* row index is unsorted or repeated (or both), thus col */
                /* is jumbled.  This is a notice, not an error condition. */
                stats [COLAMD_STATUS] = COLAMD_OK_BUT_JUMBLED ;
                stats [COLAMD_INFO1] = j ;
                stats [COLAMD_INFO2] = i ;
                (stats [COLAMD_INFO3]) ++ ;
                DEBUG1 (("symamd: row %d col %d unsorted/duplicate\n", i, j)) ;
            }

            if (i > j && mark [i] != j)
            {
                /* row k of M will contain column indices i and j */
                count [i]++ ;
                count [j]++ ;
            }

            /* mark the row as having been seen in this column */
            mark [i] = j ;

            last_row = i ;
        }
    }

    /* v2.4: removed free(mark) */

    /* === Compute column pointers of M ===================================== */

    /* use output permutation, perm, for column pointers of M */
    perm [0] = 0 ;
    for (j = 1 ; j <= n ; j++)
    {
        perm [j] = perm [j-1] + count [j-1] ;
    }
    for (j = 0 ; j < n ; j++)
    {
        count [j] = perm [j] ;
    }

    /* === Construct M ====================================================== */

    mnz = perm [n] ;
    n_row = mnz / 2 ;
    Mlen = COLAMD_recommended (mnz, n_row, n) ;
    M = (Int *) ((*allocate) (Mlen, sizeof (Int))) ;
    DEBUG0 (("symamd: M is %d-by-%d with %d entries, Mlen = %g\n",
        n_row, n, mnz, (double) Mlen)) ;

    if (!M)
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_out_of_memory ;
        (*release) ((void *) count) ;
        (*release) ((void *) mark) ;
        DEBUG0 (("symamd: allocate M (size %g) failed\n", (double) Mlen)) ;
        return (FALSE) ;
    }

    k = 0 ;

    if (stats [COLAMD_STATUS] == COLAMD_OK)
    {
        /* Matrix is OK */
        for (j = 0 ; j < n ; j++)
        {
            ASSERT (p [j+1] - p [j] >= 0) ;
            for (pp = p [j] ; pp < p [j+1] ; pp++)
            {
                i = A [pp] ;
                ASSERT (i >= 0 && i < n) ;
                if (i > j)
                {
                    /* row k of M contains column indices i and j */
                    M [count [i]++] = k ;
                    M [count [j]++] = k ;
                    k++ ;
                }
            }
        }
    }
    else
    {
        /* Matrix is jumbled.  Do not add duplicates to M.  Unsorted cols OK. */
        DEBUG0 (("symamd: Duplicates in A.\n")) ;
        for (i = 0 ; i < n ; i++)
        {
            mark [i] = -1 ;
        }
        for (j = 0 ; j < n ; j++)
        {
            ASSERT (p [j+1] - p [j] >= 0) ;
            for (pp = p [j] ; pp < p [j+1] ; pp++)
            {
                i = A [pp] ;
                ASSERT (i >= 0 && i < n) ;
                if (i > j && mark [i] != j)
                {
                    /* row k of M contains column indices i and j */
                    M [count [i]++] = k ;
                    M [count [j]++] = k ;
                    k++ ;
                    mark [i] = j ;
                }
            }
        }
        /* v2.4: free(mark) moved below */
    }

    /* count and mark no longer needed */
    (*release) ((void *) count) ;
    (*release) ((void *) mark) ;        /* v2.4: free (mark) moved here */
    ASSERT (k == n_row) ;

    /* === Adjust the knobs for M =========================================== */

    for (i = 0 ; i < COLAMD_KNOBS ; i++)
    {
        cknobs [i] = knobs [i] ;
    }

    /* there are no dense rows in M */
    cknobs [COLAMD_DENSE_ROW] = -1 ;
    cknobs [COLAMD_DENSE_COL] = knobs [COLAMD_DENSE_ROW] ;

    /* === Order the columns of M =========================================== */

    /* v2.4: colamd cannot fail here, so the error check is removed */
    (void) COLAMD_MAIN (n_row, n, (Int) Mlen, M, perm, cknobs, stats) ;

    /* Note that the output permutation is now in perm */

    /* === get the statistics for symamd from colamd ======================== */

    /* a dense column in colamd means a dense row and col in symamd */
    stats [COLAMD_DENSE_ROW] = stats [COLAMD_DENSE_COL] ;

    /* === Free M =========================================================== */

    (*release) ((void *) M) ;
    DEBUG0 (("symamd: done.\n")) ;
    return (TRUE) ;

}

/* ========================================================================== */
/* === colamd =============================================================== */
/* ========================================================================== */

/*
    The colamd routine computes a column ordering Q of a sparse matrix
    A such that the LU factorization P(AQ) = LU remains sparse, where P is
    selected via partial pivoting.   The routine can also be viewed as
    providing a permutation Q such that the Cholesky factorization
    (AQ)'(AQ) = LL' remains sparse.
*/

PUBLIC Int COLAMD_MAIN          /* returns TRUE if successful, FALSE otherwise*/
(
    /* === Parameters ======================================================= */

    Int n_row,                  /* number of rows in A */
    Int n_col,                  /* number of columns in A */
    Int Alen,                   /* length of A */
    Int A [],                   /* row indices of A */
    Int p [],                   /* pointers to columns in A */
    double knobs [COLAMD_KNOBS],/* parameters (uses defaults if NULL) */
    Int stats [COLAMD_STATS]    /* output statistics and error codes */
)
{
    /* === Local variables ================================================== */

    Int i ;                     /* loop index */
    Int nnz ;                   /* nonzeros in A */
    size_t Row_size ;           /* size of Row [], in integers */
    size_t Col_size ;           /* size of Col [], in integers */
    size_t need ;               /* minimum required length of A */
    Colamd_Row *Row ;           /* pointer into A of Row [0..n_row] array */
    Colamd_Col *Col ;           /* pointer into A of Col [0..n_col] array */
    Int n_col2 ;                /* number of non-dense, non-empty columns */
    Int n_row2 ;                /* number of non-dense, non-empty rows */
    Int ngarbage ;              /* number of garbage collections performed */
    Int max_deg ;               /* maximum row degree */
    double default_knobs [COLAMD_KNOBS] ;       /* default knobs array */
    Int aggressive ;            /* do aggressive absorption */
    int ok ;

#ifndef NDEBUG
    colamd_get_debug ("colamd") ;
#endif /* NDEBUG */

    /* === Check the input arguments ======================================== */

    if (!stats)
    {
        DEBUG0 (("colamd: stats not present\n")) ;
        return (FALSE) ;
    }
    for (i = 0 ; i < COLAMD_STATS ; i++)
    {
        stats [i] = 0 ;
    }
    stats [COLAMD_STATUS] = COLAMD_OK ;
    stats [COLAMD_INFO1] = -1 ;
    stats [COLAMD_INFO2] = -1 ;

    if (!A)             /* A is not present */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_A_not_present ;
        DEBUG0 (("colamd: A not present\n")) ;
        return (FALSE) ;
    }

    if (!p)             /* p is not present */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_p_not_present ;
        DEBUG0 (("colamd: p not present\n")) ;
        return (FALSE) ;
    }

    if (n_row < 0)      /* n_row must be >= 0 */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_nrow_negative ;
        stats [COLAMD_INFO1] = n_row ;
        DEBUG0 (("colamd: nrow negative %d\n", n_row)) ;
        return (FALSE) ;
    }

    if (n_col < 0)      /* n_col must be >= 0 */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_ncol_negative ;
        stats [COLAMD_INFO1] = n_col ;
        DEBUG0 (("colamd: ncol negative %d\n", n_col)) ;
        return (FALSE) ;
    }

    nnz = p [n_col] ;
    if (nnz < 0)        /* nnz must be >= 0 */
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_nnz_negative ;
        stats [COLAMD_INFO1] = nnz ;
        DEBUG0 (("colamd: number of entries negative %d\n", nnz)) ;
        return (FALSE) ;
    }

    if (p [0] != 0)
    {
        stats [COLAMD_STATUS] = COLAMD_ERROR_p0_nonzero ;
        stats [COLAMD_INFO1] = p [0] ;
        DEBUG0 (("colamd: p[0] not zero %d\n", p [0])) ;
        return (FALSE) ;
    }

    /* === If no knobs, set default knobs =================================== */

    if (!knobs)
    {
        COLAMD_set_defaults (default_knobs) ;
        knobs = default_knobs ;
    }

    aggressive = (knobs [COLAMD_AGGRESSIVE] != FALSE) ;

    /* === Allocate the Row and Col arrays from array A ===================== */

    ok = TRUE ;
    Col_size = COLAMD_C (n_col, &ok) ;      /* size of Col array of structs */
    Row_size = COLAMD_R (n_row, &ok) ;      /* size of Row array of structs */

    /* need = 2*nnz + n_col + Col_size + Row_size ; */
    need = t_mult (nnz, 2, &ok) ;
    need = t_add (need, n_col, &ok) ;
    need = t_add (need, Col_size, &ok) ;
    need = t_add (need, Row_size, &ok) ;

    if (!ok || need > (size_t) Alen || need > Int_MAX)
    {
        /* not enough space in array A to perform the ordering */
        stats [COLAMD_STATUS] = COLAMD_ERROR_A_too_small ;
        stats [COLAMD_INFO1] = need ;
        stats [COLAMD_INFO2] = Alen ;
        DEBUG0 (("colamd: Need Alen >= %d, given only Alen = %d\n", need,Alen));
        return (FALSE) ;
    }

    Alen -= Col_size + Row_size ;
    Col = (Colamd_Col *) &A [Alen] ;
    Row = (Colamd_Row *) &A [Alen + Col_size] ;

    /* === Construct the row and column data structures ===================== */

    if (!init_rows_cols (n_row, n_col, Row, Col, A, p, stats))
    {
        /* input matrix is invalid */
        DEBUG0 (("colamd: Matrix invalid\n")) ;
        return (FALSE) ;
    }

    /* === Initialize scores, kill dense rows/columns ======================= */

    init_scoring (n_row, n_col, Row, Col, A, p, knobs,
        &n_row2, &n_col2, &max_deg) ;

    /* === Order the supercolumns =========================================== */

    ngarbage = find_ordering (n_row, n_col, Alen, Row, Col, A, p,
        n_col2, max_deg, 2*nnz, aggressive) ;

    /* === Order the non-principal columns ================================== */

    order_children (n_col, Col, p) ;

    /* === Return statistics in stats ======================================= */

    stats [COLAMD_DENSE_ROW] = n_row - n_row2 ;
    stats [COLAMD_DENSE_COL] = n_col - n_col2 ;
    stats [COLAMD_DEFRAG_COUNT] = ngarbage ;
    DEBUG0 (("colamd: done.\n")) ;
    return (TRUE) ;
}


/* ========================================================================== */
/* === colamd_report ======================================================== */
/* ========================================================================== */

PUBLIC void COLAMD_report
(
    Int stats [COLAMD_STATS]
)
{
    print_report ("colamd", stats) ;
}


/* ========================================================================== */
/* === symamd_report ======================================================== */
/* ========================================================================== */

PUBLIC void SYMAMD_report
(
    Int stats [COLAMD_STATS]
)
{
    print_report ("symamd", stats) ;
}



/* ========================================================================== */
/* === NON-USER-CALLABLE ROUTINES: ========================================== */
/* ========================================================================== */

/* There are no user-callable routines beyond this point in the file */


/* ========================================================================== */
/* === init_rows_cols ======================================================= */
/* ========================================================================== */

/*
    Takes the column form of the matrix in A and creates the row form of the
    matrix.  Also, row and column attributes are stored in the Col and Row
    structs.  If the columns are un-sorted or contain duplicate row indices,
    this routine will also sort and remove duplicate row indices from the
    column form of the matrix.  Returns FALSE if the matrix is invalid,
    TRUE otherwise.  Not user-callable.
*/

PRIVATE Int init_rows_cols      /* returns TRUE if OK, or FALSE otherwise */
(
    /* === Parameters ======================================================= */

    Int n_row,                  /* number of rows of A */
    Int n_col,                  /* number of columns of A */
    Colamd_Row Row [],          /* of size n_row+1 */
    Colamd_Col Col [],          /* of size n_col+1 */
    Int A [],                   /* row indices of A, of size Alen */
    Int p [],                   /* pointers to columns in A, of size n_col+1 */
    Int stats [COLAMD_STATS]    /* colamd statistics */
)
{
    /* === Local variables ================================================== */

    Int col ;                   /* a column index */
    Int row ;                   /* a row index */
    Int *cp ;                   /* a column pointer */
    Int *cp_end ;               /* a pointer to the end of a column */
    Int *rp ;                   /* a row pointer */
    Int *rp_end ;               /* a pointer to the end of a row */
    Int last_row ;              /* previous row */

    /* === Initialize columns, and check column pointers ==================== */

    for (col = 0 ; col < n_col ; col++)
    {
        Col [col].start = p [col] ;
        Col [col].length = p [col+1] - p [col] ;

        if (Col [col].length < 0)
        {
            /* column pointers must be non-decreasing */
            stats [COLAMD_STATUS] = COLAMD_ERROR_col_length_negative ;
            stats [COLAMD_INFO1] = col ;
            stats [COLAMD_INFO2] = Col [col].length ;
            DEBUG0 (("colamd: col %d length %d < 0\n", col, Col [col].length)) ;
            return (FALSE) ;
        }

        Col [col].shared1.thickness = 1 ;
        Col [col].shared2.score = 0 ;
        Col [col].shared3.prev = EMPTY ;
        Col [col].shared4.degree_next = EMPTY ;
    }

    /* p [0..n_col] no longer needed, used as "head" in subsequent routines */

    /* === Scan columns, compute row degrees, and check row indices ========= */

    stats [COLAMD_INFO3] = 0 ;  /* number of duplicate or unsorted row indices*/

    for (row = 0 ; row < n_row ; row++)
    {
        Row [row].length = 0 ;
        Row [row].shared2.mark = -1 ;
    }

    for (col = 0 ; col < n_col ; col++)
    {
        last_row = -1 ;

        cp = &A [p [col]] ;
        cp_end = &A [p [col+1]] ;

        while (cp < cp_end)
        {
            row = *cp++ ;

            /* make sure row indices within range */
            if (row < 0 || row >= n_row)
            {
                stats [COLAMD_STATUS] = COLAMD_ERROR_row_index_out_of_bounds ;
                stats [COLAMD_INFO1] = col ;
                stats [COLAMD_INFO2] = row ;
                stats [COLAMD_INFO3] = n_row ;
                DEBUG0 (("colamd: row %d col %d out of bounds\n", row, col)) ;
                return (FALSE) ;
            }

            if (row <= last_row || Row [row].shared2.mark == col)
            {
                /* row index are unsorted or repeated (or both), thus col */
                /* is jumbled.  This is a notice, not an error condition. */
                stats [COLAMD_STATUS] = COLAMD_OK_BUT_JUMBLED ;
                stats [COLAMD_INFO1] = col ;
                stats [COLAMD_INFO2] = row ;
                (stats [COLAMD_INFO3]) ++ ;
                DEBUG1 (("colamd: row %d col %d unsorted/duplicate\n",row,col));
            }

            if (Row [row].shared2.mark != col)
            {
                Row [row].length++ ;
            }
            else
            {
                /* this is a repeated entry in the column, */
                /* it will be removed */
                Col [col].length-- ;
            }

            /* mark the row as having been seen in this column */
            Row [row].shared2.mark = col ;

            last_row = row ;
        }
    }

    /* === Compute row pointers ============================================= */

    /* row form of the matrix starts directly after the column */
    /* form of matrix in A */
    Row [0].start = p [n_col] ;
    Row [0].shared1.p = Row [0].start ;
    Row [0].shared2.mark = -1 ;
    for (row = 1 ; row < n_row ; row++)
    {
        Row [row].start = Row [row-1].start + Row [row-1].length ;
        Row [row].shared1.p = Row [row].start ;
        Row [row].shared2.mark = -1 ;
    }

    /* === Create row form ================================================== */

    if (stats [COLAMD_STATUS] == COLAMD_OK_BUT_JUMBLED)
    {
        /* if cols jumbled, watch for repeated row indices */
        for (col = 0 ; col < n_col ; col++)
        {
            cp = &A [p [col]] ;
            cp_end = &A [p [col+1]] ;
            while (cp < cp_end)
            {
                row = *cp++ ;
                if (Row [row].shared2.mark != col)
                {
                    A [(Row [row].shared1.p)++] = col ;
                    Row [row].shared2.mark = col ;
                }
            }
        }
    }
    else
    {
        /* if cols not jumbled, we don't need the mark (this is faster) */
        for (col = 0 ; col < n_col ; col++)
        {
            cp = &A [p [col]] ;
            cp_end = &A [p [col+1]] ;
            while (cp < cp_end)
            {
                A [(Row [*cp++].shared1.p)++] = col ;
            }
        }
    }

    /* === Clear the row marks and set row degrees ========================== */

    for (row = 0 ; row < n_row ; row++)
    {
        Row [row].shared2.mark = 0 ;
        Row [row].shared1.degree = Row [row].length ;
    }

    /* === See if we need to re-create columns ============================== */

    if (stats [COLAMD_STATUS] == COLAMD_OK_BUT_JUMBLED)
    {
        DEBUG0 (("colamd: reconstructing column form, matrix jumbled\n")) ;

#ifndef NDEBUG
        /* make sure column lengths are correct */
        for (col = 0 ; col < n_col ; col++)
        {
            p [col] = Col [col].length ;
        }
        for (row = 0 ; row < n_row ; row++)
        {
            rp = &A [Row [row].start] ;
            rp_end = rp + Row [row].length ;
            while (rp < rp_end)
            {
                p [*rp++]-- ;
            }
        }
        for (col = 0 ; col < n_col ; col++)
        {
            ASSERT (p [col] == 0) ;
        }
        /* now p is all zero (different than when debugging is turned off) */
#endif /* NDEBUG */

        /* === Compute col pointers ========================================= */

        /* col form of the matrix starts at A [0]. */
        /* Note, we may have a gap between the col form and the row */
        /* form if there were duplicate entries, if so, it will be */
        /* removed upon the first garbage collection */
        Col [0].start = 0 ;
        p [0] = Col [0].start ;
        for (col = 1 ; col < n_col ; col++)
        {
            /* note that the lengths here are for pruned columns, i.e. */
            /* no duplicate row indices will exist for these columns */
            Col [col].start = Col [col-1].start + Col [col-1].length ;
            p [col] = Col [col].start ;
        }

        /* === Re-create col form =========================================== */

        for (row = 0 ; row < n_row ; row++)
        {
            rp = &A [Row [row].start] ;
            rp_end = rp + Row [row].length ;
            while (rp < rp_end)
            {
                A [(p [*rp++])++] = row ;
            }
        }
    }

    /* === Done.  Matrix is not (or no longer) jumbled ====================== */

    return (TRUE) ;
}


/* ========================================================================== */
/* === init_scoring ========================================================= */
/* ========================================================================== */

/*
    Kills dense or empty columns and rows, calculates an initial score for
    each column, and places all columns in the degree lists.  Not user-callable.
*/

PRIVATE void init_scoring
(
    /* === Parameters ======================================================= */

    Int n_row,                  /* number of rows of A */
    Int n_col,                  /* number of columns of A */
    Colamd_Row Row [],          /* of size n_row+1 */
    Colamd_Col Col [],          /* of size n_col+1 */
    Int A [],                   /* column form and row form of A */
    Int head [],                /* of size n_col+1 */
    double knobs [COLAMD_KNOBS],/* parameters */
    Int *p_n_row2,              /* number of non-dense, non-empty rows */
    Int *p_n_col2,              /* number of non-dense, non-empty columns */
    Int *p_max_deg              /* maximum row degree */
)
{
    /* === Local variables ================================================== */

    Int c ;                     /* a column index */
    Int r, row ;                /* a row index */
    Int *cp ;                   /* a column pointer */
    Int deg ;                   /* degree of a row or column */
    Int *cp_end ;               /* a pointer to the end of a column */
    Int *new_cp ;               /* new column pointer */
    Int col_length ;            /* length of pruned column */
    Int score ;                 /* current column score */
    Int n_col2 ;                /* number of non-dense, non-empty columns */
    Int n_row2 ;                /* number of non-dense, non-empty rows */
    Int dense_row_count ;       /* remove rows with more entries than this */
    Int dense_col_count ;       /* remove cols with more entries than this */
    Int min_score ;             /* smallest column score */
    Int max_deg ;               /* maximum row degree */
    Int next_col ;              /* Used to add to degree list.*/

#ifndef NDEBUG
    Int debug_count ;           /* debug only. */
#endif /* NDEBUG */

    /* === Extract knobs ==================================================== */

    /* Note: if knobs contains a NaN, this is undefined: */
    if (knobs [COLAMD_DENSE_ROW] < 0)
    {
        /* only remove completely dense rows */
        dense_row_count = n_col-1 ;
    }
    else
    {
        dense_row_count = DENSE_DEGREE (knobs [COLAMD_DENSE_ROW], n_col) ;
    }
    if (knobs [COLAMD_DENSE_COL] < 0)
    {
        /* only remove completely dense columns */
        dense_col_count = n_row-1 ;
    }
    else
    {
        dense_col_count =
            DENSE_DEGREE (knobs [COLAMD_DENSE_COL], MIN (n_row, n_col)) ;
    }

    DEBUG1 (("colamd: densecount: %d %d\n", dense_row_count, dense_col_count)) ;
    max_deg = 0 ;
    n_col2 = n_col ;
    n_row2 = n_row ;

    /* === Kill empty columns =============================================== */

    /* Put the empty columns at the end in their natural order, so that LU */
    /* factorization can proceed as far as possible. */
    for (c = n_col-1 ; c >= 0 ; c--)
    {
        deg = Col [c].length ;
        if (deg == 0)
        {
            /* this is a empty column, kill and order it last */
            Col [c].shared2.order = --n_col2 ;
            KILL_PRINCIPAL_COL (c) ;
        }
    }
    DEBUG1 (("colamd: null columns killed: %d\n", n_col - n_col2)) ;

    /* === Kill dense columns =============================================== */

    /* Put the dense columns at the end, in their natural order */
    for (c = n_col-1 ; c >= 0 ; c--)
    {
        /* skip any dead columns */
        if (COL_IS_DEAD (c))
        {
            continue ;
        }
        deg = Col [c].length ;
        if (deg > dense_col_count)
        {
            /* this is a dense column, kill and order it last */
            Col [c].shared2.order = --n_col2 ;
            /* decrement the row degrees */
            cp = &A [Col [c].start] ;
            cp_end = cp + Col [c].length ;
            while (cp < cp_end)
            {
                Row [*cp++].shared1.degree-- ;
            }
            KILL_PRINCIPAL_COL (c) ;
        }
    }
    DEBUG1 (("colamd: Dense and null columns killed: %d\n", n_col - n_col2)) ;

    /* === Kill dense and empty rows ======================================== */

    for (r = 0 ; r < n_row ; r++)
    {
        deg = Row [r].shared1.degree ;
        ASSERT (deg >= 0 && deg <= n_col) ;
        if (deg > dense_row_count || deg == 0)
        {
            /* kill a dense or empty row */
            KILL_ROW (r) ;
            --n_row2 ;
        }
        else
        {
            /* keep track of max degree of remaining rows */
            max_deg = MAX (max_deg, deg) ;
        }
    }
    DEBUG1 (("colamd: Dense and null rows killed: %d\n", n_row - n_row2)) ;

    /* === Compute initial column scores ==================================== */

    /* At this point the row degrees are accurate.  They reflect the number */
    /* of "live" (non-dense) columns in each row.  No empty rows exist. */
    /* Some "live" columns may contain only dead rows, however.  These are */
    /* pruned in the code below. */

    /* now find the initial matlab score for each column */
    for (c = n_col-1 ; c >= 0 ; c--)
    {
        /* skip dead column */
        if (COL_IS_DEAD (c))
        {
            continue ;
        }
        score = 0 ;
        cp = &A [Col [c].start] ;
        new_cp = cp ;
        cp_end = cp + Col [c].length ;
        while (cp < cp_end)
        {
            /* get a row */
            row = *cp++ ;
            /* skip if dead */
            if (ROW_IS_DEAD (row))
            {
                continue ;
            }
            /* compact the column */
            *new_cp++ = row ;
            /* add row's external degree */
            score += Row [row].shared1.degree - 1 ;
            /* guard against integer overflow */
            score = MIN (score, n_col) ;
        }
        /* determine pruned column length */
        col_length = (Int) (new_cp - &A [Col [c].start]) ;
        if (col_length == 0)
        {
            /* a newly-made null column (all rows in this col are "dense" */
            /* and have already been killed) */
            DEBUG2 (("Newly null killed: %d\n", c)) ;
            Col [c].shared2.order = --n_col2 ;
            KILL_PRINCIPAL_COL (c) ;
        }
        else
        {
            /* set column length and set score */
            ASSERT (score >= 0) ;
            ASSERT (score <= n_col) ;
            Col [c].length = col_length ;
            Col [c].shared2.score = score ;
        }
    }
    DEBUG1 (("colamd: Dense, null, and newly-null columns killed: %d\n",
        n_col-n_col2)) ;

    /* At this point, all empty rows and columns are dead.  All live columns */
    /* are "clean" (containing no dead rows) and simplicial (no supercolumns */
    /* yet).  Rows may contain dead columns, but all live rows contain at */
    /* least one live column. */

#ifndef NDEBUG
    debug_structures (n_row, n_col, Row, Col, A, n_col2) ;
#endif /* NDEBUG */

    /* === Initialize degree lists ========================================== */

#ifndef NDEBUG
    debug_count = 0 ;
#endif /* NDEBUG */

    /* clear the hash buckets */
    for (c = 0 ; c <= n_col ; c++)
    {
        head [c] = EMPTY ;
    }
    min_score = n_col ;
    /* place in reverse order, so low column indices are at the front */
    /* of the lists.  This is to encourage natural tie-breaking */
    for (c = n_col-1 ; c >= 0 ; c--)
    {
        /* only add principal columns to degree lists */
        if (COL_IS_ALIVE (c))
        {
            DEBUG4 (("place %d score %d minscore %d ncol %d\n",
                c, Col [c].shared2.score, min_score, n_col)) ;

            /* === Add columns score to DList =============================== */

            score = Col [c].shared2.score ;

            ASSERT (min_score >= 0) ;
            ASSERT (min_score <= n_col) ;
            ASSERT (score >= 0) ;
            ASSERT (score <= n_col) ;
            ASSERT (head [score] >= EMPTY) ;

            /* now add this column to dList at proper score location */
            next_col = head [score] ;
            Col [c].shared3.prev = EMPTY ;
            Col [c].shared4.degree_next = next_col ;

            /* if there already was a column with the same score, set its */
            /* previous pointer to this new column */
            if (next_col != EMPTY)
            {
                Col [next_col].shared3.prev = c ;
            }
            head [score] = c ;

            /* see if this score is less than current min */
            min_score = MIN (min_score, score) ;

#ifndef NDEBUG
            debug_count++ ;
#endif /* NDEBUG */

        }
    }

#ifndef NDEBUG
    DEBUG1 (("colamd: Live cols %d out of %d, non-princ: %d\n",
        debug_count, n_col, n_col-debug_count)) ;
    ASSERT (debug_count == n_col2) ;
    debug_deg_lists (n_row, n_col, Row, Col, head, min_score, n_col2, max_deg) ;
#endif /* NDEBUG */

    /* === Return number of remaining columns, and max row degree =========== */

    *p_n_col2 = n_col2 ;
    *p_n_row2 = n_row2 ;
    *p_max_deg = max_deg ;
}


/* ========================================================================== */
/* === find_ordering ======================================================== */
/* ========================================================================== */

/*
    Order the principal columns of the supercolumn form of the matrix
    (no supercolumns on input).  Uses a minimum approximate column minimum
    degree ordering method.  Not user-callable.
*/

PRIVATE Int find_ordering       /* return the number of garbage collections */
(
    /* === Parameters ======================================================= */

    Int n_row,                  /* number of rows of A */
    Int n_col,                  /* number of columns of A */
    Int Alen,                   /* size of A, 2*nnz + n_col or larger */
    Colamd_Row Row [],          /* of size n_row+1 */
    Colamd_Col Col [],          /* of size n_col+1 */
    Int A [],                   /* column form and row form of A */
    Int head [],                /* of size n_col+1 */
    Int n_col2,                 /* Remaining columns to order */
    Int max_deg,                /* Maximum row degree */
    Int pfree,                  /* index of first free slot (2*nnz on entry) */
    Int aggressive
)
{
    /* === Local variables ================================================== */

    Int k ;                     /* current pivot ordering step */
    Int pivot_col ;             /* current pivot column */
    Int *cp ;                   /* a column pointer */
    Int *rp ;                   /* a row pointer */
    Int pivot_row ;             /* current pivot row */
    Int *new_cp ;               /* modified column pointer */
    Int *new_rp ;               /* modified row pointer */
    Int pivot_row_start ;       /* pointer to start of pivot row */
    Int pivot_row_degree ;      /* number of columns in pivot row */
    Int pivot_row_length ;      /* number of supercolumns in pivot row */
    Int pivot_col_score ;       /* score of pivot column */
    Int needed_memory ;         /* free space needed for pivot row */
    Int *cp_end ;               /* pointer to the end of a column */
    Int *rp_end ;               /* pointer to the end of a row */
    Int row ;                   /* a row index */
    Int col ;                   /* a column index */
    Int max_score ;             /* maximum possible score */
    Int cur_score ;             /* score of current column */
    unsigned Int hash ;         /* hash value for supernode detection */
    Int head_column ;           /* head of hash bucket */
    Int first_col ;             /* first column in hash bucket */
    Int tag_mark ;              /* marker value for mark array */
    Int row_mark ;              /* Row [row].shared2.mark */
    Int set_difference ;        /* set difference size of row with pivot row */
    Int min_score ;             /* smallest column score */
    Int col_thickness ;         /* "thickness" (no. of columns in a supercol) */
    Int max_mark ;              /* maximum value of tag_mark */
    Int pivot_col_thickness ;   /* number of columns represented by pivot col */
    Int prev_col ;              /* Used by Dlist operations. */
    Int next_col ;              /* Used by Dlist operations. */
    Int ngarbage ;              /* number of garbage collections performed */

#ifndef NDEBUG
    Int debug_d ;               /* debug loop counter */
    Int debug_step = 0 ;        /* debug loop counter */
#endif /* NDEBUG */

    /* === Initialization and clear mark ==================================== */

    max_mark = INT_MAX - n_col ;        /* INT_MAX defined in <limits.h> */
    tag_mark = clear_mark (0, max_mark, n_row, Row) ;
    min_score = 0 ;
    ngarbage = 0 ;
    DEBUG1 (("colamd: Ordering, n_col2=%d\n", n_col2)) ;

    /* === Order the columns ================================================ */

    for (k = 0 ; k < n_col2 ; /* 'k' is incremented below */)
    {

#ifndef NDEBUG
        if (debug_step % 100 == 0)
        {
            DEBUG2 (("\n...       Step k: %d out of n_col2: %d\n", k, n_col2)) ;
        }
        else
        {
            DEBUG3 (("\n----------Step k: %d out of n_col2: %d\n", k, n_col2)) ;
        }
        debug_step++ ;
        debug_deg_lists (n_row, n_col, Row, Col, head,
                min_score, n_col2-k, max_deg) ;
        debug_matrix (n_row, n_col, Row, Col, A) ;
#endif /* NDEBUG */

        /* === Select pivot column, and order it ============================ */

        /* make sure degree list isn't empty */
        ASSERT (min_score >= 0) ;
        ASSERT (min_score <= n_col) ;
        ASSERT (head [min_score] >= EMPTY) ;

#ifndef NDEBUG
        for (debug_d = 0 ; debug_d < min_score ; debug_d++)
        {
            ASSERT (head [debug_d] == EMPTY) ;
        }
#endif /* NDEBUG */

        /* get pivot column from head of minimum degree list */
        while (head [min_score] == EMPTY && min_score < n_col)
        {
            min_score++ ;
        }
        pivot_col = head [min_score] ;
        ASSERT (pivot_col >= 0 && pivot_col <= n_col) ;
        next_col = Col [pivot_col].shared4.degree_next ;
        head [min_score] = next_col ;
        if (next_col != EMPTY)
        {
            Col [next_col].shared3.prev = EMPTY ;
        }

        ASSERT (COL_IS_ALIVE (pivot_col)) ;

        /* remember score for defrag check */
        pivot_col_score = Col [pivot_col].shared2.score ;

        /* the pivot column is the kth column in the pivot order */
        Col [pivot_col].shared2.order = k ;

        /* increment order count by column thickness */
        pivot_col_thickness = Col [pivot_col].shared1.thickness ;
        k += pivot_col_thickness ;
        ASSERT (pivot_col_thickness > 0) ;
        DEBUG3 (("Pivot col: %d thick %d\n", pivot_col, pivot_col_thickness)) ;

        /* === Garbage_collection, if necessary ============================= */

        needed_memory = MIN (pivot_col_score, n_col - k) ;
        if (pfree + needed_memory >= Alen)
        {
            pfree = garbage_collection (n_row, n_col, Row, Col, A, &A [pfree]) ;
            ngarbage++ ;
            /* after garbage collection we will have enough */
            ASSERT (pfree + needed_memory < Alen) ;
            /* garbage collection has wiped out the Row[].shared2.mark array */
            tag_mark = clear_mark (0, max_mark, n_row, Row) ;

#ifndef NDEBUG
            debug_matrix (n_row, n_col, Row, Col, A) ;
#endif /* NDEBUG */
        }

        /* === Compute pivot row pattern ==================================== */

        /* get starting location for this new merged row */
        pivot_row_start = pfree ;

        /* initialize new row counts to zero */
        pivot_row_degree = 0 ;

        /* tag pivot column as having been visited so it isn't included */
        /* in merged pivot row */
        Col [pivot_col].shared1.thickness = -pivot_col_thickness ;

        /* pivot row is the union of all rows in the pivot column pattern */
        cp = &A [Col [pivot_col].start] ;
        cp_end = cp + Col [pivot_col].length ;
        while (cp < cp_end)
        {
            /* get a row */
            row = *cp++ ;
            DEBUG4 (("Pivot col pattern %d %d\n", ROW_IS_ALIVE (row), row)) ;
            /* skip if row is dead */
            if (ROW_IS_ALIVE (row))
            {
                rp = &A [Row [row].start] ;
                rp_end = rp + Row [row].length ;
                while (rp < rp_end)
                {
                    /* get a column */
                    col = *rp++ ;
                    /* add the column, if alive and untagged */
                    col_thickness = Col [col].shared1.thickness ;
                    if (col_thickness > 0 && COL_IS_ALIVE (col))
                    {
                        /* tag column in pivot row */
                        Col [col].shared1.thickness = -col_thickness ;
                        ASSERT (pfree < Alen) ;
                        /* place column in pivot row */
                        A [pfree++] = col ;
                        pivot_row_degree += col_thickness ;
                    }
                }
            }
        }

        /* clear tag on pivot column */
        Col [pivot_col].shared1.thickness = pivot_col_thickness ;
        max_deg = MAX (max_deg, pivot_row_degree) ;

#ifndef NDEBUG
        DEBUG3 (("check2\n")) ;
        debug_mark (n_row, Row, tag_mark, max_mark) ;
#endif /* NDEBUG */

        /* === Kill all rows used to construct pivot row ==================== */

        /* also kill pivot row, temporarily */
        cp = &A [Col [pivot_col].start] ;
        cp_end = cp + Col [pivot_col].length ;
        while (cp < cp_end)
        {
            /* may be killing an already dead row */
            row = *cp++ ;
            DEBUG3 (("Kill row in pivot col: %d\n", row)) ;
            KILL_ROW (row) ;
        }

        /* === Select a row index to use as the new pivot row =============== */

        pivot_row_length = pfree - pivot_row_start ;
        if (pivot_row_length > 0)
        {
            /* pick the "pivot" row arbitrarily (first row in col) */
            pivot_row = A [Col [pivot_col].start] ;
            DEBUG3 (("Pivotal row is %d\n", pivot_row)) ;
        }
        else
        {
            /* there is no pivot row, since it is of zero length */
            pivot_row = EMPTY ;
            ASSERT (pivot_row_length == 0) ;
        }
        ASSERT (Col [pivot_col].length > 0 || pivot_row_length == 0) ;

        /* === Approximate degree computation =============================== */

        /* Here begins the computation of the approximate degree.  The column */
        /* score is the sum of the pivot row "length", plus the size of the */
        /* set differences of each row in the column minus the pattern of the */
        /* pivot row itself.  The column ("thickness") itself is also */
        /* excluded from the column score (we thus use an approximate */
        /* external degree). */

        /* The time taken by the following code (compute set differences, and */
        /* add them up) is proportional to the size of the data structure */
        /* being scanned - that is, the sum of the sizes of each column in */
        /* the pivot row.  Thus, the amortized time to compute a column score */
        /* is proportional to the size of that column (where size, in this */
        /* context, is the column "length", or the number of row indices */
        /* in that column).  The number of row indices in a column is */
        /* monotonically non-decreasing, from the length of the original */
        /* column on input to colamd. */

        /* === Compute set differences ====================================== */

        DEBUG3 (("** Computing set differences phase. **\n")) ;

        /* pivot row is currently dead - it will be revived later. */

        DEBUG3 (("Pivot row: ")) ;
        /* for each column in pivot row */
        rp = &A [pivot_row_start] ;
        rp_end = rp + pivot_row_length ;
        while (rp < rp_end)
        {
            col = *rp++ ;
            ASSERT (COL_IS_ALIVE (col) && col != pivot_col) ;
            DEBUG3 (("Col: %d\n", col)) ;

            /* clear tags used to construct pivot row pattern */
            col_thickness = -Col [col].shared1.thickness ;
            ASSERT (col_thickness > 0) ;
            Col [col].shared1.thickness = col_thickness ;

            /* === Remove column from degree list =========================== */

            cur_score = Col [col].shared2.score ;
            prev_col = Col [col].shared3.prev ;
            next_col = Col [col].shared4.degree_next ;
            ASSERT (cur_score >= 0) ;
            ASSERT (cur_score <= n_col) ;
            ASSERT (cur_score >= EMPTY) ;
            if (prev_col == EMPTY)
            {
                head [cur_score] = next_col ;
            }
            else
            {
                Col [prev_col].shared4.degree_next = next_col ;
            }
            if (next_col != EMPTY)
            {
                Col [next_col].shared3.prev = prev_col ;
            }

            /* === Scan the column ========================================== */

            cp = &A [Col [col].start] ;
            cp_end = cp + Col [col].length ;
            while (cp < cp_end)
            {
                /* get a row */
                row = *cp++ ;
                row_mark = Row [row].shared2.mark ;
                /* skip if dead */
                if (ROW_IS_MARKED_DEAD (row_mark))
                {
                    continue ;
                }
                ASSERT (row != pivot_row) ;
                set_difference = row_mark - tag_mark ;
                /* check if the row has been seen yet */
                if (set_difference < 0)
                {
                    ASSERT (Row [row].shared1.degree <= max_deg) ;
                    set_difference = Row [row].shared1.degree ;
                }
                /* subtract column thickness from this row's set difference */
                set_difference -= col_thickness ;
                ASSERT (set_difference >= 0) ;
                /* absorb this row if the set difference becomes zero */
                if (set_difference == 0 && aggressive)
                {
                    DEBUG3 (("aggressive absorption. Row: %d\n", row)) ;
                    KILL_ROW (row) ;
                }
                else
                {
                    /* save the new mark */
                    Row [row].shared2.mark = set_difference + tag_mark ;
                }
            }
        }

#ifndef NDEBUG
        debug_deg_lists (n_row, n_col, Row, Col, head,
                min_score, n_col2-k-pivot_row_degree, max_deg) ;
#endif /* NDEBUG */

        /* === Add up set differences for each column ======================= */

        DEBUG3 (("** Adding set differences phase. **\n")) ;

        /* for each column in pivot row */
        rp = &A [pivot_row_start] ;
        rp_end = rp + pivot_row_length ;
        while (rp < rp_end)
        {
            /* get a column */
            col = *rp++ ;
            ASSERT (COL_IS_ALIVE (col) && col != pivot_col) ;
            hash = 0 ;
            cur_score = 0 ;
            cp = &A [Col [col].start] ;
            /* compact the column */
            new_cp = cp ;
            cp_end = cp + Col [col].length ;

            DEBUG4 (("Adding set diffs for Col: %d.\n", col)) ;

            while (cp < cp_end)
            {
                /* get a row */
                row = *cp++ ;
                ASSERT(row >= 0 && row < n_row) ;
                row_mark = Row [row].shared2.mark ;
                /* skip if dead */
                if (ROW_IS_MARKED_DEAD (row_mark))
                {
                    DEBUG4 ((" Row %d, dead\n", row)) ;
                    continue ;
                }
                DEBUG4 ((" Row %d, set diff %d\n", row, row_mark-tag_mark));
                ASSERT (row_mark >= tag_mark) ;
                /* compact the column */
                *new_cp++ = row ;
                /* compute hash function */
                hash += row ;
                /* add set difference */
                cur_score += row_mark - tag_mark ;
                /* integer overflow... */
                cur_score = MIN (cur_score, n_col) ;
            }

            /* recompute the column's length */
            Col [col].length = (Int) (new_cp - &A [Col [col].start]) ;

            /* === Further mass elimination ================================= */

            if (Col [col].length == 0)
            {
                DEBUG4 (("further mass elimination. Col: %d\n", col)) ;
                /* nothing left but the pivot row in this column */
                KILL_PRINCIPAL_COL (col) ;
                pivot_row_degree -= Col [col].shared1.thickness ;
                ASSERT (pivot_row_degree >= 0) ;
                /* order it */
                Col [col].shared2.order = k ;
                /* increment order count by column thickness */
                k += Col [col].shared1.thickness ;
            }
            else
            {
                /* === Prepare for supercolumn detection ==================== */

                DEBUG4 (("Preparing supercol detection for Col: %d.\n", col)) ;

                /* save score so far */
                Col [col].shared2.score = cur_score ;

                /* add column to hash table, for supercolumn detection */
                hash %= n_col + 1 ;

                DEBUG4 ((" Hash = %d, n_col = %d.\n", hash, n_col)) ;
                ASSERT (((Int) hash) <= n_col) ;

                head_column = head [hash] ;
                if (head_column > EMPTY)
                {
                    /* degree list "hash" is non-empty, use prev (shared3) of */
                    /* first column in degree list as head of hash bucket */
                    first_col = Col [head_column].shared3.headhash ;
                    Col [head_column].shared3.headhash = col ;
                }
                else
                {
                    /* degree list "hash" is empty, use head as hash bucket */
                    first_col = - (head_column + 2) ;
                    head [hash] = - (col + 2) ;
                }
                Col [col].shared4.hash_next = first_col ;

                /* save hash function in Col [col].shared3.hash */
                Col [col].shared3.hash = (Int) hash ;
                ASSERT (COL_IS_ALIVE (col)) ;
            }
        }

        /* The approximate external column degree is now computed.  */

        /* === Supercolumn detection ======================================== */

        DEBUG3 (("** Supercolumn detection phase. **\n")) ;

        detect_super_cols (

#ifndef NDEBUG
                n_col, Row,
#endif /* NDEBUG */

                Col, A, head, pivot_row_start, pivot_row_length) ;

        /* === Kill the pivotal column ====================================== */

        KILL_PRINCIPAL_COL (pivot_col) ;

        /* === Clear mark =================================================== */

        tag_mark = clear_mark (tag_mark+max_deg+1, max_mark, n_row, Row) ;

#ifndef NDEBUG
        DEBUG3 (("check3\n")) ;
        debug_mark (n_row, Row, tag_mark, max_mark) ;
#endif /* NDEBUG */

        /* === Finalize the new pivot row, and column scores ================ */

        DEBUG3 (("** Finalize scores phase. **\n")) ;

        /* for each column in pivot row */
        rp = &A [pivot_row_start] ;
        /* compact the pivot row */
        new_rp = rp ;
        rp_end = rp + pivot_row_length ;
        while (rp < rp_end)
        {
            col = *rp++ ;
            /* skip dead columns */
            if (COL_IS_DEAD (col))
            {
                continue ;
            }
            *new_rp++ = col ;
            /* add new pivot row to column */
            A [Col [col].start + (Col [col].length++)] = pivot_row ;

            /* retrieve score so far and add on pivot row's degree. */
            /* (we wait until here for this in case the pivot */
            /* row's degree was reduced due to mass elimination). */
            cur_score = Col [col].shared2.score + pivot_row_degree ;

            /* calculate the max possible score as the number of */
            /* external columns minus the 'k' value minus the */
            /* columns thickness */
            max_score = n_col - k - Col [col].shared1.thickness ;

            /* make the score the external degree of the union-of-rows */
            cur_score -= Col [col].shared1.thickness ;

            /* make sure score is less or equal than the max score */
            cur_score = MIN (cur_score, max_score) ;
            ASSERT (cur_score >= 0) ;

            /* store updated score */
            Col [col].shared2.score = cur_score ;

            /* === Place column back in degree list ========================= */

            ASSERT (min_score >= 0) ;
            ASSERT (min_score <= n_col) ;
            ASSERT (cur_score >= 0) ;
            ASSERT (cur_score <= n_col) ;
            ASSERT (head [cur_score] >= EMPTY) ;
            next_col = head [cur_score] ;
            Col [col].shared4.degree_next = next_col ;
            Col [col].shared3.prev = EMPTY ;
            if (next_col != EMPTY)
            {
                Col [next_col].shared3.prev = col ;
            }
            head [cur_score] = col ;

            /* see if this score is less than current min */
            min_score = MIN (min_score, cur_score) ;

        }

#ifndef NDEBUG
        debug_deg_lists (n_row, n_col, Row, Col, head,
                min_score, n_col2-k, max_deg) ;
#endif /* NDEBUG */

        /* === Resurrect the new pivot row ================================== */

        if (pivot_row_degree > 0)
        {
            /* update pivot row length to reflect any cols that were killed */
            /* during super-col detection and mass elimination */
            Row [pivot_row].start  = pivot_row_start ;
            Row [pivot_row].length = (Int) (new_rp - &A[pivot_row_start]) ;
            ASSERT (Row [pivot_row].length > 0) ;
            Row [pivot_row].shared1.degree = pivot_row_degree ;
            Row [pivot_row].shared2.mark = 0 ;
            /* pivot row is no longer dead */

            DEBUG1 (("Resurrect Pivot_row %d deg: %d\n",
                        pivot_row, pivot_row_degree)) ;
        }
    }

    /* === All principal columns have now been ordered ====================== */

    return (ngarbage) ;
}


/* ========================================================================== */
/* === order_children ======================================================= */
/* ========================================================================== */

/*
    The find_ordering routine has ordered all of the principal columns (the
    representatives of the supercolumns).  The non-principal columns have not
    yet been ordered.  This routine orders those columns by walking up the
    parent tree (a column is a child of the column which absorbed it).  The
    final permutation vector is then placed in p [0 ... n_col-1], with p [0]
    being the first column, and p [n_col-1] being the last.  It doesn't look
    like it at first glance, but be assured that this routine takes time linear
    in the number of columns.  Although not immediately obvious, the time
    taken by this routine is O (n_col), that is, linear in the number of
    columns.  Not user-callable.
*/

PRIVATE void order_children
(
    /* === Parameters ======================================================= */

    Int n_col,                  /* number of columns of A */
    Colamd_Col Col [],          /* of size n_col+1 */
    Int p []                    /* p [0 ... n_col-1] is the column permutation*/
)
{
    /* === Local variables ================================================== */

    Int i ;                     /* loop counter for all columns */
    Int c ;                     /* column index */
    Int parent ;                /* index of column's parent */
    Int order ;                 /* column's order */

    /* === Order each non-principal column ================================== */

    for (i = 0 ; i < n_col ; i++)
    {
        /* find an un-ordered non-principal column */
        ASSERT (COL_IS_DEAD (i)) ;
        if (!COL_IS_DEAD_PRINCIPAL (i) && Col [i].shared2.order == EMPTY)
        {
            parent = i ;
            /* once found, find its principal parent */
            do
            {
                parent = Col [parent].shared1.parent ;
            } while (!COL_IS_DEAD_PRINCIPAL (parent)) ;

            /* now, order all un-ordered non-principal columns along path */
            /* to this parent.  collapse tree at the same time */
            c = i ;
            /* get order of parent */
            order = Col [parent].shared2.order ;

            do
            {
                ASSERT (Col [c].shared2.order == EMPTY) ;

                /* order this column */
                Col [c].shared2.order = order++ ;
                /* collaps tree */
                Col [c].shared1.parent = parent ;

                /* get immediate parent of this column */
                c = Col [c].shared1.parent ;

                /* continue until we hit an ordered column.  There are */
                /* guarranteed not to be anymore unordered columns */
                /* above an ordered column */
            } while (Col [c].shared2.order == EMPTY) ;

            /* re-order the super_col parent to largest order for this group */
            Col [parent].shared2.order = order ;
        }
    }

    /* === Generate the permutation ========================================= */

    for (c = 0 ; c < n_col ; c++)
    {
        p [Col [c].shared2.order] = c ;
    }
}


/* ========================================================================== */
/* === detect_super_cols ==================================================== */
/* ========================================================================== */

/*
    Detects supercolumns by finding matches between columns in the hash buckets.
    Check amongst columns in the set A [row_start ... row_start + row_length-1].
    The columns under consideration are currently *not* in the degree lists,
    and have already been placed in the hash buckets.

    The hash bucket for columns whose hash function is equal to h is stored
    as follows:

        if head [h] is >= 0, then head [h] contains a degree list, so:

                head [h] is the first column in degree bucket h.
                Col [head [h]].headhash gives the first column in hash bucket h.

        otherwise, the degree list is empty, and:

                -(head [h] + 2) is the first column in hash bucket h.

    For a column c in a hash bucket, Col [c].shared3.prev is NOT a "previous
    column" pointer.  Col [c].shared3.hash is used instead as the hash number
    for that column.  The value of Col [c].shared4.hash_next is the next column
    in the same hash bucket.

    Assuming no, or "few" hash collisions, the time taken by this routine is
    linear in the sum of the sizes (lengths) of each column whose score has
    just been computed in the approximate degree computation.
    Not user-callable.
*/

PRIVATE void detect_super_cols
(
    /* === Parameters ======================================================= */

#ifndef NDEBUG
    /* these two parameters are only needed when debugging is enabled: */
    Int n_col,                  /* number of columns of A */
    Colamd_Row Row [],          /* of size n_row+1 */
#endif /* NDEBUG */

    Colamd_Col Col [],          /* of size n_col+1 */
    Int A [],                   /* row indices of A */
    Int head [],                /* head of degree lists and hash buckets */
    Int row_start,              /* pointer to set of columns to check */
    Int row_length              /* number of columns to check */
)
{
    /* === Local variables ================================================== */

    Int hash ;                  /* hash value for a column */
    Int *rp ;                   /* pointer to a row */
    Int c ;                     /* a column index */
    Int super_c ;               /* column index of the column to absorb into */
    Int *cp1 ;                  /* column pointer for column super_c */
    Int *cp2 ;                  /* column pointer for column c */
    Int length ;                /* length of column super_c */
    Int prev_c ;                /* column preceding c in hash bucket */
    Int i ;                     /* loop counter */
    Int *rp_end ;               /* pointer to the end of the row */
    Int col ;                   /* a column index in the row to check */
    Int head_column ;           /* first column in hash bucket or degree list */
    Int first_col ;             /* first column in hash bucket */

    /* === Consider each column in the row ================================== */

    rp = &A [row_start] ;
    rp_end = rp + row_length ;
    while (rp < rp_end)
    {
        col = *rp++ ;
        if (COL_IS_DEAD (col))
        {
            continue ;
        }

        /* get hash number for this column */
        hash = Col [col].shared3.hash ;
        ASSERT (hash <= n_col) ;

        /* === Get the first column in this hash bucket ===================== */

        head_column = head [hash] ;
        if (head_column > EMPTY)
        {
            first_col = Col [head_column].shared3.headhash ;
        }
        else
        {
            first_col = - (head_column + 2) ;
        }

        /* === Consider each column in the hash bucket ====================== */

        for (super_c = first_col ; super_c != EMPTY ;
            super_c = Col [super_c].shared4.hash_next)
        {
            ASSERT (COL_IS_ALIVE (super_c)) ;
            ASSERT (Col [super_c].shared3.hash == hash) ;
            length = Col [super_c].length ;

            /* prev_c is the column preceding column c in the hash bucket */
            prev_c = super_c ;

            /* === Compare super_c with all columns after it ================ */

            for (c = Col [super_c].shared4.hash_next ;
                 c != EMPTY ; c = Col [c].shared4.hash_next)
            {
                ASSERT (c != super_c) ;
                ASSERT (COL_IS_ALIVE (c)) ;
                ASSERT (Col [c].shared3.hash == hash) ;

                /* not identical if lengths or scores are different */
                if (Col [c].length != length ||
                    Col [c].shared2.score != Col [super_c].shared2.score)
                {
                    prev_c = c ;
                    continue ;
                }

                /* compare the two columns */
                cp1 = &A [Col [super_c].start] ;
                cp2 = &A [Col [c].start] ;

                for (i = 0 ; i < length ; i++)
                {
                    /* the columns are "clean" (no dead rows) */
                    ASSERT (ROW_IS_ALIVE (*cp1))  ;
                    ASSERT (ROW_IS_ALIVE (*cp2))  ;
                    /* row indices will same order for both supercols, */
                    /* no gather scatter nessasary */
                    if (*cp1++ != *cp2++)
                    {
                        break ;
                    }
                }

                /* the two columns are different if the for-loop "broke" */
                if (i != length)
                {
                    prev_c = c ;
                    continue ;
                }

                /* === Got it!  two columns are identical =================== */

                ASSERT (Col [c].shared2.score == Col [super_c].shared2.score) ;

                Col [super_c].shared1.thickness += Col [c].shared1.thickness ;
                Col [c].shared1.parent = super_c ;
                KILL_NON_PRINCIPAL_COL (c) ;
                /* order c later, in order_children() */
                Col [c].shared2.order = EMPTY ;
                /* remove c from hash bucket */
                Col [prev_c].shared4.hash_next = Col [c].shared4.hash_next ;
            }
        }

        /* === Empty this hash bucket ======================================= */

        if (head_column > EMPTY)
        {
            /* corresponding degree list "hash" is not empty */
            Col [head_column].shared3.headhash = EMPTY ;
        }
        else
        {
            /* corresponding degree list "hash" is empty */
            head [hash] = EMPTY ;
        }
    }
}


/* ========================================================================== */
/* === garbage_collection =================================================== */
/* ========================================================================== */

/*
    Defragments and compacts columns and rows in the workspace A.  Used when
    all avaliable memory has been used while performing row merging.  Returns
    the index of the first free position in A, after garbage collection.  The
    time taken by this routine is linear is the size of the array A, which is
    itself linear in the number of nonzeros in the input matrix.
    Not user-callable.
*/

PRIVATE Int garbage_collection  /* returns the new value of pfree */
(
    /* === Parameters ======================================================= */

    Int n_row,                  /* number of rows */
    Int n_col,                  /* number of columns */
    Colamd_Row Row [],          /* row info */
    Colamd_Col Col [],          /* column info */
    Int A [],                   /* A [0 ... Alen-1] holds the matrix */
    Int *pfree                  /* &A [0] ... pfree is in use */
)
{
    /* === Local variables ================================================== */

    Int *psrc ;                 /* source pointer */
    Int *pdest ;                /* destination pointer */
    Int j ;                     /* counter */
    Int r ;                     /* a row index */
    Int c ;                     /* a column index */
    Int length ;                /* length of a row or column */

#ifndef NDEBUG
    Int debug_rows ;
    DEBUG2 (("Defrag..\n")) ;
    for (psrc = &A[0] ; psrc < pfree ; psrc++) ASSERT (*psrc >= 0) ;
    debug_rows = 0 ;
#endif /* NDEBUG */

    /* === Defragment the columns =========================================== */

    pdest = &A[0] ;
    for (c = 0 ; c < n_col ; c++)
    {
        if (COL_IS_ALIVE (c))
        {
            psrc = &A [Col [c].start] ;

            /* move and compact the column */
            ASSERT (pdest <= psrc) ;
            Col [c].start = (Int) (pdest - &A [0]) ;
            length = Col [c].length ;
            for (j = 0 ; j < length ; j++)
            {
                r = *psrc++ ;
                if (ROW_IS_ALIVE (r))
                {
                    *pdest++ = r ;
                }
            }
            Col [c].length = (Int) (pdest - &A [Col [c].start]) ;
        }
    }

    /* === Prepare to defragment the rows =================================== */

    for (r = 0 ; r < n_row ; r++)
    {
        if (ROW_IS_DEAD (r) || (Row [r].length == 0))
        {
            /* This row is already dead, or is of zero length.  Cannot compact
             * a row of zero length, so kill it.  NOTE: in the current version,
             * there are no zero-length live rows.  Kill the row (for the first
             * time, or again) just to be safe. */
            KILL_ROW (r) ;
        }
        else
        {
            /* save first column index in Row [r].shared2.first_column */
            psrc = &A [Row [r].start] ;
            Row [r].shared2.first_column = *psrc ;
            ASSERT (ROW_IS_ALIVE (r)) ;
            /* flag the start of the row with the one's complement of row */
            *psrc = ONES_COMPLEMENT (r) ;
#ifndef NDEBUG
            debug_rows++ ;
#endif /* NDEBUG */
        }
    }

    /* === Defragment the rows ============================================== */

    psrc = pdest ;
    while (psrc < pfree)
    {
        /* find a negative number ... the start of a row */
        if (*psrc++ < 0)
        {
            psrc-- ;
            /* get the row index */
            r = ONES_COMPLEMENT (*psrc) ;
            ASSERT (r >= 0 && r < n_row) ;
            /* restore first column index */
            *psrc = Row [r].shared2.first_column ;
            ASSERT (ROW_IS_ALIVE (r)) ;
            ASSERT (Row [r].length > 0) ;
            /* move and compact the row */
            ASSERT (pdest <= psrc) ;
            Row [r].start = (Int) (pdest - &A [0]) ;
            length = Row [r].length ;
            for (j = 0 ; j < length ; j++)
            {
                c = *psrc++ ;
                if (COL_IS_ALIVE (c))
                {
                    *pdest++ = c ;
                }
            }
            Row [r].length = (Int) (pdest - &A [Row [r].start]) ;
            ASSERT (Row [r].length > 0) ;
#ifndef NDEBUG
            debug_rows-- ;
#endif /* NDEBUG */
        }
    }
    /* ensure we found all the rows */
    ASSERT (debug_rows == 0) ;

    /* === Return the new value of pfree ==================================== */

    return ((Int) (pdest - &A [0])) ;
}


/* ========================================================================== */
/* === clear_mark =========================================================== */
/* ========================================================================== */

/*
    Clears the Row [].shared2.mark array, and returns the new tag_mark.
    Return value is the new tag_mark.  Not user-callable.
*/

PRIVATE Int clear_mark  /* return the new value for tag_mark */
(
    /* === Parameters ======================================================= */

    Int tag_mark,       /* new value of tag_mark */
    Int max_mark,       /* max allowed value of tag_mark */

    Int n_row,          /* number of rows in A */
    Colamd_Row Row []   /* Row [0 ... n_row-1].shared2.mark is set to zero */
)
{
    /* === Local variables ================================================== */

    Int r ;

    if (tag_mark <= 0 || tag_mark >= max_mark)
    {
        for (r = 0 ; r < n_row ; r++)
        {
            if (ROW_IS_ALIVE (r))
            {
                Row [r].shared2.mark = 0 ;
            }
        }
        tag_mark = 1 ;
    }

    return (tag_mark) ;
}


/* ========================================================================== */
/* === print_report ========================================================= */
/* ========================================================================== */

PRIVATE void print_report
(
    char *method,
    Int stats [COLAMD_STATS]
)
{

    Int i1, i2, i3 ;

    PRINTF (("\n%s version %d.%d, %s: ", method,
            COLAMD_MAIN_VERSION, COLAMD_SUB_VERSION, COLAMD_DATE)) ;

    if (!stats)
    {
        PRINTF (("No statistics available.\n")) ;
        return ;
    }

    i1 = stats [COLAMD_INFO1] ;
    i2 = stats [COLAMD_INFO2] ;
    i3 = stats [COLAMD_INFO3] ;

    if (stats [COLAMD_STATUS] >= 0)
    {
        PRINTF (("OK.  ")) ;
    }
    else
    {
        PRINTF (("ERROR.  ")) ;
    }

    switch (stats [COLAMD_STATUS])
    {

        case COLAMD_OK_BUT_JUMBLED:

            PRINTF(("Matrix has unsorted or duplicate row indices.\n")) ;

            PRINTF(("%s: number of duplicate or out-of-order row indices: %d\n",
            method, i3)) ;

            PRINTF(("%s: last seen duplicate or out-of-order row index:   %d\n",
            method, INDEX (i2))) ;

            PRINTF(("%s: last seen in column:                             %d",
            method, INDEX (i1))) ;

            /* no break - fall through to next case instead */

        case COLAMD_OK:

            PRINTF(("\n")) ;

            PRINTF(("%s: number of dense or empty rows ignored:           %d\n",
            method, stats [COLAMD_DENSE_ROW])) ;

            PRINTF(("%s: number of dense or empty columns ignored:        %d\n",
            method, stats [COLAMD_DENSE_COL])) ;

            PRINTF(("%s: number of garbage collections performed:         %d\n",
            method, stats [COLAMD_DEFRAG_COUNT])) ;
            break ;

        case COLAMD_ERROR_A_not_present:

            PRINTF(("Array A (row indices of matrix) not present.\n")) ;
            break ;

        case COLAMD_ERROR_p_not_present:

            PRINTF(("Array p (column pointers for matrix) not present.\n")) ;
            break ;

        case COLAMD_ERROR_nrow_negative:

            PRINTF(("Invalid number of rows (%d).\n", i1)) ;
            break ;

        case COLAMD_ERROR_ncol_negative:

            PRINTF(("Invalid number of columns (%d).\n", i1)) ;
            break ;

        case COLAMD_ERROR_nnz_negative:

            PRINTF(("Invalid number of nonzero entries (%d).\n", i1)) ;
            break ;

        case COLAMD_ERROR_p0_nonzero:

            PRINTF(("Invalid column pointer, p [0] = %d, must be zero.\n", i1));
            break ;

        case COLAMD_ERROR_A_too_small:

            PRINTF(("Array A too small.\n")) ;
            PRINTF(("        Need Alen >= %d, but given only Alen = %d.\n",
            i1, i2)) ;
            break ;

        case COLAMD_ERROR_col_length_negative:

            PRINTF
            (("Column %d has a negative number of nonzero entries (%d).\n",
            INDEX (i1), i2)) ;
            break ;

        case COLAMD_ERROR_row_index_out_of_bounds:

            PRINTF
            (("Row index (row %d) out of bounds (%d to %d) in column %d.\n",
            INDEX (i2), INDEX (0), INDEX (i3-1), INDEX (i1))) ;
            break ;

        case COLAMD_ERROR_out_of_memory:

            PRINTF(("Out of memory.\n")) ;
            break ;

        /* v2.4: internal-error case deleted */
    }
}




/* ========================================================================== */
/* === colamd debugging routines ============================================ */
/* ========================================================================== */

/* When debugging is disabled, the remainder of this file is ignored. */

#ifndef NDEBUG


/* ========================================================================== */
/* === debug_structures ===================================================== */
/* ========================================================================== */

/*
    At this point, all empty rows and columns are dead.  All live columns
    are "clean" (containing no dead rows) and simplicial (no supercolumns
    yet).  Rows may contain dead columns, but all live rows contain at
    least one live column.
*/

PRIVATE void debug_structures
(
    /* === Parameters ======================================================= */

    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A [],
    Int n_col2
)
{
    /* === Local variables ================================================== */

    Int i ;
    Int c ;
    Int *cp ;
    Int *cp_end ;
    Int len ;
    Int score ;
    Int r ;
    Int *rp ;
    Int *rp_end ;
    Int deg ;

    /* === Check A, Row, and Col ============================================ */

    for (c = 0 ; c < n_col ; c++)
    {
        if (COL_IS_ALIVE (c))
        {
            len = Col [c].length ;
            score = Col [c].shared2.score ;
            DEBUG4 (("initial live col %5d %5d %5d\n", c, len, score)) ;
            ASSERT (len > 0) ;
            ASSERT (score >= 0) ;
            ASSERT (Col [c].shared1.thickness == 1) ;
            cp = &A [Col [c].start] ;
            cp_end = cp + len ;
            while (cp < cp_end)
            {
                r = *cp++ ;
                ASSERT (ROW_IS_ALIVE (r)) ;
            }
        }
        else
        {
            i = Col [c].shared2.order ;
            ASSERT (i >= n_col2 && i < n_col) ;
        }
    }

    for (r = 0 ; r < n_row ; r++)
    {
        if (ROW_IS_ALIVE (r))
        {
            i = 0 ;
            len = Row [r].length ;
            deg = Row [r].shared1.degree ;
            ASSERT (len > 0) ;
            ASSERT (deg > 0) ;
            rp = &A [Row [r].start] ;
            rp_end = rp + len ;
            while (rp < rp_end)
            {
                c = *rp++ ;
                if (COL_IS_ALIVE (c))
                {
                    i++ ;
                }
            }
            ASSERT (i > 0) ;
        }
    }
}


/* ========================================================================== */
/* === debug_deg_lists ====================================================== */
/* ========================================================================== */

/*
    Prints the contents of the degree lists.  Counts the number of columns
    in the degree list and compares it to the total it should have.  Also
    checks the row degrees.
*/

PRIVATE void debug_deg_lists
(
    /* === Parameters ======================================================= */

    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int head [],
    Int min_score,
    Int should,
    Int max_deg
)
{
    /* === Local variables ================================================== */

    Int deg ;
    Int col ;
    Int have ;
    Int row ;

    /* === Check the degree lists =========================================== */

    if (n_col > 10000 && colamd_debug <= 0)
    {
        return ;
    }
    have = 0 ;
    DEBUG4 (("Degree lists: %d\n", min_score)) ;
    for (deg = 0 ; deg <= n_col ; deg++)
    {
        col = head [deg] ;
        if (col == EMPTY)
        {
            continue ;
        }
        DEBUG4 (("%d:", deg)) ;
        while (col != EMPTY)
        {
            DEBUG4 ((" %d", col)) ;
            have += Col [col].shared1.thickness ;
            ASSERT (COL_IS_ALIVE (col)) ;
            col = Col [col].shared4.degree_next ;
        }
        DEBUG4 (("\n")) ;
    }
    DEBUG4 (("should %d have %d\n", should, have)) ;
    ASSERT (should == have) ;

    /* === Check the row degrees ============================================ */

    if (n_row > 10000 && colamd_debug <= 0)
    {
        return ;
    }
    for (row = 0 ; row < n_row ; row++)
    {
        if (ROW_IS_ALIVE (row))
        {
            ASSERT (Row [row].shared1.degree <= max_deg) ;
        }
    }
}


/* ========================================================================== */
/* === debug_mark =========================================================== */
/* ========================================================================== */

/*
    Ensures that the tag_mark is less that the maximum and also ensures that
    each entry in the mark array is less than the tag mark.
*/

PRIVATE void debug_mark
(
    /* === Parameters ======================================================= */

    Int n_row,
    Colamd_Row Row [],
    Int tag_mark,
    Int max_mark
)
{
    /* === Local variables ================================================== */

    Int r ;

    /* === Check the Row marks ============================================== */

    ASSERT (tag_mark > 0 && tag_mark <= max_mark) ;
    if (n_row > 10000 && colamd_debug <= 0)
    {
        return ;
    }
    for (r = 0 ; r < n_row ; r++)
    {
        ASSERT (Row [r].shared2.mark < tag_mark) ;
    }
}


/* ========================================================================== */
/* === debug_matrix ========================================================= */
/* ========================================================================== */

/*
    Prints out the contents of the columns and the rows.
*/

PRIVATE void debug_matrix
(
    /* === Parameters ======================================================= */

    Int n_row,
    Int n_col,
    Colamd_Row Row [],
    Colamd_Col Col [],
    Int A []
)
{
    /* === Local variables ================================================== */

    Int r ;
    Int c ;
    Int *rp ;
    Int *rp_end ;
    Int *cp ;
    Int *cp_end ;

    /* === Dump the rows and columns of the matrix ========================== */

    if (colamd_debug < 3)
    {
        return ;
    }
    DEBUG3 (("DUMP MATRIX:\n")) ;
    for (r = 0 ; r < n_row ; r++)
    {
        DEBUG3 (("Row %d alive? %d\n", r, ROW_IS_ALIVE (r))) ;
        if (ROW_IS_DEAD (r))
        {
            continue ;
        }
        DEBUG3 (("start %d length %d degree %d\n",
                Row [r].start, Row [r].length, Row [r].shared1.degree)) ;
        rp = &A [Row [r].start] ;
        rp_end = rp + Row [r].length ;
        while (rp < rp_end)
        {
            c = *rp++ ;
            DEBUG4 (("  %d col %d\n", COL_IS_ALIVE (c), c)) ;
        }
    }

    for (c = 0 ; c < n_col ; c++)
    {
        DEBUG3 (("Col %d alive? %d\n", c, COL_IS_ALIVE (c))) ;
        if (COL_IS_DEAD (c))
        {
            continue ;
        }
        DEBUG3 (("start %d length %d shared1 %d shared2 %d\n",
                Col [c].start, Col [c].length,
                Col [c].shared1.thickness, Col [c].shared2.score)) ;
        cp = &A [Col [c].start] ;
        cp_end = cp + Col [c].length ;
        while (cp < cp_end)
        {
            r = *cp++ ;
            DEBUG4 (("  %d row %d\n", ROW_IS_ALIVE (r), r)) ;
        }
    }
}

PRIVATE void colamd_get_debug
(
    char *method
)
{
    FILE *f ;
    colamd_debug = 0 ;          /* no debug printing */
    f = fopen ("debug", "r") ;
    if (f == (FILE *) NULL)
    {
        colamd_debug = 0 ;
    }
    else
    {
        fscanf (f, "%d", &colamd_debug) ;
        fclose (f) ;
    }
    DEBUG0 (("%s: debug version, D = %d (THIS WILL BE SLOW!)\n",
        method, colamd_debug)) ;
}

#endif /* NDEBUG */