summaryrefslogtreecommitdiff
path: root/parallel/dop7.ps
blob: eff0ac1fcec28896a7c3af4b868e08097a839c8e (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
%!PS-Adobe-2.0
%%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
%%Title: dop7.dvi
%%Pages: 14
%%PageOrder: Ascend
%%BoundingBox: 0 0 612 792
%%DocumentFonts: Helvetica Courier Helvetica-Bold Helvetica-Oblique
%%DocumentPaperSizes: Letter
%%EndComments
%DVIPSCommandLine: dvips -o dop7.ps dop7
%DVIPSParameters: dpi=600, comments removed
%DVIPSSource:  TeX output 1998.10.27:1114
%%BeginProcSet: fix-lj4si-ps-bug.pro
/fixcurrentmatrix {
    statusdict /product known {
        statusdict /product get (HP LaserJet 4Si) eq {
            /version where {
                /version get (2011.110) eq {
                    matrix currentmatrix
                    dup dup 0 get 32768 mul round 32768 div 0 exch put
                    dup dup 3 get 32768 mul round 32768 div 3 exch put
                    systemdict /setmatrix get exec
                } if
            } if
        } if 
    } if
} bind def

/setmatrix {
    systemdict /setmatrix get exec
    fixcurrentmatrix
} bind def
%%EndProcSet
%%BeginProcSet: tex.pro
/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
/@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
/FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{
/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N
string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]
N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
/rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
/base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
.1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}
if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{
cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
/eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
/IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
{}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
a}B /bos{/SS save N}B /eos{SS restore}B end
%%EndProcSet
%%BeginFont: Helvetica
% @@psencodingfile@{
%   author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry",
%   version = "0.6",
%   date = "22 June 1996",
%   filename = "8r.enc",
%   email = "kb@@mail.tug.org",
%   address = "135 Center Hill Rd. // Plymouth, MA 02360",
%   codetable = "ISO/ASCII",
%   checksum = "119     662    4424",
%   docstring = "Encoding for TrueType or Type 1 fonts to be used with TeX."
% @}
% 
% Idea is to have all the characters normally included in Type 1 fonts
% available for typesetting. This is effectively the characters in Adobe
% Standard Encoding + ISO Latin 1 + extra characters from Lucida.
% 
% Character code assignments were made as follows:
% 
% (1) the Windows ANSI characters are almost all in their Windows ANSI
% positions, because some Windows users cannot easily reencode the
% fonts, and it makes no difference on other systems. The only Windows
% ANSI characters not available are those that make no sense for
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
% (173). quotesingle and grave are moved just because it's such an
% irritation not having them in TeX positions.
% 
% (2) Remaining characters are assigned arbitrarily to the lower part
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
% 
% (3) Y&Y Lucida Bright includes some extra text characters; in the
% hopes that other PostScript fonts, perhaps created for public
% consumption, will include them, they are included starting at 0x12.
% 
% (4) Remaining positions left undefined are for use in (hopefully)
% upward-compatible revisions, if someday more characters are generally
% available.
% 
% (5) hyphen appears twice for compatibility with both ASCII and Windows.
% 
/TeXBase1Encoding [
% 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
  /.notdef /dotaccent /fi /fl
  /fraction /hungarumlaut /Lslash /lslash
  /ogonek /ring /.notdef
  /breve /minus /.notdef 
% These are the only two remaining unencoded characters, so may as
% well include them.
  /Zcaron /zcaron 
% 0x10
 /caron /dotlessi 
% (unusual TeX characters available in, e.g., Lucida Bright)
 /dotlessj /ff /ffi /ffl 
 /.notdef /.notdef /.notdef /.notdef
 /.notdef /.notdef /.notdef /.notdef
 % very contentious; it's so painful not having quoteleft and quoteright
 % at 96 and 145 that we move the things normally found there down to here.
 /grave /quotesingle 
% 0x20 (ASCII begins)
 /space /exclam /quotedbl /numbersign
 /dollar /percent /ampersand /quoteright
 /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
% 0x30
 /zero /one /two /three /four /five /six /seven
 /eight /nine /colon /semicolon /less /equal /greater /question
% 0x40
 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
% 0x50
 /P /Q /R /S /T /U /V /W
 /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore
% 0x60
 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o
% 0x70
 /p /q /r /s /t /u /v /w
 /x /y /z /braceleft /bar /braceright /asciitilde
 /.notdef % rubout; ASCII ends
% 0x80
 /.notdef /.notdef /quotesinglbase /florin
 /quotedblbase /ellipsis /dagger /daggerdbl
 /circumflex /perthousand /Scaron /guilsinglleft
 /OE /.notdef /.notdef /.notdef
% 0x90
 /.notdef /.notdef /.notdef /quotedblleft
 /quotedblright /bullet /endash /emdash
 /tilde /trademark /scaron /guilsinglright
 /oe /.notdef /.notdef /Ydieresis
% 0xA0
 /.notdef % nobreakspace
 /exclamdown /cent /sterling
 /currency /yen /brokenbar /section
 /dieresis /copyright /ordfeminine /guillemotleft
 /logicalnot
 /hyphen % Y&Y (also at 45); Windows' softhyphen
 /registered
 /macron
% 0xD0
 /degree /plusminus /twosuperior /threesuperior
 /acute /mu /paragraph /periodcentered
 /cedilla /onesuperior /ordmasculine /guillemotright
 /onequarter /onehalf /threequarters /questiondown
% 0xC0
 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
 /Egrave /Eacute /Ecircumflex /Edieresis
 /Igrave /Iacute /Icircumflex /Idieresis
% 0xD0
 /Eth /Ntilde /Ograve /Oacute
 /Ocircumflex /Otilde /Odieresis /multiply
 /Oslash /Ugrave /Uacute /Ucircumflex
 /Udieresis /Yacute /Thorn /germandbls
% 0xE0
 /agrave /aacute /acircumflex /atilde
 /adieresis /aring /ae /ccedilla
 /egrave /eacute /ecircumflex /edieresis
 /igrave /iacute /icircumflex /idieresis
% 0xF0
 /eth /ntilde /ograve /oacute
 /ocircumflex /otilde /odieresis /divide
 /oslash /ugrave /uacute /ucircumflex
 /udieresis /yacute /thorn /ydieresis
] def
%%EndFont
%%BeginProcSet: texps.pro
TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2
index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll
exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics
exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub
dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}
ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict
end definefont 3 -1 roll makefont /setfont load]cvx def}def
/ObliqueSlant{dup sin S cos div neg}B /SlantFont{4 index mul add}def
/ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def
end
%%EndProcSet
%%BeginProcSet: special.pro
TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N
/vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen
false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B
/@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit
div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{
/CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{
10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B
/@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale
true def end /@MacSetUp{userdict /md known{userdict /md get type
/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup
length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{}
N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath
clippath mark{transform{itransform moveto}}{transform{itransform lineto}
}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{
itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{
closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39
0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N
/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1
scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get
ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip
not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0
TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR
pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1
-1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg
TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg
sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr
0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add
2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp
{pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72
div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray}
N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict
maxlength dict begin /magscale true def normalscale currentpoint TR
/psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts
/psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx
psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy
scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR
/showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict
begin /SpecialSave save N gsave normalscale currentpoint TR
@SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial
{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath
}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{
end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin}
N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{
/SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX
SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X
/startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad
yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end
%%EndProcSet
TeXDict begin 40258431 52099146 1000 600 600 (dop7.dvi)
@start /Fa 134[45 1[45 45 45 45 45 45 45 45 45 45 45
45 45 1[45 45 45 45 45 45 45 45 45 38[45 10[45 45 46[{
 TeXBase1Encoding ReEncodeFont }26 75.000000 /Courier
rf /Fb 82[25 50[37 37 1[54 37 42 21 37 25 42 42 42 42
62 17 2[17 42 42 21 42 42 37 42 42 12[46 50 54 1[50 58
54 62 42 1[37 21 54 58 46 50 54 54 50 50 6[21 42 42 1[42
42 3[42 2[21 1[21 41[37 2[{ TeXBase1Encoding ReEncodeFont }51
75.000000 /Helvetica-Oblique rf /Fc 3 50 df<7FFFFFFFFFFF80FFFFFFFFFFFFC0
FFFFFFFFFFFFC07FFFFFFFFFFF803204799641>0 D<6000000006F80000000FFC000000
1F7E0000003F3F0000007E1F800000FC0FC00001F807E00003F003F00007E001F8000FC0
00FC001F80007E003F00003F007E00001F80FC00000FC1F8000007E3F0000003F7E00000
01FFC0000000FF800000007F000000007F00000000FF80000001FFC0000003F7E0000007
E3F000000FC1F800001F80FC00003F007E00007E003F0000FC001F8001F8000FC003F000
07E007E00003F00FC00001F81F800000FC3F0000007E7E0000003FFC0000001FF8000000
0F6000000006282874A841>2 D<000FF000000007F000003FFE0000003FFE0000FFFF80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>49 D E /Fd 82[28 56[28 46 32 1[51 51 51 74 23 2[23
51 1[28 46 1[46 1[46 12[51 55 2[55 8[65 51 3[60 66[{
 TeXBase1Encoding ReEncodeFont }21 83.333336 /Helvetica-Bold
rf /Fe 7 121 df<387CFEFEFE7C3807077A8614>58 D<001F8000007FC00000F0E70003
C03F0007803F000F001F000F001F001E001F003E003E003C003E007C003E007C003E00F8
007C00F8007C00F8007C00F8007C00F000F800F000F830F000F830F000F830F001F060F0
01F0607803F060780EF0C03C1CF9801FF07F8007C01E001C1B7C9924>97
D<000FE0003FF800F81C01E00E03803E07807E0F007E1E007C3E007C3C00007C00007C00
00F80000F80000F80000F80000F00000F00000F00000F00004F0000C7800187800303C00
E01E07C00FFF0003F800171B7C991E>99 D<000FC0007FF000F03803C01C07801C0F001C
1F001C1E001C3E00387C00707C07E07FFF80FFFC00F80000F80000F80000F80000F00000
F00000F0000478000C7800183800303C00E01E07C00FFF0003F800161B7C991F>101
D<00F007C001FC1FF0031E7878061EE03C061FC01C0C1F801E0C1F001E0C3F001E183F00
1F183E001F003E001F007E001F007E003E007C003E007C003E00FC003E00FC007C00F800
7C00F8007801F800F001F800F001F801E001F803C003FC078003FE1F0003E7FC0003E1F0
0007E0000007E0000007C0000007C000000FC000000FC000000F8000001F800000FFF800
00FFF800002025809922>112 D<000FE0007FF800F03C01C00E03C01E07803E07803E07
803C0F80180FE00007FF0007FFC003FFE001FFF000FFF80007F80001F83C00F87E00787E
0078FC00F0F800F07001E07003C03C0F801FFE0007F800171B7C991F>115
D<007C03C001FF0FF007079C300E03B0780C03F0F81803E1F83003E1F83003E1F06007C0
E06007C0000007C0000007C000000F8000000F8000000F8000000F8000001F0000001F00
30381F00307C1F0060FC3E0060FC3E00C0F87E00C0F06F038070C707003F83FE001F01F8
001D1B7D9926>120 D E /Ff 5 94 df<0000600000E00001C0000380000700000E0000
1E00003C0000780000780000F00001E00001E00003C00003C00007C0000780000F80000F
00000F00001F00001E00001E00003E00003E00003E00007C00007C00007C00007C00007C
00007C0000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F8
0000F80000F80000F80000F80000F80000F80000F800007C00007C00007C00007C00007C
00007C00003E00003E00003E00001E00001E00001F00000F00000F00000F800007800007
C00003C00003C00001E00001E00000F000007800007800003C00001E00000E0000070000
03800001C00000E0000060135278BD20>40 D<C00000E000007000003800001C00000E00
000F000007800003C00003C00001E00000F00000F000007800007800007C00003C00003E
00001E00001E00001F00000F00000F00000F80000F80000F800007C00007C00007C00007
C00007C00007C00003E00003E00003E00003E00003E00003E00003E00003E00003E00003
E00003E00003E00003E00003E00003E00003E00003E00003E00007C00007C00007C00007
C00007C00007C0000F80000F80000F80000F00000F00001F00001E00001E00003E00003C
00007C0000780000780000F00000F00001E00003C00003C0000780000F00000E00001C00
00380000700000E00000C0000013527CBD20>I<00000030000000000000780000000000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>43 D<FFF8FFF8FFF8FFF8F000F000F000F000F000F000F000F000F000
F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000
F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000
F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000
F000F000F000F000F000F000F000F000F000F000F000F000FFF8FFF8FFF8FFF80D5378BD
17>91 D<FFF8FFF8FFF8FFF8007800780078007800780078007800780078007800780078
007800780078007800780078007800780078007800780078007800780078007800780078
007800780078007800780078007800780078007800780078007800780078007800780078
007800780078007800780078007800780078007800780078007800780078007800780078
007800780078007800780078007800780078FFF8FFF8FFF8FFF80D537FBD17>93
D E /Fg 206[28 49[{ TeXBase1Encoding ReEncodeFont }1
50.000000 /Helvetica rf /Fh 202[32 2[32 32 32 48[{
 TeXBase1Encoding ReEncodeFont }4 58.333336 /Helvetica
rf /Fi 12 117 df<00003FC000000000FFF800000007E07C0000000F801F0000003F00
1F800C007E000F800C00FC0007C01C01F80007E01803F00007E01807E00003E0380FE000
03F0300FC00003F0301FC00003F0703F800003F0603F800003F0E07F800003F0C07F0000
03F1C07F000003F1807F000003F380FF000003F300FE000003F700FE000003FE00FE0000
03FC00FE000003FC00FC000003F800FC000003F000FC000003F000FC000003F000FC0000
03F0007E000007F0007E00000FF0003E00001DF8183F000079F8181F0000E1F8380F8007
C0F83007E03F007CF001FFF8003FC0003FC0000F802E267DA435>11
D<00000001FC0000000007FF800000001E07C00000007801E0000000E001F0000001C000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>I<00003FF00001FFF0000F
FFE0003FC000007F000000FC000003F8000007F0000007E000000FE000001FC000001FC0
00003F8000003F8000007FFFFE007FFFFF007FFFFF007F000000FF000000FE000000FE00
0000FE000000FE000000FE0000007E0000007E0000007E0000007E0000003E0000003F00
00001F0000000F8000600FC000E003E003C001F01F00007FFC00000FE0001C257DA322>
15 D<1C007F00FF80FF80FF80FF80FF807F001C000909798817>58
D<1C007F00FF80FF80FFC0FFC0FFC07FC01CC000C000C000C000C001C001800180038003
00070006000E001C003800700060000A19798817>I<0000000C0000001E0000003E0000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>
61 D<00000001FF00000000001FFFF000000000FE01FC00000003F0007E00000007C000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>79 D<7FFFFC01FFFFF800FFFF80FFFFFC01FFFFF800FFFF80FFFFFC01FFFFF000
FFFF8003FF000007FC00000FF80003FC000007F8000003E00001FC000007F8000003C000
01FC000003F8000003800001FC000003F8000003000001FC000003F8000007000001FC00
0003F8000006000001FC000007F800000C000001FC000007F800000C000001FC00000FF8
000018000001FC00000FF8000018000001FC00001BF8000030000001FC00001BF8000060
000001FC000033F8000060000001FE000073F80000C0000001FE000063F80000C0000000
FE0000C3F8000180000000FE0000C3FC000180000000FE000183FC000300000000FE0001
81FC000700000000FE000301FC000600000000FE000701FC000C00000000FE000601FC00
0C00000000FE000C01FC001800000000FE000C01FC001800000000FE001801FC00300000
0000FE001801FC007000000000FE003001FC006000000000FE003001FC00C000000000FE
006001FC00C000000000FF00E001FC018000000000FF00C001FC0180000000007F018001
FC0300000000007F018001FE0300000000007F030001FE0600000000007F030000FE0E00
000000007F060000FE0C00000000007F0E0000FE1800000000007F0C0000FE1800000000
007F180000FE3000000000007F180000FE3000000000007F300000FE6000000000007F30
0000FEE000000000007F600000FEC000000000007F600000FF8000000000007FC00000FF
8000000000007F800000FF0000000000007F800000FF0000000000003F000000FE000000
0000003F000000FE0000000000003E0000007C0000000000003E00000078000000000000
3C000000780000000000003C000000700000000000003800000070000000000000300000
00600000000000513B7CB84E>87 D<0003F0000001FFF0000001FFF0000001FFF0000000
07F000000007E000000007E000000007E00000000FE00000000FC00000000FC00000000F
C00000001FC00000001F800000001F800000001F800000003F800000003F000000003F00
0000003F000000007F000000007E0007C0007E001FF0007E00783800FE00E0F800FC01C1
FC00FC0383FC00FC0707FC01FC0E07FC01F81C07F801F83803F001F87001E003F8E00000
03F1C0000003F380000003F700000007FE00000007FE00000007FFE0000007E7F800000F
E0FE00000FC07F00000FC03F80000FC01F80001FC01FC0001F800FC0301F800FC0301F80
0FC0703F801FC0603F001F80603F001F80603F001F80E07F001F80C07E001F81C07E000F
81807E000F8380FE00078700FC0003FE00380000F800263B7CB92B>107
D<03E0007F000007F801FFE0000E3C0781F0001C3E1E00F800383F3800FC00303F7000FC
00303FE0007C00703FC0007C00603F80007C00603F80007C00E03F0000FC00C07F0000FC
00C07E0000FC00C07E0000FC00007E0001FC0000FE0001F80000FC0001F80000FC0001F8
0000FC0003F80001FC0003F00001F80003F00001F80007F00001F80007E00003F80007E0
0003F0000FE03003F0000FC03003F0001FC07007F0001F806007E0001F806007E0001F80
E007E0001F00C00FE0001F01C00FC0001F01800FC0001F03800FC0001F07001FC0000F0E
001F800007FC0007000001F0002C267EA432>110 D<000F8003F000001FE00FFC000039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>112 D<0001C0000003E0000007E0
000007E0000007E0000007E000000FE000000FC000000FC000000FC000001FC000001F80
00001F8000001F8000003F8000003F00007FFFFF807FFFFF80FFFFFF80007E0000007E00
00007E000000FE000000FC000000FC000000FC000001FC000001F8000001F8000001F800
0003F8000003F0000003F0000003F0000007F0000007E0000007E0000007E000000FE000
000FC006000FC006000FC00E001FC00C001F801C001F8018001F8038001F8070001F8060
001F80E0000F81C0000787800003FE000000F8000019357EB31E>116
D E /Fj 82[39 53[90 65 71 39 65 45 71 71 71 71 103 32
65 1[32 71 71 39 65 71 65 1[65 12[71 78 84 1[78 90 5[32
84 90 71 1[84 84 84 84 8[65 65 65 65 65 65 65 65 49[{
 TeXBase1Encoding ReEncodeFont }43 116.666672 /Helvetica-Bold
rf /Fk 82[28 24[28 28 24[42 42 42 60 42 46 23 42 28 46
46 46 46 69 18 42 18 18 46 46 23 46 46 42 46 46 3[23
1[23 1[55 1[78 55 60 51 55 60 65 55 65 60 69 46 55 1[23
60 65 51 55 60 60 55 55 5[23 23 46 46 46 46 46 46 46
46 46 46 1[23 1[23 1[32 28 28 18 35[42 42 2[{
 TeXBase1Encoding ReEncodeFont }74 83.333336 /Helvetica
rf /Fl 82[22 51[33 1[48 33 37 18 33 22 1[37 37 37 55
15 33 1[15 37 37 18 37 37 33 37 37 12[41 44 48 1[44 52
48 1[37 2[18 48 52 41 44 48 48 1[44 7[37 37 37 37 37
37 37 37 37 37 18 18 1[18 40[33 33 2[{ TeXBase1Encoding ReEncodeFont }
53 66.666664 /Helvetica rf /Fm 1 4 df<006000007000006000006000406020E060
70F861F07E67E01FFF8007FE0000F00007FE001FFF807E67E0F861F0E060704060200060
0000600000700000600014157B9620>3 D E /Fn 82[25 21[75
42 25[19 1[37 37 37 54 37 42 21 37 25 42 42 42 42 62
17 37 17 17 42 42 21 42 42 37 42 42 3[21 1[21 46 50 1[71
50 54 46 50 54 58 50 58 54 62 42 50 37 21 54 58 46 50
54 54 50 50 6[21 42 42 42 42 42 42 42 42 42 42 21 21
1[21 1[29 25 25 17 35[37 37 2[{ TeXBase1Encoding ReEncodeFont }77
75.000000 /Helvetica rf /Fo 139[25 42 29 14[42 46 42
31[54 65[{ TeXBase1Encoding ReEncodeFont }7 75.000000
/Helvetica-Bold rf /Fp 134[60 3[60 1[60 60 1[60 1[60
60 1[60 3[60 1[60 60 60 1[60 32[60 17[60 46[{
 TeXBase1Encoding ReEncodeFont }15 100.000000 /Courier
rf /Fq 1 4 df<000C0000001E0000001E0000001E0000001E0000001E0000601E018078
1E0780FC0C0FC07F0C3F803F8C7F0007CCF80001FFE000007F8000001E0000007F800001
FFE00007CCF8003F8C7F007F0C3F80FC0C0FC0781E0780601E0180001E0000001E000000
1E0000001E0000001E0000000C00001A1D7C9E23>3 D E /Fr 134[50
2[50 55 28 50 33 1[55 55 55 83 22 2[22 55 1[28 55 55
50 1[55 8[66 94 66 72 1[66 2[66 1[72 83 55 66 1[28 1[78
61 1[72 72 1[66 6[28 55 55 55 55 55 55 55 55 55 55 3[28
2[33 33 40[{ TeXBase1Encoding ReEncodeFont }48 100.000000
/Helvetica rf /Fs 1 4 df<00007000000000F800000000F800000000F800000000F8
00000000F800000000F800000000F800000000F80000000070000078007000F07C007001
F0FF007007F87F80700FF03FE0703FE00FF0707F8003F870FE0000FE73F800003F77E000
000FFF80000003FE00000000F800000003FE0000000FFF8000003F77E00000FE73F80003
F870FE000FF0707F803FE0703FE07F80700FF0FF007007F87C007001F078007000F00000
7000000000F800000000F800000000F800000000F800000000F800000000F800000000F8
00000000F800000000700000252B7AAD32>3 D E /Ft 82[47 50[71
4[78 39 71 47 78 78 78 78 118 31 2[31 78 1[39 78 1[71
78 78 12[86 94 2[94 110 10[102 2[94 65[{ TeXBase1Encoding ReEncodeFont }
25 141.666672 /Helvetica rf end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: Letter

%%EndSetup
%%Page: 1 1
1 0 bop 181 739 a Ft(State\255of\255the\255Art)43 b(in)d(Parallel)i
(Search)e(T)-16 b(echniques)42 b(for)e(Discrete)1264
967 y(Optimization)h(Problems)2722 915 y Fs(\003)1217
1297 y Fr(Ananth)29 b(Grama)1864 1261 y Fq(\003)1904
1297 y Fr(,)f(and)h(V)n(ipin)g(Kumar)2693 1261 y Fq(\003\003)1209
1551 y(\003)1249 1587 y Fr(Department)g(of)f(Computer)h(Sciences)1015
1733 y(Purdue)h(University)-7 b(,)27 b(W)n(est)h(Lafayette,)g(IN)f
(47907)538 1878 y Fp([email protected])p Fr(,)c(Ph:)40
b(\(765\))29 b(494)g(6964,)g(F)-5 b(AX:)28 b(\(765\))h(494)g(0739)1216
2132 y Fq(\003\003)1291 2169 y Fr(Department)g(of)f(Computer)h(Science)
924 2314 y(University)f(of)g(Minnesota,)h(Minneapolis,)h(MN)e(55455)568
2459 y Fp([email protected])p Fr(,)c(Ph:)39 b(\(612\))29
b(624)g(8023,)g(F)-5 b(AX:)28 b(\(612\))h(625)g(0572)1842
2916 y Fo(Abstract)363 3102 y Fn(Discrete)k(optimization)e(problems)h
(arise)i(in)f(a)h(variety)f(of)h(domains)f(such)h(as)g(VLSI)f(design,)j
(transportation,)251 3216 y(scheduling)18 b(and)j(management,)d(and)i
(design)g(optimization.)28 b(V)l(ery)21 b(often,)f(these)g(problems)g
(are)h(solved)f(using)g(state)251 3331 y(space)29 b(search)h
(techniques.)55 b(Due)30 b(to)g(the)f(high)g(computational)d
(requirements)i(and)h(inherent)f(parallel)f(nature)h(of)251
3445 y(search)21 b(techniques,)f(there)h(has)g(been)g(a)h(great)e(deal)
h(of)h(interest)e(in)h(the)g(development)f(of)h(parallel)e(search)j
(methods)251 3559 y(since)33 b(the)h(dawn)f(of)h(parallel)d(computing.)
68 b(Signi\002cant)32 b(advances)h(have)g(been)g(made)g(in)h(the)f(use)
h(of)g(powerful)251 3673 y(heuristics)21 b(and)g(parallel)e(processing)
i(to)h(solve)g(large)f(scale)h(discrete)f(optimization)e(problems.)33
b(Problem)21 b(instances)251 3787 y(that)j(were)g(considered)f
(computationally)f(intractable)g(only)j(a)g(few)g(years)g(back)g(are)g
(routinely)e(solved)h(currently)g(on)251 3901 y(server\255class)f
(symmetric)h(multi\255processors)d(and)h(small)h(workstation)f
(clusters.)36 b(Parallel)21 b(game\255playing)f(programs)251
4015 y(are)d(challenging)c(the)k(best)g(human)g(minds)g(at)g(games)h
(like)f(chess.)28 b(In)17 b(this)g(paper)l(,)f(we)i(describe)e(the)h
(state)g(of)g(the)g(art)g(in)251 4130 y(parallel)d(algorithms)h(used)i
(for)g(solving)f(discrete)g(optimization)f(problems.)26
b(W)o(e)16 b(address)h(heuristic)e(and)i(non\255heuristic)251
4244 y(techniques)c(for)j(searching)e(graphs)h(as)h(well)f(as)h(trees,)
g(and)f(speedup)f(anomalies)g(in)i(parallel)d(search)i(that)g(are)h
(caused)251 4358 y(by)k(the)g(inherent)e(speculative)g(nature)h(of)h
(search)g(techniques.)p 43 4427 1560 4 v 127 4481 a Fm(\003)163
4504 y Fl(This)f(work)h(is)g(sponsored)g(by)h(the)f(by)g(Army)h
(Research)f(Of)o(\002ce)h(contract)g(DA/DAAG55\25598\2551\2550441,)h
(NSF)d(Grant)i(EIA9876014,)g(and)f(by)g(Army)43 4603
y(High)30 b(Performance)h(Computing)g(Research)f(Center)h(under)f(the)g
(auspices)h(of)f(the)h(Department)g(of)g(the)f(Army)-5
b(,)35 b(Army)c(Research)g(Laboratory)43 4701 y(cooperative)20
b(agreement)g(number)g(DAAH04\25595\2552\2550003/contract)h(number)f
(DAAH04\25595\255C\2550008,)g(the)g(content)g(of)g(which)g(does)f(not)h
(necessarily)43 4800 y(re\003ect)f(the)g(position)f(or)h(the)g(policy)f
(of)h(the)g(government,)g(and)g(no)f(of)o(\002cial)g(endorsement)h
(should)g(be)f(inferred.)1970 5610 y Fk(1)p eop
%%Page: 2 2
2 1 bop 43 345 a Fj(1)116 b(Introduction)43 630 y Fk(The)21
b(generalized)h(discrete)f(optimization)i(problem)f(\(DOP\))e(deals)h
(with)h(minimizing)h(an)e(objective)h(function)g(of)f(a)g(set)g(of)43
755 y(variables)e(that)f(are)h(allowed)g(only)g(discrete)f(values.)31
b(An)19 b(instance)f(of)h(this)f(class)g(is)g(the)h(integer)g
(programming)g(problem)43 879 y(in)26 b(which)g(only)g(integer)g
(solutions)g(are)g(admissible)h(for)e(a)h(linear)h(programming)f
(\(LP\))g(model.)41 b(Other)25 b(instances)g(arise)43
1004 y(in)d(diverse)f(domains)h(such)e(as)h(VLSI)h(design)g
(\(placement)g(of)g(VLSI)g(components)f(on)h(a)f(die)h(to)g(minimize)g
(silicon)g(area\),)43 1128 y(robotics)k(\(robot)h(motion)g(planning\),)
i(and)e(logistics)g(\(visiting)g(all)g(vertices)f(of)g(a)h(graph)g
(while)g(minimizing)i(edge)e(cost\).)43 1253 y(Most)f(such)g(problems)h
(are)g(NP)f(complete;)k(hence)c(their)h(solution)h(time)f(increases)f
(exponentially)i(in)f(the)g(size)f(of)h(the)43 1377 y(problem)h(for)e
(all)i(known)f(algorithms.)44 b(One)27 b(may)f(argue)i(that)f(it)g(is)g
(pointless)g(to)g(apply)g(parallel)i(processing)d(to)h(these)43
1502 y(problems)21 b(since)g(we)g(can)f(never)h(reduce)g(their)g
(worst\255case)e(run)i(time)h(to)f(a)g(polynomial)h(without)g(using)f
(an)g(exponential)43 1626 y(number)37 b(of)f(processors.)69
b(However)-5 b(,)40 b(the)c(average\255time)h(complexity)f(of)h
(heuristic)f(search)f(algorithms)i(for)f(some)43 1751
y(problems)31 b(is)f(polynomial)i([38)q(,)e(46)q(].)54
b(Furthermore,)32 b(there)f(are)f(heuristic)h(search)e(algorithms)i
(that)g(\002nd)g(suboptimal)43 1875 y(solutions)25 b(for)g(speci\002c)f
(problems)h(in)g(polynomial)i(time.)38 b(In)25 b(such)f(cases,)g
(bigger)h(problem)h(instances)e(can)h(be)g(solved)43
2000 y(using)31 b(parallel)h(computers.)55 b(Many)30
b(DOPs)g(\(such)g(as)g(robot)h(motion)g(planning,)k(speech)30
b(understanding,)k(and)d(task)43 2124 y(scheduling\))25
b(require)f(real\255time)i(solutions.)36 b(For)24 b(these)g
(applications,)i(parallel)g(processing)e(may)g(be)h(the)f(only)h(way)f
(to)43 2249 y(obtain)j(acceptable)f(performance.)39 b(Problems)26
b(for)g(which)f(optimal)i(solutions)f(are)g(highly)g(desirable)g(can)g
(sometimes)43 2374 y(be)c(solved)g(for)f(moderate)i(sized)e(instances)g
(in)i(a)f(reasonable)g(amount)h(of)e(time)i(by)e(using)h(parallel)i
(search)d(techniques)43 2498 y(\(for)i(example,)h(VLSI)g
(\003oor\255plan)f(optimization\).)43 2683 y(Due)32 b(to)f(the)h(high)g
(computational)h(requirement)f(and)g(inherent)g(parallel)h(nature)e(of)
h(search)e(techniques,)k(there)d(has)43 2807 y(been)f(great)g(interest)
g(in)g(the)g(development)h(of)f(parallel)h(search)e(methods)h(since)f
(the)h(dawn)g(of)g(parallel)h(computing.)43 2932 y(In)24
b(fact,)g(parallel)h(search)e(programs)h(were)f(among)i(the)f(\002rst)f
(non\255numerical)i(programs)e(written)h(for)g(early)g(generation)43
3056 y(parallel)c(machines)e(such)f(as)h(HEP)h(and)f(C.mmp.)31
b(There)18 b(are)g(many)g(characteristics)f(of)h(search)f(techniques)i
(that)f(make)43 3181 y(their)28 b(parallel)i(formulations)f(very)d(dif)
o(ferent)j(from)e(those)h(of)g(traditional)i(numerical)f(algorithms,)h
(but)e(much)g(closer)f(to)43 3305 y(other)22 b(non\255numerical)g
(algorithms.)32 b(Parallel)24 b(search)c(algorithms)i(rely)f(on)h
(dynamic)f(data)h(structures,)e(which)i(makes)f(it)43
3430 y(dif)o(\002cult)j(to)h(use)f(static)h(work)f(distribution)h
(techniques.)38 b(There)24 b(is)g(potential)j(for)d(speculative)h(work)
f(both)h(in)g(serial)g(and)43 3554 y(parallel)30 b(search)e
(algorithms.)49 b(Hence)28 b(load)i(balancing)f(algorithms)h(for)e
(parallel)i(search)e(must)g(consider)g(qualitative)43
3679 y(information)34 b(as)e(well.)61 b(Some)33 b(search)e(algorithms)i
(require)g(use)f(of)g(large)h(global)h(hash)e(structures,)h(which)g
(can)f(be)43 3803 y(hard)27 b(to)g(implement)i(on)f(losely)f(coupled)h
(parallel)g(architectures.)44 b(T)m(remendous)27 b(advances)f(have)h
(been)h(made)f(both)43 3928 y(in)d(theoretical)h(and)f(applied)h
(aspects)e(of)h(parallel)h(search)e(techniques)h([11)q(].)34
b(These)23 b(techniques)h(have)g(been)g(used)g(to)43
4052 y(solve)19 b(problems)g(that)h(are)f(well)h(beyond)g(the)f(scope)g
(of)g(conventional)h(serial)g(processors.)29 b(In)19
b(this)h(paper)-5 b(,)20 b(we)g(provide)f(a)43 4177 y(brief)i(overview)
g(of)g(commonly)g(used)g(sequential)h(search)e(methods,)h(and)h
(present)e(the)i(state\255of\255the\255art)e(in)h(the)g(parallel)43
4302 y(formulations)j(of)g(various)e(search)g(techniques.)43
4701 y Fj(2)116 b(Overview)32 b(of)g(Search)f(T)-9 b(echniques)43
4986 y Fk(V)k(ery)24 b(often)h(a)f(discrete)g(optimization)i(problem)f
(is)f(formulated)h(as)f(the)h(problem)g(of)f(\002nding)h(a)f(path)g(in)
h(a)f(graph)h(from)f(a)43 5110 y(designated)j(initial)i(node)e(to)f
(one)h(of)g(several)f(possible)g(goals)h([38)q(,)f(20].)42
b(Such)27 b(a)f(graph)h(is)f(called)h(a)g(state)f(space.)42
b(In)43 5235 y(some)22 b(problems,)g(any)g(solution)h(path)f(between)g
(the)h(initial)g(node)g(and)f(a)g(goal)g(node)h(is)e(acceptable,)i
(whereas)f(in)g(other)43 5359 y(problems)f(a)g(cost)f(is)h(associated)g
(with)h(each)e(path)i(and)f(a)g(minimum)i(cost)d(solution)i(is)f
(desired.)32 b(In)21 b(many)g(applications,)1970 5610
y(2)p eop
%%Page: 3 3
3 2 bop 43 344 a Fk(it)29 b(is)g(possible)h(to)f(compute)g(a)g(lower)g
(bound)h(on)f(the)g(cost)f(of)i(a)f(solution)h(path)f(that)g(passes)f
(through)i(a)f(given)g(state.)43 469 y(This)23 b(cost)f(is)h(also)h
(called)g(a)f(heuristic)g(estimate.)43 653 y(T)-5 b(wo)31
b(commonly)g(used)f(techniques)h(for)g(searching)f(state)h(spaces)e
(are)i(depth\255\002rst)e(and)i(best\255\002rst)e(search.)54
b(Depth\255)43 778 y(\002rst)32 b(search)g(techniques)h(start)f(from)h
(an)g(initial)i(state)e(and)g(generate)h(a)f(set)f(of)i(successor)c
(states.)61 b(One)33 b(of)g(these)43 903 y(most)28 b(recently)g
(generated)g(states)g(is)g(then)g(selected)h(for)e(expansion)i(at)f
(each)g(step.)47 b(Many)28 b(variants)g(of)g(this)g(simple)43
1027 y(DFS)34 b(technique)g(exist.)64 b(In)34 b(one)g(such)f(variant,)j
(the)e(successor)d(states)i(of)h(a)g(node)g(are)g(ordered)g(on)f(the)h
(basis)g(of)43 1152 y(their)i(likelihood)h(of)f(yielding)h(a)e
(solution.)70 b(This)36 b(search)e(technique)j(is)e(referred)g(to)g(as)
g(directed\255DFS)h(or)f(ordered)43 1276 y(backtracking.)c(In)20
b(many)g(applications,)j(early)d(branches)g(explored)h(by)f(DFS)g
(techniques)h(may)f(lead)h(to)g(large)g(subtrees)43 1401
y(containing)27 b(no)f(solutions)g(when)g(solutions)h(may)e(be)h
(available)i(close)d(to)h(the)g(root)g(on)g(alternate)g(branches.)40
b(In)26 b(these)43 1525 y(cases,)33 b(it)f(is)g(useful)h(to)f(limit)h
(the)g(depth)f(in)h(terms)e(of)h(cost)g(or)f(number)i(of)f(edges,)i(to)
e(which)g(exhaustive)g(search)f(is)43 1650 y(performed.)j(If)23
b(no)h(desirable)g(solution)h(is)e(found)h(in)g(this)f(limited)j
(search,)c(the)i(limit)h(is)e(revised)g(and)h(search)f(performed)43
1774 y(again.)50 b(The)29 b(limit)h(can)f(be)g(de\002ned)g(in)g(terms)f
(of)h(tree)g(depth)g(\(depth\255bounded)h(DFS\),)e(or)h(in)g(terms)f
(of)h(solution)h(cost)43 1899 y(\(IDA*\).)39 b(In)25
b(depth\255\002rst)f(search,)h(if)h(the)g(lower)f(bound)h(of)g(a)f
(node)h(is)f(larger)g(than)h(the)g(cost)e(of)h(the)h(best)f(solution)h
(found)43 2023 y(thus)17 b(far)-5 b(,)19 b(then)e(the)h(node)f(can)g
(be)g(pruned.)31 b(This)17 b(technique)h(is)f(referred)g(to)g(as)g
(depth\255\002rst)f(branch\255and\255bound)h(\(DFBB\).)43
2208 y(In)j(all)h(algorithms)g(based)f(on)h(the)f(DFS)g(technique,)i
(once)e(a)g(successor)e(node)j Fi(n)e Fk(is)h(selected)h(for)f
(expansion,)h(its)f(sibling)43 2332 y(nodes)29 b(are)f(expanded)h(only)
g(after)g(the)f(entire)i(tree)e(rooted)h(at)g(node)g
Fi(n)f Fk(is)g(expanded,)j(even)d(if)h(successors)d(of)j
Fi(n)f Fk(are)43 2457 y(much)g(less)f(promising)h(than)g(the)g
(siblings)g(of)g Fi(n)p Fk(.)45 b(Thus,)28 b(these)g(algorithms)g(make)
g(use)f(of)h(the)g(heuristic)f(information)43 2581 y(only)21
b(on)g(a)g(local)g(scale)f(to)h(order)g(the)g(successors)d(of)i(a)h
(node.)32 b(DFBB)21 b(and)h(IDA*)f(also)g(make)f(a)h(limited)i(use)d
(of)h(heuristic)43 2706 y(information)j(on)f(a)f(global)i(scale)f(as)f
(they)g(prune)h(a)f(node)h(if)g(it)g(is)g(less)f(promising)h(than)g(a)f
(previously)h(found)g(solution)g(or)43 2830 y(bound.)34
b(As)23 b(we)h(will)g(discuss)f(later)-5 b(,)24 b(this)g(limited)h(use)
e(of)h(heuristics)f(makes)g(it)g(easy)g(to)h(develop)g(parallel)h
(formulations)43 2955 y(of)e(DFS)g(algorithms.)43 3140
y(In)h(contrast)g(to)g(DFS,)g(best\255\002rst)f(search)g(algorithms)i
(expand)f(the)h(most)f(promising)g(node)h(currently)f(available)h(on)g
(the)43 3264 y(search)i(frontier)-5 b(,)30 b(and)f(thus)e(make)h
(complete)h(use)f(of)g(the)g(available)i(heuristic)e(information.)49
b(The)28 b(search)f(frontier)h(is)43 3389 y(maintained)g(as)d(a)g
(priority)h(queue,)h(and)f(is)f(commonly)h(referred)f(to)g(as)g(the)h
(OPEN)g(list)2893 3359 y Fh(1)2930 3389 y Fk(.)39 b(A)26
b(well)h(known)e(algorithm)i(of)43 3513 y(this)i(type)g(is)g(A*,)i
(also)e(referred)f(to)h(as)g(best\255\002rst)e(branch)h(and)i(bound)f
(\(BFBB\).)g(A)h(major)f(price)g(paid)g(for)g(this)g(use)g(of)43
3638 y(heuristic)c(information)h(in)f(a)f(global)i(context)f(is)f(that)
h(BFS)g(requires)f(memory)h(proportional)h(to)e(the)h(size)f(of)h(the)g
(search)43 3762 y(frontier)-5 b(,)38 b(which)c(is)g(often)h
(exponential)h(in)e(the)h(depth)g(of)f(the)g(solution.)67
b(In)34 b(contrast,)i(all)g(DFS)e(algorithms)h(require)43
3887 y(memory)23 b(proportional)h(to)g(the)f(depth)h(of)f(the)h
(solution)g(path.)43 4071 y(In)19 b(many)g(applications,)i(identical)g
(states)d(can)h(be)g(reached)g(from)g(multiple)i(paths.)31
b(Re\255expansion)19 b(of)g(these)g(states)f(can)43 4196
y(be)26 b(avoided)f(if)h(the)g(search)e(technique)i(checks)e(every)g
(generated)i(node)f(for)g(possible)h(replications.)40
b(BFS)25 b(strategies)43 4320 y(support)d(replication)h(checking)e(by)h
(maintaining)i(a)e(list)h(of)f(all)g(the)h(expanded)f(nodes)g(\(and)g
(their)g(costs\))f(on)h(a)g(list)g(called)43 4445 y(CLOSED)384
4415 y Fh(1)421 4445 y Fk(.)31 b(DFS)19 b(based)g(techniques)g(do)h
(not)f(support)g(replication)h(checking,)g(and)f(thus)g(essentially)g
(unroll)h(the)g(graph)43 4569 y(into)28 b(a)f(tree.)44
b(Unrolling)29 b(graphs)d(into)i(trees)f(can)f(have)h(overheads)g
(ranging)h(from)f(a)g(constant)g(factor)f(to)h(exponential)43
4694 y(depending)d(on)f(the)g(structure)f(of)h(the)f(graph.)33
b(In)23 b(the)g(latter)g(case,)f(DFS)h(techniques)g(may)f(be)h
(unsuitable,)h(as)e(they)h(will)43 4819 y(need)28 b(to)g(expand)h(many)
e(more)h(nodes)g(than)g(BFS.)h(Finally)-6 b(,)29 b(simple)g(and)f
(directed)g(DFS)g(are)g(generally)g(used)g(to)g(\002nd)43
4943 y(the)e(\002rst)f(solution)i(that)f(they)g(encounter)g(in)g(the)h
(search)e(space,)h(whereas)f(DFBB,)i(IDA*,)g(and)g(BFS)f(\002nd)g(a)g
(least)g(cost)43 5068 y(solution.)34 b(This)22 b(taxonomy)h(of)h
(search)e(characteristics)g(is)h(illustrated)h(in)g(Figure)f(1.)p
43 5139 1560 4 v 131 5196 a Fg(1)163 5219 y Fl(OPEN)17
b(and)g(CLOSED)g(data)h(structures)g(are)f(called)g(lists)h(for)f
(historical)g(reasons.)26 b(In)18 b(fact,)h(OPEN)e(list)h(is)f
(typically)g(implemented)g(using)g(a)g(heap)43 5318 y(and)h(CLOSED)h
(list)f(is)h(implemented)g(as)f(a)h(hash)f(table.)1970
5610 y Fk(3)p eop
%%Page: 4 4
4 3 bop 43 344 a Fk(Although)36 b(a)e(central)g(component)h(of)f(most)g
(parallel)i(search)d(algorithms)i(is)f(a)g(dynamic)g(load)h(balancing)g
(scheme,)43 469 y(the)j(parallel)g(formulation)h(is)e(impacted)h(by)f
(the)g(nature)h(of)f(the)h(search)e(space,)k(the)e(use)f(of)g
(heuristics,)j(memory)43 593 y(requirements,)20 b(and)f(the)f(desired)h
(solution)g(quality)-6 b(.)32 b(In)18 b(the)h(following)h(sections,)f
(we)f(discuss)f(the)i(parallel)h(formulations)43 718
y(developed)k(for)f(various)g(search)f(algorithms.)205
2054 y @beginspecial 0 @llx 0 @lly 455 @urx 154 @ury
4320 @rwi @setspecial
%%BeginDocument: search.eps
%Magnification: 0.80
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {0 setgray} bind def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
-72.0 233.0 translate
1 -1 scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
n 0 792 m 0 0 l 612 0 l 612 792 l cp clip
 0.04800 0.04800 sc
7.500 slw
% Polyline
n 6600 2100 m 6675 2100 l 6675 4275 l 6600 4275 l gs col-1 s gr 
% Polyline
n 6600 4500 m 6675 4500 l 6675 4800 l 6600 4800 l gs col-1 s gr 
% Polyline
n 8400 2100 m 8475 2100 l 8475 2475 l 8400 2475 l gs col-1 s gr 
% Polyline
n 8400 2700 m 8475 2700 l 8475 3075 l 8400 3075 l gs col-1 s gr 
% Polyline
n 8400 3300 m 8475 3300 l 8475 4275 l 8400 4275 l gs col-1 s gr 
% Polyline
n 8400 4500 m 8475 4500 l 8475 4800 l 8400 4800 l gs col-1 s gr 
% Polyline
n 9900 2100 m 9975 2100 l 9975 3075 l 9900 3075 l gs col-1 s gr 
% Polyline
n 9900 3300 m 9975 3300 l 9975 4800 l 9900 4800 l gs col-1 s gr 
% Polyline
n 5025 2100 m 5100 2100 l 5100 4275 l 5025 4275 l gs col-1 s gr 
% Polyline
n 5025 4500 m 5100 4500 l 5100 4800 l 5025 4800 l gs col-1 s gr 
/Helvetica ff 180.00 scf sf
1500 2400 m
gs 1 -1 sc (Simple DFS \(Backtracking\)) col-1 sh gr
/Helvetica ff 180.00 scf sf
1500 3000 m
gs 1 -1 sc (Directed DFS \(Ordered Backtracking\)) col-1 sh gr
/Helvetica ff 180.00 scf sf
1500 3600 m
gs 1 -1 sc (Depth-First Branch and Bound) col-1 sh gr
/Helvetica ff 180.00 scf sf
1500 4200 m
gs 1 -1 sc (IDA*) col-1 sh gr
/Helvetica ff 180.00 scf sf
1500 4800 m
gs 1 -1 sc (Best-First Search / A*) col-1 sh gr
/Helvetica ff 180.00 scf sf
6975 3000 m
gs 1 -1 sc (Linear) col-1 sh gr
/Helvetica ff 180.00 scf sf
6975 3240 m
gs 1 -1 sc (in) col-1 sh gr
/Helvetica ff 180.00 scf sf
6975 3480 m
gs 1 -1 sc (Depth) col-1 sh gr
/Helvetica ff 180.00 scf sf
6975 4725 m
gs 1 -1 sc (Exponential) col-1 sh gr
/Helvetica ff 180.00 scf sf
6675 2040 m
gs 1 -1 sc (Requirement) col-1 sh gr
/Helvetica ff 180.00 scf sf
8775 2325 m
gs 1 -1 sc (None) col-1 sh gr
/Helvetica ff 180.00 scf sf
8775 2925 m
gs 1 -1 sc (Local) col-1 sh gr
/Helvetica ff 180.00 scf sf
8775 3750 m
gs 1 -1 sc (Partially) col-1 sh gr
/Helvetica ff 180.00 scf sf
8775 3990 m
gs 1 -1 sc (Global) col-1 sh gr
/Helvetica ff 180.00 scf sf
8775 4725 m
gs 1 -1 sc (Global) col-1 sh gr
/Helvetica ff 180.00 scf sf
8475 1800 m
gs 1 -1 sc (Heuristic) col-1 sh gr
/Helvetica ff 180.00 scf sf
9975 1800 m
gs 1 -1 sc (Solution) col-1 sh gr
/Helvetica ff 180.00 scf sf
9975 2040 m
gs 1 -1 sc (Quality) col-1 sh gr
/Helvetica ff 180.00 scf sf
10275 2625 m
gs 1 -1 sc (Any) col-1 sh gr
/Helvetica ff 180.00 scf sf
10275 2865 m
gs 1 -1 sc (Solution) col-1 sh gr
/Helvetica ff 180.00 scf sf
10275 4050 m
gs 1 -1 sc (Best) col-1 sh gr
/Helvetica ff 180.00 scf sf
10275 4290 m
gs 1 -1 sc (Solution) col-1 sh gr
/Helvetica ff 180.00 scf sf
6675 1800 m
gs 1 -1 sc (Memory) col-1 sh gr
/Helvetica ff 180.00 scf sf
5100 1800 m
gs 1 -1 sc (Nature of) col-1 sh gr
/Helvetica ff 180.00 scf sf
5100 2040 m
gs 1 -1 sc (Search Space) col-1 sh gr
/Helvetica ff 180.00 scf sf
5400 3075 m
gs 1 -1 sc (Tree) col-1 sh gr
/Helvetica ff 180.00 scf sf
5400 3315 m
gs 1 -1 sc (Search) col-1 sh gr
/Helvetica ff 180.00 scf sf
5400 4575 m
gs 1 -1 sc (Graph/Tree) col-1 sh gr
/Helvetica ff 180.00 scf sf
5400 4815 m
gs 1 -1 sc (Search) col-1 sh gr
$F2psEnd
rs
%%EndDocument
 @endspecial 603 2262 a(Figure)h(1:)33 b(V)-6 b(arious)23
b(enumerative)h(search)e(procedures)g(and)i(their)g(characteristics.)43
2701 y Fj(3)116 b(Depth\255First)31 b(Search)43 2986
y Fk(Depth\255\002rst)21 b(search)h(is)h(particularly)f(amenable)i(to)f
(parallel)h(processing)e(primarily)h(because)g(a)f(subtree)h(rooted)f
(at)h(any)43 3110 y(node)h(can)f(be)h(searched)f(relatively)h
(independently)h(of)e(the)h(search)f(at)g(other)h(nodes)f(or)h
(subtrees.)32 b(In)24 b(fact,)f(searching)43 3235 y(dif)o(ferent)39
b(subtrees)g(concurrently)f(requires)h(either)g(no)h(communication)g
(\(e.g.,)j(backtracking,)f(IDA*\))d(or)g(minimal)43 3359
y(communication)33 b(\(e.g.,)g(DFBB\).)f(This)f(is)h(due)g(to)f(the)h
(fact)f(that)h(DFS)g(techniques)g(do)f(not)h(make)g(use)f(of)g
(heuristics)43 3484 y(on)d(a)g(global)i(scale.)46 b(It)28
b(would)h(therefore)f(seem)g(that)h(by)e(statically)h(assigning)h
(nodes)f(in)g(a)g(tree)g(to)h(processors,)e(it)h(is)43
3608 y(possible)22 b(to)g(derive)g(parallel)i(formulations.)33
b(However)-5 b(,)22 b(for)g(most)f(applications,)j(trees)d(generated)i
(by)e(DFS)h(tend)h(to)f(be)43 3733 y(highly)g(irregular)-5
b(,)22 b(and)f(it)g(is)g(very)f(dif)o(\002cult)g(to)h(estimate)h(the)f
(work)f(associated)h(with)g(a)g(node)h(a\255priori.)31
b(Hence,)22 b(any)e(static)43 3857 y(allocation)j(of)e(subtrees)g(to)g
(processors)f(is)h(bound)h(to)f(result)g(in)h(signi\002cant)g(load)g
(imbalance)g(among)g(processors.)30 b(The)43 3982 y(core)25
b(of)g(parallel)i(formulations)g(of)e(DFS)h(algorithms)g(is)f(thus)g(a)
h(dynamic)f(load)h(balancing)h(technique)f(that)g(minimizes)43
4106 y(inter\255processor)c(communication)i(and)g(processor)d(idling.)
43 4291 y(A)26 b(number)g(of)g(load)g(distribution)h(techniques)f(have)
g(been)g(developed)g(for)g(parallel)h(DFS)e([25)q(,)g(41)q(,)g(22)q(,)g
(17)q(,)g(21)q(].)39 b(Load)43 4415 y(balancing)26 b(techniques)f(may)f
(be)h(receiver)f(initiated)j(or)d(sender)g(initiated.)39
b(In)25 b(receiver)f(initiated)i(techniques,)g(when)f(a)43
4540 y(processor)g(becomes)h(idle,)i(it)f(requests)f(a)h(selected)f
(processor)f(for)h(work.)42 b(Many)26 b(dif)o(ferent)h(selection)g
(policies)g(have)43 4665 y(been)19 b(proposed)g(such)f(as)g(use)h(of)g
(a)f(centralized)h(server)-5 b(,)19 b(random)g(polling)i([12],)f
(nearest)e(neighbor)i([25],)g(global)g(round\255)43 4789
y(robin)i([25)q(],)g(and)g(asynchronous)e(round)i(robin)g([12,)g(25].)
32 b(In)22 b(contrast)f(to)h(receiver)f(initiated)i(load)g(balancing,)g
(in)f(sender)43 4914 y(initiated)e(schemes,)e(a)g(processor)f(sends)g
(work)g(to)i(selected)f(processors)e(as)h(it)i(is)e(generated.)32
b(Selection)19 b(strategies)f(for)43 5038 y(sender)h(initiated)i(work)e
(transfers)f(include)j(hierarchical)e(techniques)h(\(with)g
(super\255servers,)d(servers)h(and)h(clients\))g([13)q(],)43
5163 y(random)k(sampling)h([25)q(],)f(and)g(near\255neighbor)h
(techniques)f([41].)33 b(The)22 b(quanta)i(of)f(work)f(farmed)h(out)h
(to)f(processors)d(in)43 5287 y(sender)27 b(and)h(receiver)e(initiated)
j(schemes)e(ranges)g(from)g(a)g(single)h(state)g(\(node)f(in)h(the)g
(tree\))e(to)i(forests)e(of)i(subtrees)1970 5610 y(4)p
eop
%%Page: 5 5
5 4 bop 43 344 a Fk(resulting)21 b(from)g(stack)f(splitting)i([12,)f
(25].)32 b(Schemes)21 b(for)f(fault)i(tolerance)f(using)g
(acknowledgements)h(have)e(also)h(been)43 469 y(built)j(into)g(load)g
(balancing)h(techniques)e([12)q(].)43 653 y(The)34 b(performance)f(and)
h(scalability)g(of)g(these)f(techniques)h(is)f(often)i(dependent)f(on)g
(the)g(underlying)g(architecture.)43 778 y(Many)e(of)g(these)f
(techniques)h(are,)i(in)f(principle)f(scalable,)j(i.e.,)f(they)e
(result)g(in)g(linear)h(speedup)f(on)g(increasing)g(the)43
903 y(number)c(of)f(processors)f Fi(p)h Fk(as)g(long)h(as)f(the)g(size)
g(of)h(the)g(search)e(space)h(grows)f(fast)i(enough)g(with)g
Fi(p)p Fk(.)45 b(It)27 b(is)g(desirable)43 1027 y(that)21
b(this)f(required)h(rate)f(of)h(growth)f(of)h(problem)g(size)f(\(also)g
(referred)g(to)h(as)f(the)g(Isoef)o(\002ciency)f(metric)h([24)q(]\))g
(be)g(as)g(small)43 1152 y(as)k(possible)h(since)g(it)g(allows)g(the)g
(use)f(of)h(a)f(larger)h(number)g(of)g(processors)d(ef)o(fectively)i
(for)h(solving)g(a)f(given)h(problem)43 1276 y(instance.)43
1461 y(The)17 b(irregular)h(and)g(dynamic)f(nature)g(of)h(the)f(search)
g(space)f(makes)h(it)h(dif)o(\002cult)f(to)g(precisely)g(compute)g(the)
h(performance)43 1585 y(and)34 b(scalability)h(of)f(parallel)i(DFS.)e
(However)-5 b(,)37 b(for)d(trees)f(with)i(branching)f(factor)g(greater)
g(than)g(1)25 b Ff(+)f Fi(\017)34 b Fk(\(for)f(a)h(small)43
1710 y(constant)28 b Fi(\017)p Fk(\),)h(it)g(is)g(often)g(possible)g
(to)f(partition)i(the)f(computation)g(associated)g(with)g(a)f
(collection)h(of)g(nodes)g(into)g(two)43 1834 y(parts)21
b(such)h(that)g(the)g(smaller)h(of)f(these)f(two)i(parts)e(can)h(be)g
(guaranteed)h(to)f(have)f(a)h(minimum)i(\002xed)d(constant)h(fraction)
43 1959 y(of)38 b(the)g(computation)h(associated)f(with)g(the)g
(original)i(node.)77 b(This)37 b(property)h(has)f(been)i(used)f(to)g
(derive)f(parallel)43 2083 y(formulations)c(of)g(DFS)f(with)h(isoef)o
(\002ciency)e(of)h Fi(O)r Ff(\()p Fi(p)14 b Fk(log)1856
2043 y Fh(2)1906 2083 y Fi(p)p Ff(\))32 b Fk(on)h(hypercube)f(and)g
Fi(O)r Ff(\()p Fi(p)2866 2053 y Fh(1)p Fe(:)p Fh(5)2969
2083 y Fk(log)15 b Fi(p)p Ff(\))32 b Fk(on)h(mesh)f(connected)43
2208 y(parallel)25 b(computers)d([25)q(].)43 2392 y(The)27
b(parallel)h(formulation)g(of)f(DFS)g(described)g(above)g(can)f(be)h
(applied)i(to)e(DFBB)g(with)g(one)g(minor)h(modi\002cation.)44
b(In)43 2517 y(DFBB,)19 b(we)g(need)h(to)f(keep)g(all)h(the)f
(processors)e(informed)j(of)f(the)g(current)f(best)h(solution.)32
b(On)19 b(a)g(shared)f(address)h(space)43 2641 y(architecture,)29
b(this)f(can)f(be)h(done)h(easily)f(by)f(maintaining)k(a)d(global)h
(best)f(solution)g(in)h(globally)g(accessible)e(memory)-6
b(.)43 2766 y(On)22 b(a)f(message)h(passing)g(computer)-5
b(,)22 b(this)g(can)g(be)g(done)g(by)g(allowing)h(each)f(processor)e
(to)i(maintain)i(the)e(current)f(best)43 2891 y(solution)33
b(known)f(to)g(it.)60 b(Whenever)32 b(a)g(processor)f(\002nds)g(a)h
(solution)h(path)g(better)f(than)h(the)f(current)f(best)h(known,)j(it)
43 3015 y(broadcasts)25 b(it)h(to)g(all)h(the)f(other)g(processors)d
(which)j(update)h(\(if)f(necessary\))d(their)j(current)g(best)f
(solution)i(path.)41 b(Note)43 3140 y(that)20 b(if)h(a)f(processor)s(')
o(s)e(current)h(best)h(solution)h(path)f(is)g(worse)g(than)g(the)g
(global)i(best)e(solution)h(path,)g(then)f(it)h(only)f(af)o(fects)43
3264 y(the)25 b(ef)o(\002ciency)f(of)h(the)h(search)e(but)h(not)h(its)f
(correctness.)36 b(The)25 b(overhead)g(for)g(maintaining)i(the)f
(current)e(best)h(solution)43 3389 y(tends)20 b(to)g(be)h(a)f(small)h
(fraction)f(of)g(the)h(overhead)f(for)g(dynamic)g(load)h(balancing.)32
b(Parallel)22 b(formulations)f(of)g(DFBB)f(have)43 3513
y(been)k(shown)f(to)g(yield)h(near)f(linear)h(speedups)f(for)g(many)g
(problems)h(and)f(architectures)g([45,)g(27,)h(8,)f(7,)g(1].)43
3839 y Fd(T)-6 b(ermination)22 b(of)h(Parallel)h(Search)43
4089 y Fk(If)30 b(only)g(one)g(solution)h(is)f(desired)g(to)g(a)g
(problem)h(instance)f(being)h(solved)e(using)i(depth\255\002rst)d
(search,)j(then)g(as)e(soon)43 4214 y(as)g(any)f(processor)g(\002nds)g
(a)h(solution,)j(search)c(at)h(other)g(processors)e(needs)i(to)g(be)h
(terminated.)51 b(On)29 b(some)f(parallel)43 4338 y(architectures,)23
b(it)h(is)g(possible)g(for)g(one)g(processor)e(to)i(interrupt)g(other)g
(processors)e(and)i(send)g(them)g(a)g(special)g(signal.)43
4463 y(If)29 b(such)f(a)h(mechanism)h(is)e(not)i(available,)i(then)d
(each)g(processor)e(needs)i(to)g(perform)g(a)g(periodic)h(check)e(to)h
(see)f(if)i(a)43 4587 y(solution)24 b(has)f(been)h(found.)43
4772 y(If)c(all)g(the)g(solutions)g(\(or)f(all)h(best)g(solutions\))f
(to)h(a)g(problem)g(instance)g(are)f(desired,)i(then)f(parallel)h(DFS)e
(searches)f(a)i(well)43 4896 y(de\002ned)28 b(portion)h(of)f(the)h
(entire)g(search)e(tree)h(\(determined)h(by)e(the)i(speci\002c)e
(algorithm)i(and)f(heuristic)h(used\).)46 b(Since)43
5021 y(this)22 b(search)e(space)h(is)g(dynamically)h(allocated)h(to)e
(processors,)f(it)i(is)f(not)h(suf)o(\002cient)e(to)i(poll)g(each)f
(processor)f(serially)i(to)43 5146 y(check)f(if)i(they)g(have)f
(exhausted)g(their)h(work.)32 b(The)22 b(reason)g(is)h(that)g(a)f
(processor)f(may)h(pick)g(up)h(work)f(after)g(it)h(has)f(been)43
5270 y(polled.)52 b(Hence)29 b(a)h(distributed)g(termination)h
(detection)f(algorithm)g(is)f(needed)i(to)e(ensure)g(that)h(all)g
(processors)d(have)1970 5610 y(5)p eop
%%Page: 6 6
6 5 bop 43 344 a Fk(indeed)24 b(exhausted)f(the)h(search)e(space)h
([5].)43 744 y Fj(4)116 b(Best\255First)31 b(Search)43
1029 y Fk(Best\255\002rst)k(search)g(techniques)i(are)f(more)g(dif)o
(\002cult)g(to)h(parallelize)h(for)e(the)h(following)h(two)e(reasons:)
58 b(\(i\))36 b(they)g(use)43 1153 y(heuristics)19 b(on)i(a)f(global)h
(scale,)f(making)h(it)f(harder)g(to)g(utilize)h(the)f(full)h(power)f
(of)g(heuristics)f(in)i(a)f(parallel)h(setting;)h(and)e(\(ii\))43
1278 y(they)h(need)h(to)g(maintain)h(the)f(CLOSED)g(list)f(if)h(the)g
(search)f(space)f(is)i(a)f(graph.)32 b(W)o(e)22 b(\002rst)e(discuss)g
(parallel)j(formulations)43 1402 y(of)g(best\255\002rst)f(search)g(in)i
(the)f(context)g(of)g(state\255space)f(trees,)h(and)h(subsequently)f
(discuss)f(the)h(case)g(for)g(graphs.)43 1587 y(BFS)f(methods)g(may)f
(appear)h(inherently)g(serial)g(since)f(they)g(require)h(expansion)g
(of)g(the)f(most)h(promising)g(node)g(on)g(the)43 1711
y(OPEN)27 b(list)h(at)f(each)g(step.)44 b(However)-5
b(,)28 b(it)g(is)f(possible)g(that)h(many)f(nodes)g(on)g(the)g(OPEN)h
(list)f(are)g(equally)h(promising,)43 1836 y(in)f(which)g(case,)h(they)
e(can)h(all)h(be)f(expanded)g(concurrently)-6 b(.)43
b(If)27 b(the)g(number)g(of)g(such)f(\(equally\))i(promising)f(nodes)g
(is)43 1960 y(fewer)22 b(than)h(the)g(number)f(of)h(available)h
(processors,)c(then)j(the)g(other)f(less)g(promising)h(nodes)g
(available)g(on)g(OPEN)g(list)43 2085 y(can)h(be)h(searched)f(by)h(the)
g(idle)g(processors.)35 b(These)25 b(less)f(promising)h(nodes)g(may)f
(not)h(be)g(searched)f(by)h(serial)f(BFS,)43 2209 y(and)29
b(thus)f(form)h(speculative)f(work.)48 b(The)28 b(ef)o(fective)g
(degree)h(of)g(concurrency)d(of)j(such)f(a)g(formulation)i(is)e
(essentially)43 2334 y(equal)22 b(to)g(the)f(average)g(number)h(of)f
(nodes)g(available)i(on)f(the)f(OPEN)h(list)f(such)g(that)g(their)h
(cost)e(is)h(less)g(than)h(that)f(of)h(the)43 2458 y(optimal)f
(solution.)32 b(Since)20 b(the)g(cost)f(of)g(the)h(optimal)h(solution)f
(is)g(unknown,)g(a)g(good)f(strategy)g(is)g(to)h(always)f(expand)h
(only)43 2583 y(the)k Fi(p)p Fk(\255most)f(promising)i(nodes)f
(currently)f(available)j(on)e(the)g(OPEN)g(list;)h(where)f
Fi(p)f Fk(is)h(the)g(number)h(of)f(processors.)32 b(As)43
2707 y(long)24 b(as)f Fi(p)g Fk(is)g(no)h(more)f(than)h(the)g(ef)o
(fective)f(degree)g(of)h(concurrency)-6 b(,)21 b(all)k(processors)c
(will)j(tend)g(to)g(search)e(nodes)i(that)43 2832 y(will)j(also)e(be)h
(searched)f(by)g(serial)h(BFS.)g(The)f(ef)o(fective)g(degree)h(of)g
(concurrency)d(is)i(usually)h(dependent)h(on)f(the)f(size)43
2957 y(of)e(the)g(search)e(space)h(\(larger)h(spaces)e(have)i(higher)g
(degree)g(of)g(concurrency\))d(and)j(the)g(heuristic)g(function)g
(\(stronger)43 3081 y(heuristics)g(lead)h(to)f(smaller)h(degree)f(of)g
(concurrency\).)43 3266 y(The)28 b(basic)g(data)g(structure)f(of)h
(BFS,)h(the)f(OPEN)g(list,)i(is)e(a)g(priority)g(queue)g(that)h(must)f
(support)g(removal)g(of)g(the)g(best)43 3390 y(entry)i(in)g(the)h
(queue)g(and)g(insertion)f(of)h(nodes)f(into)h(the)g(queue.)54
b(The)30 b(data)h(structure)e(most)h(often)h(used)f(for)g(this)g(is)43
3515 y(a)h(heap.)56 b(The)31 b(simplest)g(parallel)i(formulation)f(of)f
(BFS)h(uses)e(multiple)j(processors)28 b(working)j(on)g(a)g(heap)h
(stored)e(in)43 3639 y(globally)h(addressable)e(memory)-6
b(.)51 b(Each)29 b(processor)f(locks)g(the)i(heap)g(and)g(picks)e(out)i
(the)g(current)e(best)h(node.)52 b(The)43 3764 y(heap)32
b(is)g(unlocked)g(and)g(the)g(node)g(is)g(expanded)g(to)g(yield)g(its)g
(children.)58 b(These)32 b(children)g(are)f(reinserted)h(into)h(the)43
3888 y(heap)24 b(with)f(appropriate)h(locking.)32 b(The)23
b(use)g(of)g(a)g(global)i(heap)e(is)g(a)g(source)f(of)h(contention.)33
b(If)23 b(the)h(time)f(taken)g(to)g(lock,)43 4013 y(remove,)h(and)h
(unlock)f(the)g(top)g(element)i(of)e(the)g(heap)h(is)f
Fi(t)1947 4025 y Fe(access)2164 4013 y Fk(and)h(time)g(for)f(expansion)
g(is)g Fi(t)3145 4025 y Fe(exp)3252 4013 y Fk(,)h(then)f(the)h(speedup)
43 4137 y(of)e(the)h(formulation)h(is)e(bounded)h(by)f
Ff(\()p Fi(t)1310 4149 y Fe(access)1522 4137 y Ff(+)18
b Fi(t)1635 4149 y Fe(exp)1742 4137 y Ff(\))p Fi(=t)1846
4149 y Fe(access)2040 4137 y Fk(.)43 4322 y(Many)26 b(techniques)h
(have)g(been)g(developed)g(to)g(ef)o(fectively)f(reduce)g
Fi(t)2327 4334 y Fe(access)2547 4322 y Fk(and)h(thus)f(increase)g(the)h
(available)h(paral\255)43 4446 y(lelism.)33 b(These)22
b(techniques)h(support)g(multiple)h(operations)f(on)g(heaps)f(stored)h
(in)g(shared)f(memory)g(while)i(maintaining)43 4571 y(the)35
b(strict)e(deletion)j(and)f(insertion)g(ordering)g([42].)66
b(The)34 b(performance)h(of)f(these)g(schemes)g(scales)f(to)i(dozens)f
(of)43 4695 y(processors)21 b(but)j(is)f(ultimately)h(limited)h(by)e
(the)g(locking)h(and)f(access)f(times)h(as)g(speci\002ed)g(above.)43
4880 y(Another)k(way)e(of)i(alleviating)h(the)e(contention)h(overhead)f
(associated)g(with)h(a)f(single)h(global)g(heap)g(is)f(to)g(use)g
(multiple)43 5005 y(heaps.)38 b(The)24 b(concept)h(of)g
Fi(p)f Fk(processors)f(sharing)i(a)g(single)g(heap)h(can)e(be)h
(relaxed)g(to)g Fi(k)j Fk(processors)23 b(sharing)i(a)f(heap)43
5129 y(with)g Fi(p=k)i Fk(heaps)d(in)h(all.)35 b(The)23
b(number)h Fi(k)i Fk(is)d(selected)h(by)f(studying)h(the)f(overheads)h
(associated)f(with)h(communication)43 5254 y(and)18 b(locking,)i
(contention,)g(and)e(node)h(expansion)f(time.)32 b(In)18
b(the)g(extreme)g(case,)g(each)g(processor)f(has)g(its)h(own)g(heap.)32
b(A)1970 5610 y(6)p eop
%%Page: 7 7
7 6 bop 43 344 a Fk(simple)20 b(parallel)g(formulation)g(based)f(on)g
(multiple)h(heaps)f(starts)f(with)h(the)g(initial)i(state)d(in)h(a)g
(single)h(heap.)31 b(As)19 b(additional)43 469 y(nodes)i(are)h
(generated,)g(they)f(are)g(farmed)h(out)g(to)f(other)g(heaps)h(\(using)
f(schemes)f(similar)i(to)g(load)g(balancing)g(in)g(DFS\).)43
593 y(Once)17 b(all)i(the)g(heaps)f(have)g(been)g(populated,)j
(processors)16 b(can)h(perform)h(BFS)h(on)f(individual)i(heaps.)31
b(This)17 b(formulation)43 718 y(has)g(a)g(signi\002cant)g(drawback)f
(in)h(that)h(although)g(it)g(balances)f(computation)h(among)f
(processors,)g(it)g(does)g(not)g(guarantee)43 842 y(the)k(quality)h(of)
f(nodes)g(individual)i(processors)c(explore.)32 b(Consequently)-6
b(,)21 b(processors)e(may)i(spend)g(considerable)g(time)43
967 y(exploring)j(parts)f(of)g(the)g(search)g(space)f(that)i(may)f(not)
g(be)h(explored)f(by)g(the)g(serial)h(formulation.)43
1152 y(T)-9 b(o)27 b(alleviate)j(this)d(drawback)g(of)h(excess)e(work)h
(performed)h(by)g(the)g(parallel)h(formulation,)i(one)d(must)f(ensure)h
(that)g(all)43 1276 y(heaps)35 b(have)f(a)h(share)f(of)h(the)g(most)f
(promising)h(nodes)g(globally)h(available.)68 b(This)35
b(process,)h(also)e(referred)h(to)f(as)43 1401 y(quality)22
b(equalization,)i(ensures)d(that)h(the)g(best)f(nodes)h(available)h
(globally)g(are)e(evenly)h(distributed)g(across)e(the)i(heaps.)43
1525 y(Since)27 b(the)g(quality)h(of)f(the)g(nodes)g(changes)f(as)h
(computation)h(proceeds,)f(quality)g(equalization)i(must)e(be)g
(performed)43 1650 y(periodically)-6 b(.)50 b(A)29 b(variety)f(of)h
(triggers)f(and)h(mechanisms)g(have)f(been)i(used)e(for)h(quality)g
(equalization)i([30,)e(6,)g(45,)g(4].)43 1774 y(A)k(simple)g
(triggering)g(mechanism)g(tracks)e(the)i(best)f(node)h(in)g(the)g
(system.)59 b(The)32 b(best)g(node)h(in)g(the)g(local)g(heap)g(is)43
1899 y(compared)21 b(to)g(the)g(best)g(node)g(in)h(the)f(system)f(and)h
(if)g(it)g(is)g(considerably)g(worse,)g(an)g(equalization)i(process)c
(is)i(initiated.)43 2023 y(Alternately)-6 b(,)24 b(an)g(equalization)h
(process)c(may)i(be)h(initiated)h(periodically)-6 b(.)43
2208 y(Quality)26 b(equalization)i(requires)e(movement)h(of)f(nodes)g
(between)h(heaps.)41 b(This)26 b(movement)h(of)f(nodes)g(can)g(be)g
(struc\255)43 2332 y(tured)33 b(\(into)f(a)h(ring,)i(or)d(via)h(a)f
(shared)g(blackboard\))h(or)f(randomized.)61 b(In)32
b(randomized)h(equalization,)k(a)c(processor)43 2457
y(periodically)i(selects)f(a)g(random)h(heap,)i(picks)c(up)i(the)f
(best)g(node)h(from)f(the)h(heap,)i(and)e(inserts)e(it)i(into)g(the)f
(local)43 2581 y(heap.)56 b(Alternately)-6 b(,)33 b(heaps)e(can)f(be)h
(organized)g(into)h(hierarchical)f(structures.)53 b(At)31
b(the)g(lower)g(levels,)i(equalization)43 2706 y(is)28
b(performed)h(more)g(frequently)g(than)g(the)g(higher)g(levels)f(in)h
(the)g(hierarchy)-6 b(.)48 b(Several)29 b(triggering)g(mechanisms)g
(and)43 2830 y(redistribution)c(strategies)f(have)g(been)g(explored)h
([6,)f(47].)35 b(It)24 b(has)g(been)g(shown)g(that)g(these)g(schemes)f
(are)h(capable)h(of)43 2955 y(providing)k(linearly)g(increasing)f
(speedups)h(with)f(number)h(of)f(processors)e(if)j(the)f(problem)h
(size)f(is)g(increased)g(appro\255)43 3080 y(priately)-6
b(.)41 b(Speci\002cally)-6 b(,)26 b(scalable)g(parallel)i(formulations)
e(of)g(best\255\002rst)e(tree)i(search)f(with)h(isoef)o(\002ciency)e
(of)i Fi(O)r Ff(\()p Fi(p)14 b Fk(log)3819 3039 y Fh(2)3869
3080 y Fi(p)p Ff(\))43 3204 y Fk(on)26 b(hypercube)f(architectures)g
(have)h(been)g(developed)h([30].)41 b(Using)25 b(these)h(schemes,)g
(speedups)f(in)i(excess)d(of)i(950)43 3329 y(have)20
b(been)g(demonstrated)g(on)h(1024)f(processor)e(hypercubes)h(in)h(the)g
(context)g(of)g(TSPs)g(formulated)h(as)e(best\255\002rst)f(tree)43
3453 y(search)k(problems)i([6].)43 3779 y Fd(Best\255First)f(Search)h
(of)e(State)h(Space)h(Graphs)43 4029 y Fk(Ef)o(\002cient)19
b(parallel)j(implementation)g(of)e(BFS)g(on)g(state)g(space)f(graphs)g
(requires)h(a)g(distributed)g(mechanism)g(that)g(allows)43
4154 y(many)h(processors)e(to)j(perform)f(duplicate)i(checking)d(or)h
(insertions)h(into)g(the)f(CLOSED)h(list)g(concurrently)-6
b(.)30 b(The)22 b(nature)43 4278 y(of)h(implementation)j(of)e(this)f
(mechanism)g(is)g(dependent)i(on)e(the)h(underlying)g(parallel)g
(architecture.)43 4463 y(In)d(shared)f(address)g(space)g(machines,)h
(insertion)g(of)g(nodes)f(into)i(the)e(closed)h(list)g(requires)f
(locking)h(of)f(the)h(list.)32 b(If)21 b(there)43 4587
y(is)26 b(a)h(single)g(lock)f(associated)g(with)h(the)f(entire)h(list,)
h(the)e(list)h(must)f(be)h(locked)f(approximately)h(as)f(many)g(times)h
(as)f(the)43 4712 y(total)32 b(number)g(of)f(nodes)g(expanded.)57
b(This)31 b(represents)f(a)h(serial)h(bottleneck.)56
b(The)32 b(bottleneck)f(can)g(be)g(alleviated)43 4836
y(by)25 b(associating)g(multiple)i(locks)d(with)i(the)g(closed)f(list.)
38 b(Processors)24 b(lock)g(only)i(relevant)f(parts)g(of)g(the)h
(closed)f(list)g(into)43 4961 y(which)d(the)h(node)g(is)f(being)h
(inserted.)32 b(Message)22 b(passing)g(formulations)i(of)e(this)g
(parallel)i(search)d(technique)i(distribute)43 5086 y(the)35
b(closed)f(list)h(across)e(the)i(processors.)65 b(Each)35
b(processor)e(expands)h(a)h(node)g(and)g(hashes)f(the)h(successors)d
(to)43 5210 y(appropriate)38 b(processors)c(via)j(a)g(message.)73
b(The)37 b(destination)h(processor)d(checks)g(for)i(replication)h(and)f
(sends)f(a)43 5335 y(message)19 b(back)g(specifying)h(whether)f(the)h
(node)g(exists)f(in)h(the)g(closed)f(list)h(or)g(not.)31
b(Such)20 b(a)f(protocol)h(requires)g(two\255way)1970
5610 y(7)p eop
%%Page: 8 8
8 7 bop 43 344 a Fk(cooperation)24 b(between)f(various)f(processors.)31
b(Each)23 b(processor)e(must)i(periodically)g(check)f(for)h(incoming)h
(messages,)43 469 y(process)e(them)h(and)h(send)f(appropriate)h
(responses.)43 653 y(A)29 b(simpler)g(parallel)h(formulation)g(uses)d
(the)i(hash)f(function)h(both)g(for)g(replication)g(checking)f(and)h
(for)f(qualitative)j(and)43 778 y(quantitative)26 b(load)g(balance.)38
b(In)25 b(this)g(formulation,)i(the)e(successors)d(of)j(a)g(node)g(are)
g(hashed)g(to)g(appropriate)g(proces\255)43 903 y(sors.)36
b(The)25 b(destination)h(processor)d(checks)g(for)h(replication)i(as)f
(before.)37 b(However)-5 b(,)25 b(instead)g(of)g(sending)h(a)f
(response)43 1027 y(back,)18 b(it)h(inserts)e(the)i(successor)c(nodes)j
(into)h(its)f(own)g(open)h(list.)31 b(This)17 b(has)h(two)g(main)h
(advantages:)30 b(\(i\))18 b(it)g(eliminates)i(the)43
1152 y(need)f(for)f(a)g(return)g(message)g(and)h(wait)g(at)f(the)h
(source)e(processor;)h(and)h(\(ii\))f(the)h(randomized)g(hash)f
(function)h(serves)d(as)43 1276 y(the)24 b(method)g(for)g(balancing)h
(load)f(qualitatively)h(and)f(quantitatively)-6 b(.)35
b(These)23 b(schemes)g(have)h(been)g(studied)g(by)f(many)43
1401 y(researchers)g([33)q(,)i(30].)38 b(Assuming)26
b(a)f(perfectly)g(random)g(hash)g(function,)i(it)e(has)g(been)h(shown)f
(that)g(if)h(the)f(number)h(of)43 1525 y(nodes)d(originating)i(at)f
(each)f(processor)e(grows)i(as)f Fi(O)r Ff(\()p Fk(log)16
b Fi(p)p Ff(\))p Fk(,)23 b(then)h(each)f(processor)e(will)k(have)e
(asymptotically)g(equal)43 1650 y(number)h(of)f(nodes)g(after)g(the)h
(hash)f(operation)h([33].)43 1834 y(One)31 b(of)g(the)g(major)h
(drawbacks)e(of)h(graph)g(search)f(techniques)h(such)g(as)f(BFS)i(is)f
(that)g(their)g(memory)g(requirement)43 1959 y(grows)c(linearly)h(with)
g(the)g(search)f(space.)44 b(For)27 b(large)h(problems,)h(this)f
(memory)f(requirement)h(becomes)g(prohibitive.)43 2083
y(Many)17 b(limited\255memory)i(variants)e(of)g(heuristic)g(search)g
(have)g(been)h(developed.)31 b(These)17 b(techniques)h(rely)f(on)g
(retraction)43 2208 y(or)28 b(delayed)h(expansion)f(of)g(less)g
(promising)h(nodes)f(to)g(reduce)g(memory)g(requirement.)48
b(In)29 b(the)f(parallel)i(processing)43 2332 y(context,)23
b(retractions)f(lead)i(to)g(additional)h(communication)g(and)e
(indexing)h(for)f(parent\255child)h(relationships)g([9].)43
2732 y Fj(5)116 b(Game)32 b(T)-6 b(ree)32 b(Search)43
3017 y Fk(Min\255max)27 b(techniques)g(such)f(as)h Fi(\013)21
b Fc(\000)f Fi(\014)31 b Fk(game)c(tree)g(search)f(are)g(extensively)h
(used)g(in)g(game\255playing)h(programs.)43 b(In)43 3141
y(this)28 b(algorithm,)j(the)d(minimax)h(value)g(of)f(a)g(node)h(is)f
(determined)h(for)f(a)g(given)h(search)e(window)h Ff([)p
Fi(\013;)14 b(\014)t Ff(])p Fk(.)48 b(This)28 b(window)43
3266 y(serves)19 b(to)h(prune)g(signi\002cant)g(parts)g(of)g(the)g
(tree)g(that)h(do)f(not)g(ef)o(fect)g(the)g(minimax)h(evaluation)h(of)e
(the)g(root.)32 b(Initially)-6 b(,)22 b(this)43 3390
y(window)27 b(is)f(set)g(to)g Ff([)p Fc(\0001)p Fi(;)14
b Fc(1)p Ff(])p Fk(.)41 b(The)26 b(\002rst)e(child)j(of)f(a)g(node)h
(is)f(searched)g(with)g(the)h(same)f(window)g(as)g(the)g(parent.)42
b(The)43 3515 y(window)26 b(of)f(other)h(children)g(is)f(narrowed)g(on)
h(the)f(basis)g(of)h(results)e(from)i(previously)e(visited)i(children.)
39 b(This)25 b(process)43 3639 y(is)f(continued)h(until)g(a)f
(speci\002ed)g(depth)g(is)g(reached,)g(or)g(the)g(branch)g(has)g(no)g
(successors.)32 b(V)-6 b(ariants)24 b(of)h(this)f(algorithm)43
3764 y(are)f(based)g(on)h(iterative)f(deepening,)i(which)f(guides)f
(search)g(based)g(on)g(results)g(from)g(previous)g(iterations.)43
3948 y(Min\255max)31 b(techniques)f(present)g(considerable)h
(challenges)h(for)e(parallel)i(processing.)53 b(Conceptually)-6
b(,)33 b Fi(\013)23 b Fc(\000)f Fi(\014)34 b Fk(game)43
4073 y(tree)20 b(search)f(algorithm)i(can)e(also)h(be)h(viewed)f(as)f
(a)h(depth\255\002rst)f(branch\255and\255bound)h(algorithm)h([23)q(])e
(that)i(searches)d(for)43 4197 y(a)25 b(maximum)g(payof)o(f)f(strategy)
g(among)h(all)h(strategies)f(represented)f(in)h(the)g(game)g(tree.)37
b(However)-5 b(,)25 b(game)g(trees)f(tend)43 4322 y(to)30
b(be)f(strongly)g(ordered.)51 b(Therefore,)31 b(naive)f(parallel)h
(formulations)f(may)f(expand)h(a)f(large)h(number)g(of)f(nodes)h(that)
43 4446 y(are)23 b(not)g(expanded)g(by)g(the)g(serial)g(formulation.)34
b(Second,)24 b(the)f(information)h(about)g(the)f(current)f(best)h
(strategy)f(cannot)43 4571 y(be)27 b(captured)f(by)h(a)f(single)i
(node,)g(as)e(the)h(aim)g(is)g(to)f(\002nd)h(a)f(best)h(winning)h
(strategy)e(rather)g(than)h(a)g(winning)h(position.)43
4695 y(Ef)o(fective)h(propagation)i(of)f(this)g(information)h(to)f
(other)f(processors)f(searching)h(disjoint)i(parts)e(of)h(the)g(search)
f(space)43 4820 y(is)e(much)g(harder)f(compared)h(to)g(sharing)g(of)g
(the)g(cost)f(of)h(current)g(best)f(solution)i(in)f(traditional)i(DFBB)
e(algorithms.)45 b(A)43 4945 y(number)22 b(of)g(researchers)e(have)i
(investigated)h(parallel)g(formulations)g(of)f Fi(\013)16
b Fc(\000)f Fi(\014)26 b Fk([10,)c(34,)g(40].)32 b(The)22
b(problem)h(of)f(excess)43 5069 y(node)f(expansions)g(is)f(handled)i
(by)e(using)h(a)g(prioritized)g(work)f(distribution)i(in)f(which)g(the)
g(tree)g(is)f(expanded)h(left)h(to)f(right)43 5194 y(in)30
b(a)g(sweeping)h(pattern)f(also)g(referred)f(to)h(as)g(the)g(\223Y)-8
b(oung)31 b(Brothers)e(W)m(ait)i(Concept\224.)52 b(Using)30
b(these)g(optimizations)43 5318 y(along)24 b(with)f(dynamic)f(load)i
(balancing,)g(researchers)d(have)h(demonstrated)h(speedups)g(of)f(as)h
(much)f(as)g(two)h(orders)f(of)1970 5610 y(8)p eop
%%Page: 9 9
9 8 bop 43 344 a Fk(magnitude)25 b(on)e(a)g(variety)g(of)g(parallel)i
(platforms.)43 529 y(The)36 b(utility)h(of)f(parallel)i(processing)d
(has)h(been)h(demonstrated)f(in)h(the)f(context)g(of)g(a)g(number)g(of)
h(games,)i(and)d(in)43 653 y(particular)-5 b(,)29 b(Chess.)44
b(W)o(ork)26 b(on)h(large)h(scale)f(parallel)h Fi(\013)22
b Fc(\000)e Fi(\014)31 b Fk(search)26 b(led)i(to)g(the)f(development)h
(of)g(deep)g(thought)g([15])43 778 y(in)37 b(1990.)73
b(This)37 b(program)f(was)g(capable)h(of)g(playing)g(chess)f(at)g
(grandmaster)h(level.)73 b(Subsequent)38 b(advances)d(in)43
903 y(the)30 b(use)f(of)g(dedicated)i(hardware,)g(parallel)f
(processing,)h(and)f(algorithms)g(resulted)f(in)h(the)g(development)g
(of)g(IBM')o(s)43 1027 y(Deep)c(Blue)g([16,)g(14])f(that)h(beat)g(the)g
(reigning)g(world)g(champion)g(Gary)e(Kasparov)-6 b(.)38
b(Feldmann)27 b(et)e(al.)h([40)q(])f(developed)43 1152
y(a)33 b(distributed)g(chess)e(program)i(that)f(is)h(acknowledged)g(to)
f(be)h(one)g(of)f(the)h(best)f(computer)h(chess)e(players)h(based)43
1276 y(entirely)24 b(on)f(general)h(purpose)f(hardware.)43
1675 y Fj(6)116 b(Speedup)31 b(Anomalies)h(in)h(Parallel)f
(Formulations)g(of)g(Search)f(Algorithms)43 1960 y Fk(In)21
b(parallel)i(search)d(algorithms,)j(the)e(speedup)h(can)f(dif)o(fer)g
(greatly)g(from)g(one)h(execution)f(to)h(another)-5 b(.)32
b(This)21 b(is)g(because)43 2085 y(the)32 b(portions)g(of)f(the)h
(search)f(space)g(examined)h(by)g(dif)o(ferent)f(processors)f(are)h
(determined)i(dynamically)f(and)g(can)43 2209 y(be)27
b(dif)o(ferent)g(for)g(dif)o(ferent)g(executions.)44
b(A)27 b(number)g(of)g(researchers)e(have)i(observed)f(and)h(analyzed)g
(the)g(existence)43 2334 y(of)j(these)h(speedup)f(anomalies.)55
b(An)31 b(acceleration)g(anomaly)g(manifests)f(itself)h(in)g(the)f
(form)h(of)f(a)g(speedup)h(greater)43 2458 y(than)e Fi(p)g
Fk(on)g Fi(p)g Fk(processors)e(when)i(parallel)i(search)d(does)g(less)h
(work)f(than)h(the)h(serial)f(search)f(algorithm)i([26)q(,)f(29].)50
b(A)43 2583 y(speedup)18 b(of)f(less)g(than)g Fi(p)p
Fk(,)h(attributed)g(to)g(excess)d(work)h(done)i(by)f(the)g(parallel)i
(formulation)g(compared)e(to)g(the)h(sequential)43 2707
y(formulation,)34 b(is)c(termed)h(as)e(a)i(deceleration)g(anomaly)g
([26)q(,)f(28].)54 b(The)30 b(ratio)h Fi(W)2712 2719
y Fe(p)2751 2707 y Fi(=W)2871 2719 y Fe(s)2906 2707 y
Fk(,)h(where)f Fi(W)3296 2719 y Fe(p)3365 2707 y Fk(and)f
Fi(W)3611 2719 y Fe(s)3677 2707 y Fk(are)h(the)43 2832
y(number)24 b(of)f(nodes)g(searched)g(by)f(parallel)j(and)f(serial)f
(formulations,)h(is)f(called)h(the)g(search)e(overhead)h(factor)-5
b(.)43 3017 y(Speedup)28 b(anomalies)h(can)e(easily)g(occur)f(in)i
(parallel)g(depth\255\002rst)f(search)f(for)h(\002nding)g(any)g
(solution,)j(as)c(certain)i(exe\255)43 3141 y(cution)i(sequences)f(may)
g(\002nd)g(a)h(solution)g(much)f(earlier)h(than)g(others.)51
b(However)-5 b(,)31 b(it)f(appears)f(natural)h(that,)i(on)e(the)43
3266 y(average,)22 b(the)f(search)f(overhead)i(factor)e(for)h(parallel)
i(DFS)e(should)h(always)f(be)g(greater)g(than)h(or)f(equal)h(to)g(one.)
32 b(Other\255)43 3390 y(wise)23 b(the)h(serial)f(search)g(algorithm)h
(is)f(not)h(optimal;)h(i.e.,)f(the)f(emulation)i(of)f(parallel)g(DFS)f
(on)h(a)f(single)h(processor)e(will)43 3515 y(be)k(faster)g(than)h
(sequential)g(DFS.)f(Surprisingly)-6 b(,)28 b(there)e(are)g
(situations,)h(in)g(which)f(parallel)i(DFS)e(can)g(have)g(a)g(search)43
3639 y(overhead)i(factor)f(of)g(less)g(than)h(1)g(on)g(the)f(average,)i
(implying)g(that)f(the)g(serial)f(search)g(algorithm)i(in)f(the)g
(situation)g(is)43 3764 y(suboptimal.)43 b(Kumar)27 b(and)g(Rao)f([43])
h(show)f(that)g(if)h(no)g(heuristic)f(information)i(is)e(available)i
(to)e(order)g(the)h(successors)43 3888 y(of)i(a)f(node,)j(then)d(on)h
(the)g(average,)h(the)e(speedup)h(obtained)h(by)e(parallel)i(DFS)e(is)g
(superlinear)h(if)g(the)g(distribution)h(of)43 4013 y(solutions)d(is)g
(non\255uniform.)43 b(This)27 b(model)g(is)g(validated)h(by)e
(experiments)g(on)h(synthetic)f(state\255space)g(trees)g(modeling)43
4137 y(the)j(hackers)e(problem,)k(the)e(15\255puzzle)f(problem)i(and)f
(the)g Fi(n)p Fk(\255Queens)e(problem.)50 b(In)29 b(all)g(these)g
(experiments,)h(serial)43 4262 y(and)i(parallel)h(DFS)e(do)g(not)h(use)
f(any)g(heuristic)h(ordering,)i(and)d(select)g(successors)e
(arbitrarily)-6 b(.)57 b(The)31 b(basic)g(reason)43 4386
y(for)23 b(this)h(phenomenon)h(is)e(that)h(parallel)h(search)e(can)g
(invest)g(resources)f(into)j(multiple)g(regions)f(of)f(the)h(search)f
(frontier)43 4511 y(concurrently)-6 b(.)35 b(When)26
b(the)e(solution)i(density)e(in)h(dif)o(ferent)g(regions)g(of)f(the)h
(search)f(frontier)g(is)h(non\255uniform)g(and)g(these)43
4635 y(non\255uniformities)32 b(are)f(not)g(known)g(a)g(priori,)i(then)
f(sequential)g(search)e(has)g(an)i(equal)f(chance)g(of)g(searching)g(a)
g(low)43 4760 y(density)19 b(region)h(\(and)f(thus)g(be)h(forced)f(to)g
(expand)h(many)f(nodes\))g(or)g(a)g(high)h(density)f(region.)32
b(On)19 b(the)h(contrary)-6 b(,)18 b(parallel)43 4885
y(search)25 b(can)h(search)f(all)h(regions)g(at)g(the)h(same)f(time,)h
(ensuring)f(that)g(at)g(least)h(some)e(processors)f(are)i(searching)g
(the)43 5009 y(high)e(density)f(regions.)43 5194 y(If)k(good)g
(heuristics)f(are)g(available)i(to)f(order)f(the)g(successors)e(of)j
(nodes)f(in)h(DFS,)g(then)g(the)f(search)g(overhead)g(factor)43
5318 y(of)k(parallel)h(DFS)f(may)g(be)g(greater)g(than)g(one.)53
b(The)30 b(reason)g(is)f(that)i(some)f(processors)d(may)j(search)f
(unpromising)1970 5610 y(9)p eop
%%Page: 10 10
10 9 bop 43 344 a Fk(areas)25 b(of)g(the)h(search)e(space.)38
b(This)25 b(phenomenon)i(is)e(most)g(visible)h(in)g(the)f(context)g(of)
g Fi(\013)20 b Fc(\000)g Fi(\014)29 b Fk(search)24 b(of)i(game)f(trees)
43 469 y(for)f(which)h(strong)e(ordering)i(heuristics)f(are)g
(available.)38 b(Kale)25 b(and)g(Saletore)g([19])g(present)f(a)g
(scheme)g(that)h(combines)43 593 y(the)h(information)h(about)f(the)g
(ordering)h(and)f(depth)g(of)g(nodes)f(to)h(perform)g(load)g(balancing)
h(among)f(processors.)38 b(This)43 718 y(scheme)15 b(largely)g
(maintains)h(the)f(depth\255\002rst)f(heuristic)h(order)f(of)i(search)d
(across)h(processors,)g(and)i(results)e(in)h(consistent)43
842 y(linear)32 b(speedup.)55 b(Even)31 b(in)g(the)g(presence)f(of)h
(good)g(ordering)h(heuristics,)g(parallel)g(DFS)f(may)f(lead)i(to)e
(superlinear)43 967 y(speedup.)i(For)21 b(example,)h(this)f(can)f
(happen)i(if)f(there)g(is)g(a)g(non\255zero)g(probability)h(that)f(the)
g(heuristic)g(can)g(guide)h(search)43 1092 y(to)h(large)h(regions)f
(that)h(do)f(not)h(contain)f(any)g(solutions)h([43].)43
1276 y(In)k(parallel)h(BFS,)g(the)f(search)f(overhead)g(can)h(be)g
(greater)g(than)g(one)g(if)h(some)e(processors)f(expand)i(nodes)g(that)
g(are)43 1401 y(never)f(expanded)h(by)f(serial)h(BFS)g(\(e.g.,)g(nodes)
g(that)g(have)f(higher)h(bound)g(than)g(the)g(cost)f(of)g(the)h
(optimal)h(solution\).)43 1525 y(In)j(fact,)h(it)f(can)f(be)h(shown)f
(that)h(for)f(any)h(given)g(BFS)f(instance,)j(there)e(exists)e(a)i
(constant)f Fi(k)j Fk(such)d(that)h(expanding)43 1650
y(more)25 b(than)g Fi(k)j Fk(nodes)d(in)g(parallel)h(from)f(a)g(global)
h(open)g(list)f(leads)g(to)g(wasted)g(computation)h([26].)38
b(This)24 b(situation)i(gets)43 1774 y(worse)j(with)i(distributed)g
(open)f(lists)g(since)g(expanded)g(nodes)h(have)e(locally)i(minimum)h
(heuristics)d(that)h(are)g(not)h(the)43 1899 y(best)23
b(nodes)g(across)f(all)i(open)g(lists.)43 2083 y(The)29
b(search)f(overhead)g(for)h(parallel)h(BFS)g(can)e(also)h(be)g(less)f
(than)i(one.)49 b(This)29 b(can)f(happen)i(when)f(multiple)i(nodes)43
2208 y(on)c(the)g(open)g(list)h(have)e(identical)i(heuristic)f(values,)
h(but)f(require)g(dif)o(ferent)g(amount)h(of)f(search)e(to)i(detect)g
(a)g(solution)43 2332 y(\(or)h(to)h(\002nd)g(out)g(that)g(the)g(node)h
(does)e(not)i(lead)f(to)g(optimal)i(solution\).)50 b(Hence,)30
b(if)f(a)g(parallel)h(algorithm)h(happens)e(to)43 2457
y(explore)h(the)h(better)g(nodes)f(\002rst,)h(then)g(it)f(can)g(result)
h(in)f(acceleration)h(anomalies,)j(and)c(vice)g(versa.)53
b(It)30 b(is)h(possible)43 2581 y(to)39 b(derive)g(properties)f(of)h
(the)g(heuristic)g(function)g(such)f(that)h(superlinearity)h(ef)o
(fects)d(in)j(parallel)g(BFS)f(are)f(made)43 2706 y(deterministic)24
b([29,)g(32].)43 3105 y Fj(7)116 b(Role)32 b(of)g(Heuristics)43
3390 y Fk(Heuristics)e(form)g(the)h(most)g(important)g(component)g(of)g
(search)f(techniques,)j(and)e(parallel)h(formulations)f(of)g(search)43
3515 y(algorithms)22 b(must)f(be)h(viewed)g(in)g(the)f(context)g(of)h
(these)f(heuristics.)32 b(In)21 b(BFS)h(techniques,)g(heuristics)f
(focus)g(search)f(by)43 3639 y(lowering)j(the)f(ef)o(fective)g
(branching)g(factor)-5 b(.)32 b(In)22 b(DFBB)g(methods,)h(heuristics)e
(provide)h(better)h(bounds,)f(and)g(thus)g(serve)43 3764
y(to)h(prune)h(the)f(search)f(space.)43 3948 y(Often,)f(there)g(is)g(a)
f(tradeof)o(f)h(between)g(the)g(strength)g(of)g(the)g(heuristic)f(and)h
(the)g(ef)o(fective)g(size)f(of)h(search)e(space.)32
b(Better)43 4073 y(heuristics)21 b(result)g(in)h(smaller)g(search)f
(spaces)f(but)i(are)f(also)h(more)f(expensive)g(to)h(compute.)32
b(For)21 b(example,)h(an)g(impor\255)43 4197 y(tant)27
b(application)h(of)e(strong)g(heuristics)g(is)g(in)h(the)g(computation)
g(of)g(bounds)f(for)g(mixed)h(integer)g(programming)g(\(MIP\).)43
4322 y(Mixed)36 b(integer)g(programming)h(has)e(seen)g(signi\002cant)h
(advances)f(over)g(the)g(years)g([18].)70 b(Whereas)35
b(\002fteen)h(years)43 4446 y(back,)c(MIP)f(problems)f(with)h(100)g
(integer)g(variables)g(were)f(considered)g(challenging,)k(today)-6
b(,)32 b(many)f(problems)f(with)43 4571 y(up)25 b(to)g(1000)h(integer)f
(variables)g(can)g(be)g(solved)g(on)g(workstation)f(class)g(machines)h
(using)g(branch\255and\255cut)g(methods.)43 4695 y(Largest)35
b(known)h(instances)f(of)h(TSPs)g(and)g(QAPs)f(have)h(been)g(solved)f
(using)h(branch\255and\255bound)g(with)g(powerful)43
4820 y(heuristics)e([3,)h(35].)66 b(The)34 b(presence)g(of)h(ef)o
(fective)f(heuristics)g(may)g(prune)h(the)f(search)g(space)g
(considerably)-6 b(.)66 b(For)43 4945 y(example,)24 b(when)f(Padberg)h
(and)f(Rinaldi)h(introduced)g(the)f(branch\255and\255cut)f(algorithm)i
(in)g(1987,)f(they)g(used)g(it)g(to)g(solve)43 5069 y(a)i(532)g(city)e
(TSP)-11 b(,)26 b(which)e(was)g(the)h(largest)g(TSP)g(solved)f
(optimally)i(at)f(that)g(time.)37 b(Subsequent)25 b(improvements)g(to)g
(the)43 5194 y(method)d(led)g(to)f(the)g(solution)i(of)e(a)g(a)g(2,392)
h(city)f(problem)g([36)q(].)32 b(More)21 b(recently)-6
b(,)21 b(using)g(cutting)h(planes,)g(problems)f(with)43
5318 y(over)f(7,000)h(cities)g(have)f(been)i(solved)e([18)q(])g(on)h
(serial)g(machines.)32 b(However)-5 b(,)21 b(for)f(many)h(problems)g
(of)f(interest,)i(the)f(re\255)1947 5610 y(10)p eop
%%Page: 11 11
11 10 bop 43 344 a Fk(duced)21 b(search)f(space)g(still)i(requires)f
(the)g(use)g(of)g(parallelism)i([3,)e(35,)g(44,)g(31)q(].)31
b(Use)21 b(of)g(powerful)h(heuristics)e(combined)43 469
y(with)26 b(ef)o(fective)f(parallel)i(processing)d(has)h(enabled)i(the)
f(solution)g(of)g(extremely)f(large)g(problems)h([35].)39
b(For)25 b(example,)43 593 y(QAP)k(problems)f(from)h(the)f(Nugent)h
(and)g(Eschermann)g(test)f(suites)g(with)h(up)f(to)h(4)p
Fi(:)p Fk(8)22 b Fc(\002)f Fk(10)3028 563 y Fh(10)3125
593 y Fk(nodes)28 b(\(Nugent22)h(with)43 718 y(Gilmore\255Lawler)f
(bound\))g(were)f(solved)g(on)g(a)h(NEC)f(Cenju\2553)h(parallel)h
(computer)e(in)h(under)g(nine)g(days)e([3].)45 b(Another)43
842 y(dazzling)31 b(demonstration)h(of)f(this)g(was)g(presented)g(by)f
(the)h(IBM)h(Deep)f(Blue.)57 b(Deep)31 b(blue)h(used)e(a)h(combination)
i(of)43 967 y(dedicated)h(hardware)e(for)h(generating)h(and)f
(evaluating)h(board)f(positions)g(and)h(parallel)g(search)e(of)h(the)g
(game)g(tree)43 1092 y(using)24 b(an)f(IBM)h(SP2)g(to)f(beat)h(the)f
(current)g(world)g(chess)f(champion,)i(Garry)e(Kasparov)g([16)q(,)h
(14].)43 1276 y(Heuristics)g(have)h(several)g(important)g(implications)
i(for)e(exploiting)h(parallelism.)36 b(Strong)25 b(heuristics)e(narrow)
h(the)g(state)43 1401 y(space)18 b(and)h(thus)g(reduce)f(the)h
(concurrency)e(available)j(in)f(the)g(search)f(space.)31
b(Use)18 b(of)h(powerful)g(heuristics)f(pose)h(other)43
1525 y(computational)30 b(challenges)f(for)e(parallel)j(processing)d
(as)h(well.)48 b(For)27 b(example,)j(in)e(branch\255and\255cut)g
(methods,)h(a)f(cut)43 1650 y(generated)e(at)g(a)g(certain)f(state)h
(may)f(be)h(required)g(by)f(other)h(states.)39 b(Therefore,)26
b(in)g(addition)h(to)f(balancing)h(load,)g(the)43 1774
y(parallel)g(branch\255and\255cut)e(formulation)i(must)e(also)h
(partition)h(cuts)e(among)h(processors)d(so)j(that)g(processors)d
(working)43 1899 y(in)h(certain)f(LP)h(domains)f(have)g(access)f(to)h
(the)h(desired)f(cuts)f([2)q(,)h(27,)g(8].)43 2083 y(In)35
b(addition)h(to)f(inter\255node)g(parallelism)h(that)f(has)f(been)h
(discussed)f(until)i(this)e(point,)39 b(intra\255node)c(parallelism)h
(can)43 2208 y(become)21 b(a)h(viable)g(option)g(if)f(the)h(heuristic)f
(is)g(computationally)i(expensive.)31 b(For)21 b(example,)h(the)f
(assignment)h(heuristic)43 2332 y(of)27 b(Pekny)g(et)h(al.)45
b(has)27 b(been)h(ef)o(fectively)f(parallelized)i(for)e(solving)g
(large)h(instances)f(of)g(TSPs)g([35)q(].)44 b(If)28
b(the)f(degree)h(of)43 2457 y(inter\255node)h(parallelism)h(is)d
(small,)j(this)f(form)f(of)g(parallelism)i(provides)d(a)i(desirable)f
(alternative.)49 b(Another)28 b(example)43 2581 y(of)e(this)h(is)f(in)g
(the)h(solution)g(of)g(MIP)f(problems)h(using)f(branch\255and\255cut)g
(methods.)42 b(Branch\255and\255cut)25 b(methods)i(rely)f(on)43
2706 y(LP)f(relaxation)h(for)f(generating)h(lower)f(bounds)g(of)g
(partially)h(instantiated)g(solutions)g(followed)g(by)e(generation)i
(of)f(valid)43 2830 y(inequalities)33 b([18].)55 b(These)30
b(LP)h(relaxations)f(constitute)h(a)g(major)g(part)f(of)h(the)g
(overall)g(computation)g(time.)55 b(Many)31 b(of)43 2955
y(the)25 b(industrial)i(codes)d(rely)h(on)g(Simplex)i(to)e(solve)g(the)
g(LP)h(problem)g(since)f(it)g(can)g(adapt)h(the)f(solution)i(to)e
(added)h(rows)43 3080 y(and)e(columns.)35 b(While)26
b(interior)f(point)f(methods)h(are)f(better)g(suited)g(to)g
(parallelism,)i(they)e(tend)h(to)f(be)g(less)g(ef)o(\002cient)f(for)43
3204 y(reoptimizing)k(LP)f(solutions)g(with)f(added)i(rows)d(and)i
(columns)f(\(in)h(branch\255and\255cut)e(methods\).)40
b(LP)25 b(relaxation)i(using)43 3329 y(Simplex)j(has)e(been)i(shown)e
(to)h(parallelize)i(well)e(on)g(small)h(numbers)e(of)h(processors)e
(but)i(ef)o(forts)f(to)g(scale)h(to)g(larger)43 3453
y(numbers)e(of)h(processors)d(have)i(not)h(been)f(successful.)44
b(LP)27 b(based)h(branch)f(and)g(bound)h(methods)g(have)f(also)g(been)
43 3578 y(used)g(for)f(solving)h(quadratic)g(assignment)g(problems)g
(using)g(iterative)h(solvers)d(such)h(as)g(preconditioned)j(Conjugate)
43 3702 y(Gradient)37 b(to)h(approximately)f(compute)h(the)f(interior)h
(point)g(directions)f([37].)74 b(These)37 b(methods)g(have)g(been)h
(used)43 3827 y(to)30 b(compute)f(lower)h(bounds)g(using)g(linear)g
(programs)f(with)h(over)f(150,000)h(constraints)f(and)h(300,000)h
(variables)e(for)43 3951 y(solving)20 b(QAPs.)32 b(These)19
b(and)i(other)f(iterative)h(solvers)e(parallelize)j(very)c(ef)o
(fectively)i(to)g(a)h(large)f(number)h(of)f(processors.)43
4076 y(A)25 b(general)g(parallel)g(framework)f(for)g(computing)h
(heuristic)f(solutions)h(to)f(problems)h(is)f(presented)g(by)g
(Pramanick)h(and)43 4200 y(Kuhl)f([39)q(].)43 4600 y
Fj(8)116 b(Concluding)32 b(Remarks)43 4885 y Fk(Great)e(advances)g
(have)h(been)h(made)f(in)g(the)g(development)h(of)f(scalable)g
(parallel)i(formulations)f(of)f(various)f(search)43 5009
y(methods)h(in)h(the)f(past)g(two)g(decades.)56 b(Performance)31
b(and)g(scalability)h(of)f(load)h(balancing)g(techniques)g(for)e
(parallel)43 5134 y(depth\255\002rst)36 b(search)g(methods)i(is)f
(largely)g(understood,)k(and)d(it)g(is)f(unlikely)g(that)h(any)f(new)g
(methods)h(will)g(perform)43 5258 y(signi\002cantly)32
b(better)g(than)g(the)g(existing)g(methods.)59 b(Parallel)34
b(best\255\002rst)c(search)g(methods)j(require)f(qualitative)h(load)
1947 5610 y(11)p eop
%%Page: 12 12
12 11 bop 43 344 a Fk(balancing)39 b(in)g(addition)h(to)e(quantitative)
h(load)g(balancing)g(used)f(in)h(depth\255\002rst)e(search,)k(and)d
(are)g(thus)g(harder)g(to)43 469 y(analyze.)h(However)-5
b(,)25 b(many)g(methods)h(for)f(qualitative)i(load)f(balancing)g(have)f
(been)h(developed)g(and)g(shown)f(to)g(have)43 593 y(provably)h(good)i
(performance)e(under)h(a)g(reasonable)g(set)g(of)f(assumptions.)43
b(For)27 b(a)f(wide)i(variety)e(of)h(problems,)g(both)43
718 y(best\255\002rst)33 b(and)i(depth\255\002rst)e(search)h(methods)h
(have)f(shown)h(excellent)g(performance)g(and)g(scalability)g(up)g(to)f
(1024)43 842 y(processors)22 b([45)q(,)i(25)q(,)g(1,)h(40,)f(30)q(,)g
(44)q(].)36 b(Attention)26 b(has)e(now)h(shifted)g(to)g(the)f
(development)i(of)f(software)f(environments)43 967 y(that)g(allow)g
(practitioners)f(to)g(solve)g(real)h(problems)f(on)h(parallel)g
(machines)g([4].)43 1152 y(It)33 b(is)g(important)h(to)f(note)g(that)h
(parallel)g(processing)f(can)f(only)h(provide)h(speed)f(up)g
(proportional)h(to)f(the)g(number)h(of)43 1276 y(processors.)g(Hence,)
24 b(parallel)i(search)d(can)h(be)h(used)f(to)h(solve)f
(signi\002cantly)g(bigger)h(instances)f(of)g(the)h(problem)g(only)43
1401 y(if)i(the)g(search)f(space)g(grows)g(as)h(a)g(small)g(exponent)g
(of)g(problem)h(size.)42 b(Hence,)28 b(challenges)g(remain)f(in)g
(discovering)43 1525 y(better)c(heuristics)g(that)h(reduce)f(the)g
(growth)g(of)h(the)f(search)g(space.)32 b(Indeed,)24
b(for)f(many)g(problems,)g(gains)h(due)f(to)h(new)43
1650 y(powerful)31 b(heuristics)e(may)g(be)i(much)e(higher)i(than)f
(those)g(due)g(to)g(massively)f(parallel)j(formulations)e(using)h
(existing)43 1774 y(heuristics.)h(These)23 b(powerful)h(heuristics)e
(may)h(also)h(of)o(fer)e(new)i(opportunities)g(for)f(exploiting)i
(parallelism.)43 2154 y Fj(References)85 2371 y Fn([1])40
b(S.)19 b(Arvindam,)f(V)o(ipin)f(Kumar)l(,)h(and)g(V)-7
b(.)18 b(Nageshwara)f(Rao.)25 b(Floorplan)16 b(optimization)f(on)k
(multiprocessors.)24 b(In)18 b Fb(Proceedings)f(of)209
2486 y(the)j(1989)f(International)d(Conference)i(on)i(Computer)e
(Design)p Fn(,)h(1989.)85 2625 y([2])40 b(R.)17 b(E.)h(Bixby)-6
b(,)18 b(W)l(.)f(Cook,)g(A.)g(Cox,)h(and)e(E.)h(Lee.)22
b(Parallel)15 b(mixed)i(integer)e(programming.)20 b(T)-8
b(echnical)15 b(Report)h(CRPC)h(TR)h(95554,)209 2739
y(Center)h(for)h(Research)g(on)g(Parallel)d(Computation,)h(Research)h
(Monograph,)f(1995.)85 2878 y([3])40 b(Adrian)20 b(Brungger)l(,)f
(Ambros)i(Marzetta,)g(Jens)h(Clausen,)e(and)g(Michael)h(Perregaard.)28
b(Solving)20 b(large\255scale)f(qap)i(problems)f(in)209
2992 y(parallel)e(with)h(the)h(search)g(library)f(zram.)29
b Fb(Journal)19 b(of)h(Parallel)e(and)h(Distributed)f(Computing)p
Fn(,)g(50:157\226169,)e(1998.)85 3131 y([4])40 b(B.)57
b(L.)g(Cun)f(and)g(C.)i(Roucairol.)123 b(BOB:)57 b(A)g(uni\002ed)f
(platform)f(for)i(implementing)d(branch\255and\255bound)e(like)k
(al\255)209 3246 y(gorithms.)132 b(T)-8 b(echnical)58
b(Report)g(N95/16,)68 b(Universiti)58 b(de)h(V)l(ersailles)f(Saint)h
(Quentin,)67 b(1995.)132 b(A)o(vailable)57 b(from)209
3360 y Fa(http://www.prism.uvsq.fr/english/paralle)o(l/cr/b)o(ob)p
2371 3360 23 4 v 21 w(us.html)p Fn(.)85 3499 y([5])40
b(E.)29 b(W)l(.)f(Dijkstra,)i(W)l(.)e(H.)g(Seijen,)h(and)e(A.)i(J.)g
(M.)g(V)-6 b(an)28 b(Gasteren.)49 b(Derivation)26 b(of)i(a)h
(termination)c(detection)h(algorithm)g(for)i(a)209 3613
y(distributed)18 b(computation.)25 b Fb(Information)18
b(Processing)h(Letters)p Fn(,)g(16\2555:217\226219,)d(1983.)85
3752 y([6])40 b(S.)28 b(Dutt)f(and)f(N.)i(R.)f(Mahapatra.)45
b(Scalable)25 b(load\255balancing)e(strategies)j(for)h(parallel)d(a*)k
(algorithms.)45 b Fb(Journal)26 b(of)h(Parallel)209 3866
y(and)f(Distributed)e(Computing)p Fn(,)h(22\(3\):488\226505,)e(Sept.)i
(1994.)43 b(Special)25 b(Issue)i(on)e(Scalability)g(of)h(Parallel)e
(Algorithms)g(and)209 3981 y(Architectures.)85 4120 y([7])40
b(Jonathan)13 b(Eckstein.)19 b(Parallel)13 b(branch\255and\255bound)d
(methods)k(for)h(mixed\255integer)d(programming)h(on)h(the)h(cm\2555.)
20 b Fb(SIAM)15 b(Journal)209 4234 y(on)20 b(Optimization)p
Fn(,)e(4\(4\):794\226814,)e(1994.)85 4373 y([8])40 b(Jonathan)14
b(Eckstein.)22 b(Distributed)13 b(versus)k(centralized)d(storage)h(and)
g(control)g(for)h(parallel)e(branch)h(and)g(bound:)25
b(Mixed)16 b(integer)209 4487 y(programming)i(on)i(the)g(cm\2555.)28
b Fb(Computational)17 b(Optimization)h(and)h(Applications)p
Fn(,)f(7\(2\):199\226220,)e(1997.)85 4626 y([9])40 b(M.)22
b(Evett,)f(James)h(Hendler)l(,)d(Ambujashka)h(Mahanti,)g(and)g(Dana)g
(Nau.)31 b(PRA*:)f(A)22 b(memory\255limited)d(heuristic)h(search)h
(proce\255)209 4741 y(dure)i(for)h(the)f(connection)e(machine.)37
b(In)24 b Fb(Proceedings)d(of)j(the)f(Third)g(Symposium)g(on)h(the)f
(Frontiers)f(of)i(Massively)g(Parallel)209 4855 y(Computation)p
Fn(,)17 b(pages)j(145\226149,)d(1990.)43 4994 y([10])40
b(Chris)17 b(Ferguson)e(and)h(Richard)g(Korf.)22 b(Distributed)15
b(tree)h(search)h(and)f(its)h(application)d(to)j(alpha\255beta)d
(pruning.)20 b(In)d Fb(Proceedings)209 5108 y(of)j(the)g(1988)f
(National)f(Conference)g(on)i(Arti\002cial)f(Intelligence)p
Fn(,)d(August)k(1988.)43 5247 y([11])40 b(A.)33 b(Ferreira)d(and)i(P)
-10 b(.)33 b(Pardalos,)g(editors.)59 b Fb(Solving)31
b(Combinatorial)e(Optimization)h(Problems)i(in)g(Parallel:)50
b(Methods)31 b(and)209 5361 y(T)-7 b(echniques)p Fn(,)18
b(volume)i(1054)f(of)h Fb(LNCS)g(State\255of\255the\255Art)c(Surveys)p
Fn(.)29 b(Springer\255V)l(erlag,)16 b(1996.)1947 5610
y Fk(12)p eop
%%Page: 13 13
13 12 bop 43 344 a Fn([12])40 b(Raphael)27 b(A.)i(Finkel)e(and)h(Udi)g
(Manber)l(.)50 b(DIB)29 b(\255)g(a)g(distributed)d(implementation)f(of)
j(backtracking.)50 b Fb(ACM)29 b(T)-6 b(ransactions)28
b(of)209 459 y(Programming)18 b(Languages)g(and)h(Systems)p
Fn(,)i(9)g(No.)f(2:235\226256,)c(April)j(1987.)43 606
y([13])40 b(M.)30 b(Furuichi,)f(K.)h(T)-8 b(aki,)31 b(and)d(N.)h
(Ichiyoshi.)52 b(A)29 b(multi\255level)e(load)h(balancing)f(scheme)i
(for)g(OR\255parallel)e(exhaustive)h(search)209 720 y(programs)20
b(on)g(the)h(Multi\255PSI.)28 b(In)20 b Fb(Proceedings)f(of)i(the)f
(Second)g(ACM)h(SIGPLAN)f(Symposium)h(on)f(Principles)f(and)h(Practice)
209 834 y(of)g(Parallel)e(Programming)p Fn(,)g(pages)h(50\22659,)g
(1990.)43 982 y([14])40 b(Scott)19 b(Hamilton)d(and)i(Lee)g(Garber)l(.)
24 b(Deep)18 b(Blue')o(s)f(hardware\255software)e(synergy)-6
b(.)26 b Fb(IEEE)18 b(Computer)p Fn(,)g(30\(10\):29\22635,)c(October)
209 1096 y(1997.)43 1243 y([15])40 b(F)-8 b(.\255H.)21
b(Hsu.)33 b(Large)20 b(scale)i(parallelization)17 b(of)22
b(alpha\255beta)c(search:)31 b(An)22 b(algorithmic)d(and)i
(architectural)f(study)h(with)h(computer)209 1357 y(chess.)29
b(T)-8 b(echnical)18 b(report,)h(Carnegie)f(Mellon)h(University)-6
b(,)20 b(Pittsburgh,)e(P)-6 b(A,)20 b(1990.)27 b(Ph.D.)20
b(Thesis.)43 1505 y([16])40 b(F)-8 b(.\255H.)19 b(Hsu,)h(M.)f(S.)h
(Campbell,)d(and)i(A.)h(J.)g(Hoane.)25 b(Deep)19 b(Blue)f(system)j
(overview)l(.)26 b(In)19 b Fb(Conference)e(proceedings)g(of)i(the)g
(1995)209 1619 y(International)d(Conference)i(on)i(Supercomputing,)d
(Barcelona,)h(Spain)p Fn(,)g(pages)h(240\226244,)f(1995.)43
1766 y([17])40 b(V)o(irendra)24 b(K.)i(Janakiram,)h(Dharma)e(P)-10
b(.)27 b(Agrawal,)f(and)f(Ram)i(Mehrotra.)43 b(A)26 b(randomized)e
(parallel)g(backtracking)h(algorithm.)209 1880 y Fb(IEEE)20
b(T)-6 b(ransactions)19 b(on)h(Computers)p Fn(,)f(C\25537)h(\(12\),)f
(1988.)43 2028 y([18])40 b(L.)28 b(Johnson,)h(G.)g(Nemhauser)l(,)f(and)
g(M.)g(Savelsbergh.)48 b(Progress)27 b(in)h(integer)e(programming:)42
b(An)28 b(exposition.)48 b(T)-8 b(echnical)209 2142 y(report,)35
b(School)c(of)i(Industrial)e(and)h(Systems)i(Engineering,)e(Georgia)f
(Institute)g(of)h(T)-8 b(echnology)i(,)34 b(1997.)60
b(A)o(vailable)31 b(from)209 2256 y Fa(http://akula.isye.gatech.edu/)40
b(mwps/mwps.html)p Fn(.)43 2403 y([19])g(L.)19 b(V)-7
b(.)20 b(Kale)e(and)g(V)-7 b(.)20 b(Saletore.)k(Parallel)17
b(state\255space)h(search)h(for)g(a)g(\002rst)h(solution)d(with)i
(consistent)f(linear)g(speedups.)25 b Fb(Interna\255)209
2517 y(tional)19 b(Journal)f(of)i(Parallel)e(Programming)p
Fn(,)g(3,)i(August)g(1990.)43 2665 y([20])40 b(L.)20
b(N.)h(Kanal)e(and)g(V)o(ipin)g(Kumar)l(.)27 b Fb(Search)20
b(in)g(Arti\002cial)f(Intelligence)p Fn(.)25 b(Springer\255V)l(erlag,)
15 b(New)20 b(Y)-7 b(ork,)21 b(NY)-10 b(,)20 b(1988.)43
2812 y([21])40 b(R.)21 b(Karp)f(and)g(Y)-10 b(.)21 b(Zhang.)28
b(Randomized)18 b(parallel)g(algorithms)h(for)h(backtrack)h(search)g
(and)f(branch\255and\255bound)15 b(computation.)209 2926
y Fb(Journal)k(of)h(ACM)p Fn(,)h(40:765\226789,)16 b(1993.)43
3074 y([22])40 b(George)20 b(Karypis)g(and)g(V)o(ipin)e(Kumar)l(.)29
b(Unstructured)19 b(T)m(ree)h(Search)g(on)g(SIMD)h(Parallel)d
(Computers.)28 b Fb(IEEE)21 b(T)-6 b(ransactions)19 b(on)209
3188 y(Parallel)f(and)h(Distributed)f(Systems)p Fn(,)j(5\(10\),)e
(October)h(1994.)43 3335 y([23])40 b(V)-7 b(.)24 b(Kumar)f(and)g(L.)g
(N.)h(Kanal.)36 b(A)24 b(general)d(branch\255and\255bound)e
(formulations)i(for)j(understanding)19 b(and)k(synthesizing)f(and/or)
209 3449 y(tree)e(search)g(procedures.)26 b Fb(Arti\002cial)20
b(Intelligence)p Fn(,)c(21:179\226198,)g(1983.)43 3597
y([24])40 b(V)o(ipin)23 b(Kumar)l(,)h(Ananth)e(Grama,)j(Anshul)e
(Gupta,)h(and)g(George)e(Karypis.)39 b Fb(Introduction)21
b(to)j(Parallel)e(Computing:)33 b(Algorithm)209 3711
y(Design)19 b(and)h(Analysis)p Fn(.)28 b(Benjamin)18
b(Cummings/)i(Addison)e(W)o(esley)i(\(ISBN)g(0\2558053\2553170\2550\),)
15 b(Redwod)k(City)-6 b(,)21 b(1994.)43 3858 y([25])40
b(V)o(ipin)17 b(Kumar)l(,)h(Ananth)f(Grama,)h(and)g(V)-7
b(.)19 b(Nageshwara)d(Rao.)25 b(Scalable)17 b(load)g(balancing)f
(techniques)g(for)j(parallel)c(computers.)209 3972 y
Fb(Journal)k(of)h(Parallel)e(and)h(Distributed)f(Computing)p
Fn(,)g(22\(1\):60\22679,)e(July)21 b(1994.)43 4120 y([26])40
b(T)-8 b(.)27 b(H.)h(Lai)e(and)h(Sartaj)f(Sahni.)46 b(Anomalies)25
b(in)i(parallel)e(branch)i(and)f(bound)g(algorithms.)45
b Fb(Communications)25 b(of)i(the)g(ACM)p Fn(,)209 4234
y(pages)19 b(594\226602,)f(1984.)43 4381 y([27])40 b(Eva)19
b(K.)h(Lee)e(and)g(John)h(E.)g(Mitchell.)25 b(Computational)15
b(experience)i(of)i(an)f(interior\255point)d(algorithm)i(in)i(a)g
(parallel)d(branch\255and\255)209 4495 y(cut)21 b(framework.)27
b(In)20 b Fb(Proceedings)e(for)i(SIAM)h(Conference)d(on)i(Parallel)d
(Processing)j(for)g(Scienti\002c)f(Computing)p Fn(,)f(1997.)43
4643 y([28])40 b(G.)28 b(J.)g(Li)f(and)g(B.)g(W)l(.)h(W)m(ah.)46
b(Computational)24 b(ef)o(\002ciency)j(of)g(combinatorial)e(or\255tree)
h(searches.)47 b Fb(IEEE)27 b(T)-6 b(rans.)27 b(on)g(Software)209
4757 y(Engineering)p Fn(,)16 b(16\(1\):13\22631,)h(Jan.)j(1990.)43
4904 y([29])40 b(Guo\255Jie)22 b(Li)h(and)f(Benjamin)f(W)l(.)h(W)m(ah.)
35 b(Coping)21 b(with)i(anomalies)e(in)h(parallel)e
(branch\255and\255bound)e(algorithms.)34 b Fb(IEEE)23
b(T)-6 b(rans\255)209 5019 y(actions)20 b(on)g(Computers)p
Fn(,)e(C\22635,)i(June)f(1986.)43 5166 y([30])40 b(N.)18
b(R.)f(Mahapatra)e(and)i(S.)h(Dutt.)k(Scalable)16 b(global)f(and)i
(local)f(hashing)g(strategies)g(for)h(duplicate)e(pruning)g(in)i
(parallel)d(a*)k(graph)209 5280 y(search.)28 b Fb(IEEE)20
b(T)-6 b(rans.)20 b(Parallel)e(and)i(Distributed)d(Systems)p
Fn(,)k(8\(7\),)f(July)g(1997.)1947 5610 y Fk(13)p eop
%%Page: 14 14
14 13 bop 43 344 a Fn([31])40 b(B.)19 b(Mans,)g(T)-8
b(.)18 b(Mautor)l(,)g(and)g(C.)h(Roucairol.)k(A)c(parallel)d(depth)i
(\002rst)h(search)g(branch)e(and)h(bound)f(for)h(the)g(quadratic)f
(assignment)209 459 y(problem.)27 b Fb(European)18 b(Journal)g(of)i
(Operational)d(Research)p Fn(,)j(81\(3\):617\226628,)15
b(1995.)43 606 y([32])40 b(B.)21 b(Mans)g(and)e(C.)i(Roucairol.)27
b(Performances)19 b(of)h(parallel)e(branch)i(and)f(bound)g(algorithms)g
(with)h(best\255\002rst)g(search.)29 b Fb(Discrete)209
720 y(Applied)18 b(Mathematics)p Fn(,)h(66\(1\):57\22676,)e(April)i
(1996.)43 867 y([33])40 b(G.)19 b(Manzini)f(and)g(M.)h(Somalvico.)25
b(Probabilistic)16 b(performance)h(analysis)h(of)h(heuristic)e(search)i
(using)f(parallel)e(hash)i(tables.)25 b(In)209 982 y
Fb(Proceedings)18 b(of)i(the)g(International)c(Symposium)k(on)g
(Arti\002cial)f(Intelligence)e(and)i(Mathematics)p Fn(,)g(1990.)43
1129 y([34])40 b(T)-8 b(.)16 b(A.)g(Marsland)f(and)g(M.)h(Campbell.)j
(Parallel)14 b(search)h(of)h(strongly)f(ordered)f(game)h(trees.)21
b Fb(Computing)14 b(Surveys)p Fn(,)j(14:533\226551,)209
1243 y(1982.)43 1390 y([35])40 b(D.)24 b(L.)g(Miller)e(and)h(J.)h(F)-8
b(.)24 b(Pekny)-6 b(.)38 b(The)23 b(role)f(of)i(performance)e(metrics)h
(for)h(parallel)d(mathematical)g(programming)h(algorithms.)209
1505 y Fb(ORSA)f(Journal)d(on)i(Computing)p Fn(,)e(5\(1\),)i(1993.)43
1652 y([36])40 b(M.)33 b(Padberg)e(and)g(G.)i(Rinaldi.)59
b(A)33 b(branch\255and\255cut)c(algorithm)h(for)j(the)f(resolution)e
(of)i(large\255scale)e(symmetric)k(traveling)209 1766
y(salesman)20 b(problems.)27 b Fb(SIAM)20 b(Rev)-6 b(.)p
Fn(,)21 b(33:60\226100,)16 b(1991.)43 1913 y([37])40
b(P)-10 b(.)20 b(Pardalos,)e(Y)-10 b(.)20 b(Li,)f(K.G.)g(Ramakrishna,)f
(and)g(M.G.C.)i(Resende.)25 b(Lower)18 b(bounds)g(for)i(the)e
(quadratic)f(assignment)h(problem.)209 2028 y Fb(Annals)c(of)h
(Operations)f(Research)p Fn(,)h(50:387\226411,)d(1994.)18
b(Special)c(V)l(olume)g(on)h(Applications)d(of)j(Combinatorial)d
(Optimization.)43 2175 y([38])40 b(Judea)23 b(Pearl.)38
b Fb(Heuristics\255Intelligent)19 b(Search)24 b(Strategies)e(for)i
(Computer)f(Problem)f(Solving)p Fn(.)38 b(Addison\255W)o(esley)-6
b(,)22 b(Reading,)209 2289 y(MA,)f(1984.)43 2437 y([39])40
b(Ira)35 b(Pramanick)f(and)h(Jon)f(G.)i(Kuhl.)66 b(An)35
b(inherently)e(parallel)f(method)i(for)g(heuristic)g
(problem\255solving:)55 b(Part)34 b(i\255general)209
2551 y(framework.)28 b Fb(IEEE)20 b(T)-6 b(ransactions)19
b(on)h(Parallel)d(and)j(Distributed)e(Systems)p Fn(,)j(6\(10\),)e
(October)g(1995.)43 2698 y([40])40 b(B.)18 b(Monien)e(R.)i(Feldmann,)e
(P)-10 b(.)18 b(Mysliwietz.)24 b(Studying)16 b(overheads)g(in)h
(massively)h(parallel)d(min/max\255tree)h(evaluation.)21
b(In)d Fb(Proc.)209 2812 y(of)i(the)g(6th)g(ACM)g(Symposium)g(on)g
(Parallel)e(Algorithms)h(and)g(Architectures)p Fn(,)g(pages)g
(94\226103,)f(1994.)43 2960 y([41])40 b(A.)26 b(G.)g(Ranade.)40
b(Optimal)24 b(speedup)f(for)j(backtrack)f(search)g(on)g(a)h
(butter\003y)e(network.)41 b(In)26 b Fb(Proceedings)d(of)i(the)g(Third)
f(ACM)209 3074 y(Symposium)c(on)g(Parallel)e(Algorithms)h(and)g
(Architectures)p Fn(,)g(1991.)43 3221 y([42])40 b(V)-7
b(.)30 b(Nageshwara)e(Rao)h(and)g(V)-7 b(.)30 b(Kumar)l(.)53
b(Concurrent)28 b(access)j(of)f(priority)e(queues.)52
b Fb(IEEE)30 b(T)-6 b(ransactions)28 b(on)i(Computers)p
Fn(,)209 3335 y(C\25537\(12\),)18 b(1988.)43 3483 y([43])40
b(V)-7 b(.)23 b(Nageshwara)d(Rao)i(and)g(V)o(ipin)f(Kumar)l(.)33
b(On)23 b(the)f(ef)o(\002cicency)g(of)h(parallel)c(backtracking.)33
b Fb(IEEE)23 b(T)-6 b(ransactions)21 b(on)h(Parallel)209
3597 y(and)f(Distributed)f(Systems)p Fn(,)j(4\(4\):427\226437,)17
b(April)k(1993.)31 b(Also)22 b(available)d(as)j(T)-8
b(echnical)20 b(Report)h(TR)h(90\25555,)e(Department)g(of)209
3711 y(Computer)f(Science,)g(University)h(of)g(Minnesota,)e
(Minneapolis,)g(MN.)43 3858 y([44])40 b(C.)26 b(Roucairol.)41
b(A)26 b(parallel)d(branch)h(and)h(bound)f(algorithm)f(for)j(the)f
(quadratic)e(assignment)i(problem.)41 b Fb(Discr)m(.)26
b(Appl.)e(Math.)p Fn(,)209 3972 y(18:211\226225,)16 b(1987.)43
4120 y([45])40 b(B.)29 b(Monien)e(S.)i(T)-8 b(schvke,)31
b(R.)e(L|ling.)50 b(Solving)27 b(the)h(traveling)f(salesman)h(problem)f
(with)h(a)h(distributed)d(branch\255and\255bound)209
4234 y(algorithm)21 b(on)j(a)f(1024)f(processor)h(network.)36
b(In)23 b Fb(Proc.)h(of)f(the)g(9th)g(International)c(Parallel)i
(Processing)i(Symposium)p Fn(,)g(pages)209 4348 y(182\226189,)18
b(Santa)h(Barbara,)f(CA,)i(April)f(1995.)43 4495 y([46])40
b(Benjamin)20 b(W)l(.)i(W)m(ah,)f(Gou\255Jie)g(Li,)h(and)f(C.)i(F)-8
b(.)21 b(Y)l(u.)33 b(Multiprocessing)19 b(of)j(combinatorial)d(search)j
(problems.)31 b Fb(IEEE)22 b(Computer)p Fn(,)209 4610
y(June)e(1985.)43 4757 y([47])40 b(Benjamin)20 b(W)l(.)i(W)m(ah)f(and)g
(Y)-10 b(.)22 b(W)l(.)g(Eva)g(Ma.)33 b(MANIP\227a)21
b(multicomputer)f(architecture)g(for)h(solving)g(combinatorial)e
(extremum\255)209 4871 y(search)h(problems.)27 b Fb(IEEE)20
b(T)-6 b(ransactions)19 b(on)h(Computers)p Fn(,)f(c\22633,)h(May)g
(1984.)1947 5610 y Fk(14)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF