20:
555:
to compare. Additionally, it may be possible to sort a partial order using a number of comparisons that is better than this analysis would suggest, because it may not be possible for this worst-case behavior to occur at each step of a sorting algorithm. In this direction, it has been conjectured that
532:
of elements may yield as little information as possible, by resolving the comparison in a way that leaves as many linear extensions as possible compatible with the comparison result. The 1/3β2/3 conjecture states that, at each step, one may choose a comparison to perform that reduces the remaining
292:
in the linear extension. In other words, it is a total order compatible with the partial order. If a finite partially ordered set is totally ordered, then it has only one linear extension, but otherwise it will have more than one. The 1/3β2/3 conjecture states that one can choose two elements
657:) for posets of width three is 14/39, and computer searches have shown that no smaller value is possible for width-3 posets with nine or fewer elements. Another related conjecture, again based on computer searches, states that there is a gap between 1/3 and the other possible values of Ξ΄(
44:
a set of items then, no matter what comparisons may have already been performed, it is always possible to choose the next comparison in such a way that it will reduce the number of possible sorted orders by a factor of 2/3 or better. Equivalently, in every finite
709:
If this conjecture is true, it would imply the 1/3β2/3 conjecture: the first of the two comparisons must be between a pair that splits the remaining comparisons by at worst a 1/3β2/3 ratio. The gold partition conjecture would also imply that a partial order with
27:
of one-element and three-element partial orders. Among its 27 linear extensions, the bottom left element occurs prior to the bottom right element in 9 out of 27. Partial orders with this structure are the only known extreme cases for the 1/3β2/3
331:
on the linear extensions in which each possible linear extension is equally likely to be chosen. The 1/3β2/3 conjecture states that, under this probability distribution, there exists a pair of elements
672:
Marcin
Peczarski has formulated a "gold partition conjecture" stating that in each partial order that is not a total order one can find two consecutive comparisons such that, if
1594:
1565:
272:. A total order is a partial order in which every pair of elements is comparable. A linear extension of a finite partial order is a sequential ordering of the elements of
486:) to extend its definition to certain infinite partial orders; in that context, they show that their bounds are optimal, in that there exist infinite partial orders with
440:
two, partial orders of height two, partial orders with at most 11 elements, partial orders in which each element is incomparable to at most six others,
1833:
391:
of the total number of linear extensions. In this notation, the 1/3β2/3 conjecture states that every finite partial order that is not total has
570:
A closely related class of comparison sorting problems was considered by
Fredman in 1976, among them the problem of comparison sorting a set
1239:
1720:
590:
is not necessarily generated as the set of linear extensions of a partial order. Despite this added generality, Fredman showed that
328:
641:
should tend to 1/2. In particular, they expect that only partially ordered sets of width two can achieve the worst case value
1129:
537:
linear extensions of the partial order given by the initial information, the sorting problem can be completed in at most log
441:
244:; they are the only known extreme cases for the conjecture and can be proven to be the only extreme cases with width two.
238:
218:
24:
1642:
1320:
1150:
Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture",
846:
to the problem of sorting partially ordered data and hence to the 1/3β2/3 conjecture, it is not mentioned in that paper.
356:
436:
The 1/3β2/3 conjecture is known to be true for certain special classes of partial orders, including partial orders of
425:, and remains unsolved; being called "one of the most intriguing problems in the combinatorial theory of posets."
1410:
1316:
1278:
509:
352:
482:
Their results improve previous weaker bounds of the same type. They use the probabilistic interpretation of Ξ΄(
233:
of elements, one of the two elements occurs earlier than the other in at most 1/3 of the linear extensions of
1161:
463:
In 1995, Graham
Brightwell, Stefan Felsner, and William Trotter proved that, for any finite partial order
265:
1823:
46:
221:
of three-element partial orders and of one-element partial orders, such as the one in the figure. Then
1828:
1739:
1570:
1541:
1357:(1989), "Optimal algorithms in convex programming decomposition and sorting", in Jaravlev, J. (ed.),
1166:
685:
of the comparisons have been made, then (in each of the four possible outcomes of the comparisons)
269:
1802:
1776:
1755:
1729:
1702:
1668:
1631:
1605:
1526:
1501:
Peczarski, Marcin (2019), "The worst balanced partially ordered setsβladders with broken rungs",
1489:
1458:
1427:
1393:
1376:
1342:
1304:
1222:
1187:
1115:
1084:
324:
261:
721:
comparisons; the name of the conjecture is derived from this connection with the golden ratio.
1767:
Zaguia, Imed (2019), "The 1/3β2/3 conjecture for ordered sets whose cover graph is a forest",
1680:
520:, for which some partial order information is already known in the form of a partial order on
323:
There is an alternative and equivalent statement of the 1/3β2/3 conjecture in the language of
1786:
1747:
1694:
1660:
1615:
1510:
1481:
1450:
1419:
1385:
1371:
1354:
1334:
1296:
1267:
1214:
1171:
1138:
1107:
1076:
725:
408:
62:
1798:
1627:
1522:
1247:
1183:
301:
such that, among this set of possible linear extensions, between 1/3 and 2/3 of them place
1794:
1685:
1651:
1623:
1518:
1472:
1441:
1325:
1255:
1243:
1234:
Felsner, Stefan; Trotter, William T. (1993), "Balancing pairs in partially ordered sets",
1205:
1179:
1152:
1098:
1067:
513:
421:
412:
257:
41:
176:
have the desired property, showing that this partial order obeys the 1/3β2/3 conjecture.
1743:
1646:
1062:
607:
1142:
1817:
1806:
1672:
1635:
1530:
1397:
1346:
1308:
1287:
1271:
1226:
1200:
1088:
1058:
650:
419:
in 1984. It was listed as a featured unsolved problem at the founding of the journal
1706:
1493:
1462:
1374:(1968), "A finite partially ordered set and its corresponding set of permutations",
1191:
1119:
1759:
1431:
564:
512:
and Saks proposed the following application for the problem: suppose one wishes to
33:
1514:
610:. This same bound applies as well to the case of partial orders and shows that log
225:
forms an extreme case for the 1/3β2/3 conjecture in the sense that, for each pair
179:
This example shows that the constants 1/3 and 2/3 in the conjecture are tight; if
1405:
1282:
416:
183:
is any fraction strictly between 1/3 and 2/3, then there would not exist a pair
50:
1790:
1619:
1485:
1454:
1470:
Peczarski, Marcin (2008), "The gold partition conjecture for 6-thin posets",
547:
However, this analysis neglects the complexity of selecting the optimal pair
445:
437:
241:
460:-element partial orders that obey the 1/3β2/3 conjecture approaches 100%.
19:
449:
533:
number of linear extensions by a factor of 2/3; therefore, if there are
1698:
1664:
1389:
1338:
1300:
1218:
1175:
1111:
1096:
Brightwell, Graham R. (1989), "Semiorders and the 1/3β2/3 conjecture",
1080:
801:
799:
797:
795:
653:
stated this explicitly as a conjecture. The smallest known value of Ξ΄(
881:
879:
1423:
1781:
1610:
1751:
1734:
1127:
Brightwell, Graham R. (1999), "Balanced pairs in partial orders",
18:
1258:(1976), "How good is the information theory bound in sorting?",
1408:(1984), "The information-theoretic bound is good for merging",
524:. In the worst case, each additional comparison between a pair
1238:, Bolyai Society Mathematical Studies, vol. 1, Budapest:
1439:
Peczarski, Marcin (2006), "The gold partition conjecture",
61:
with the property that at least 1/3 and at most 2/3 of the
740:, to calculate the proportion of the linear extensions of
629:
In 1984, Kahn and Saks conjectured that, in the limit as
1364:
967:
805:
771:
681:
denotes the number of linear extensions remaining after
885:
387:
in a number of linear extensions that is between Ξ΄ and
1575:
1546:
309:, and symmetrically between 1/3 and 2/3 of them place
1573:
1544:
348:
in a random linear extension is between 1/3 and 2/3.
237:. Partial orders with this structure are necessarily
1285:(1991), "Balancing extensions via Brunn-Minkowski",
782:
780:
1588:
1559:
1714:Zaguia, Imed (2012), "The 1/3-2/3 Conjecture for
1042:
428:A survey of the conjecture was produced in 1999.
199:in a number of partial orderings that is between
1683:(1992), "Balance theorems for height-2 posets",
983:
714:linear extensions can be sorted in at most log
1647:"Balancing linear extensions of ordered sets"
1361:(in Russian), Moscow: Nauka, pp. 161β205
563:comparisons may suffice, where Ο denotes the
210:times the total number of partial orderings.
8:
661:): whenever a partial order does not have Ξ΄(
367:, to be the largest real number Ξ΄ such that
1679:Trotter, William T.; Gehrlein, William V.;
816:
814:
81:The partial order formed by three elements
979:
943:
855:
93:with a single comparability relationship,
1780:
1733:
1609:
1574:
1572:
1545:
1543:
1165:
1030:
975:
919:
904:
842:However, despite the close connection of
831:
411:in 1968, and later made independently by
407:The 1/3β2/3 conjecture was formulated by
1365:Brightwell, Felsner & Trotter (1995)
1006:
971:
968:Brightwell, Felsner & Trotter (1995)
820:
806:Brightwell, Felsner & Trotter (1995)
772:Brightwell, Felsner & Trotter (1995)
767:
995:
886:Trotter, Gehrlein & Fishburn (1992)
843:
760:
1323:(1984), "Balancing poset extensions",
1203:(1991), "Counting linear extensions",
955:
931:
915:
913:
900:
898:
896:
894:
866:
786:
637:) for partially ordered sets of width
168:in the third. Therefore, the pair of
7:
1018:
456:goes to infinity, the proportion of
164:in only two of them, and later than
1721:Electronic Journal of Combinatorics
1538:Sah, Ashwin (2021), "Improving the
1236:Combinatorics, Paul ErdΕs is eighty
870:
625:Generalizations and related results
1596:conjecture for width two posets",
633:tends to infinity, the value of Ξ΄(
148:In all three of these extensions,
53:, there exists a pair of elements
16:Unsolved problem on partial orders
14:
1240:JΓ‘nos Bolyai Mathematical Society
363:), for any partially ordered set
252:A partially ordered set is a set
1834:Unsolved problems in mathematics
728:, given a finite partial order
329:uniform probability distribution
1589:{\displaystyle {\tfrac {2}{3}}}
1560:{\displaystyle {\tfrac {1}{3}}}
1359:Computers and Decision Problems
1043:Brightwell & Winkler (1991)
340:such that the probability that
36:, a branch of mathematics, the
442:series-parallel partial orders
23:A partial order formed by the
1:
1515:10.1080/10586458.2017.1368050
1143:10.1016/S0012-365X(98)00311-2
606:|) comparisons, expressed in
104:has three linear extensions,
1272:10.1016/0304-3975(76)90078-5
1260:Theoretical Computer Science
984:Felsner & Trotter (1993)
578:is known to lie in some set
276:, with the property that if
284:in the partial order, then
65:of the partial order place
1850:
1791:10.1007/s11083-018-9469-0
1620:10.1007/s00493-020-4091-3
1486:10.1007/s11083-008-9081-9
1455:10.1007/s11083-006-9033-1
1411:SIAM Journal on Computing
574:when the sorted order of
1503:Experimental Mathematics
980:Kahn & Linial (1991)
665:) exactly 1/3, it has Ξ΄(
544:additional comparisons.
1199:Brightwell, Graham R.;
732:and a pair of elements
621:) comparisons suffice.
594:can be sorted using log
40:states that, if one is
1590:
1561:
1007:Kahn & Saks (1984)
972:Kahn & Saks (1984)
821:Kahn & Saks (1984)
768:Kahn & Saks (1984)
516:a totally ordered set
29:
1718:-free ordered sets",
1649:, Unsolved problems,
1591:
1562:
47:partially ordered set
22:
1571:
1542:
1242:, pp. 145β157,
1130:Discrete Mathematics
213:More generally, let
1744:2011arXiv1107.5626Z
1063:"A note on merging"
582:of permutations of
467:that is not total,
327:. One may define a
1699:10.1007/BF00419038
1681:Fishburn, Peter C.
1665:10.1007/BF00333138
1586:
1584:
1557:
1555:
1390:10.1007/BF01111312
1377:Mathematical Notes
1339:10.1007/BF00565647
1301:10.1007/BF01275670
1219:10.1007/BF00383444
1176:10.1007/BF01110378
1112:10.1007/BF00353656
1081:10.1007/BF00333131
452:. In the limit as
325:probability theory
219:series composition
42:comparison sorting
38:1/3β2/3 conjecture
30:
25:series composition
1583:
1554:
1355:Khachiyan, Leonid
944:Brightwell (1989)
856:Brightwell (1999)
602:| + O(|
288:must come before
63:linear extensions
1841:
1809:
1784:
1762:
1737:
1709:
1675:
1638:
1613:
1595:
1593:
1592:
1587:
1585:
1576:
1566:
1564:
1563:
1558:
1556:
1547:
1533:
1496:
1465:
1434:
1400:
1372:Kislitsyn, S. S.
1362:
1349:
1311:
1274:
1250:
1229:
1194:
1169:
1145:
1122:
1091:
1046:
1040:
1034:
1031:Peczarski (2019)
1028:
1022:
1016:
1010:
1004:
998:
993:
987:
976:Khachiyan (1989)
965:
959:
953:
947:
941:
935:
929:
923:
920:Peczarski (2008)
917:
908:
905:Peczarski (2006)
902:
889:
883:
874:
864:
858:
853:
847:
840:
834:
832:Kislitsyn (1968)
829:
823:
818:
809:
803:
790:
784:
775:
765:
708:
648:
500:
498:
497:
492:) = 1/2 −
481:
479:
478:
473:) β₯ 1/2 −
409:Sergey Kislitsyn
398:
390:
344:is earlier than
320:
256:together with a
209:
195:is earlier than
160:is earlier than
152:is earlier than
147:
132:
118:
103:
1849:
1848:
1844:
1843:
1842:
1840:
1839:
1838:
1814:
1813:
1766:
1713:
1678:
1641:
1569:
1568:
1540:
1539:
1537:
1500:
1469:
1438:
1424:10.1137/0213049
1404:
1370:
1353:
1315:
1277:
1254:
1233:
1198:
1149:
1126:
1095:
1057:
1054:
1049:
1041:
1037:
1029:
1025:
1017:
1013:
1005:
1001:
994:
990:
966:
962:
954:
950:
942:
938:
930:
926:
918:
911:
903:
892:
884:
877:
865:
861:
854:
850:
841:
837:
830:
826:
819:
812:
804:
793:
785:
778:
766:
762:
758:
717:
706:
699:
692:
686:
680:
642:
627:
617: + O(
613:
597:
559:
540:
514:comparison sort
506:
495:
493:
487:
476:
474:
468:
434:
432:Partial results
413:Michael Fredman
405:
392:
388:
314:
258:binary relation
250:
239:series-parallel
204:
134:
119:
105:
94:
79:
51:totally ordered
17:
12:
11:
5:
1847:
1845:
1837:
1836:
1831:
1826:
1816:
1815:
1812:
1811:
1775:(2): 335β347,
1764:
1711:
1676:
1639:
1582:
1579:
1553:
1550:
1535:
1509:(2): 181β184,
1498:
1467:
1436:
1418:(4): 795β801,
1402:
1384:(5): 798β801,
1368:
1363:. As cited by
1351:
1333:(2): 113β126,
1313:
1295:(4): 363β368,
1275:
1266:(4): 355β361,
1256:Fredman, M. L.
1252:
1231:
1213:(3): 225β242,
1201:Winkler, Peter
1196:
1167:10.1.1.38.7841
1160:(4): 327β349,
1147:
1137:(1β3): 25β52,
1124:
1106:(4): 369β380,
1093:
1075:(3): 257β264,
1059:Aigner, Martin
1053:
1050:
1048:
1047:
1035:
1023:
1011:
999:
996:Fredman (1976)
988:
960:
948:
936:
924:
909:
890:
875:
859:
848:
844:Fredman (1976)
835:
824:
810:
791:
776:
759:
757:
754:
715:
704:
697:
690:
676:
669:) β₯ 0.348843.
626:
623:
611:
608:big O notation
595:
557:
538:
505:
502:
433:
430:
404:
401:
249:
246:
78:
75:
15:
13:
10:
9:
6:
4:
3:
2:
1846:
1835:
1832:
1830:
1827:
1825:
1822:
1821:
1819:
1808:
1804:
1800:
1796:
1792:
1788:
1783:
1778:
1774:
1770:
1765:
1761:
1757:
1753:
1752:10.37236/2345
1749:
1745:
1741:
1736:
1731:
1727:
1723:
1722:
1717:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1687:
1682:
1677:
1674:
1670:
1666:
1662:
1658:
1654:
1653:
1648:
1644:
1643:Saks, Michael
1640:
1637:
1633:
1629:
1625:
1621:
1617:
1612:
1607:
1604:(1): 99β126,
1603:
1599:
1598:Combinatorica
1580:
1577:
1551:
1548:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1499:
1495:
1491:
1487:
1483:
1480:(2): 91β103,
1479:
1475:
1474:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1443:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1412:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1378:
1373:
1369:
1366:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1327:
1322:
1321:Saks, Michael
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1289:
1288:Combinatorica
1284:
1280:
1276:
1273:
1269:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1207:
1202:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1168:
1163:
1159:
1155:
1154:
1148:
1144:
1140:
1136:
1132:
1131:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1100:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1069:
1064:
1060:
1056:
1055:
1051:
1044:
1039:
1036:
1032:
1027:
1024:
1020:
1015:
1012:
1008:
1003:
1000:
997:
992:
989:
985:
981:
977:
973:
969:
964:
961:
957:
956:Zaguia (2019)
952:
949:
945:
940:
937:
933:
932:Zaguia (2012)
928:
925:
921:
916:
914:
910:
906:
901:
899:
897:
895:
891:
887:
882:
880:
876:
872:
869:, Theorem 2;
868:
867:Linial (1984)
863:
860:
857:
852:
849:
845:
839:
836:
833:
828:
825:
822:
817:
815:
811:
807:
802:
800:
798:
796:
792:
788:
787:Aigner (1985)
783:
781:
777:
773:
769:
764:
761:
755:
753:
751:
748:earlier than
747:
743:
739:
735:
731:
727:
722:
720:
713:
703:
696:
689:
684:
679:
675:
670:
668:
664:
660:
656:
652:
651:Martin Aigner
646:
640:
636:
632:
624:
622:
620:
616:
609:
605:
601:
593:
589:
585:
581:
577:
573:
568:
566:
562:
554:
550:
545:
543:
536:
531:
527:
523:
519:
515:
511:
503:
501:
491:
485:
472:
466:
461:
459:
455:
451:
447:
443:
439:
431:
429:
426:
424:
423:
418:
414:
410:
402:
400:
396:
386:
383:earlier than
382:
378:
374:
370:
366:
362:
358:
354:
349:
347:
343:
339:
335:
330:
326:
321:
318:
312:
308:
305:earlier than
304:
300:
296:
291:
287:
283:
279:
275:
271:
267:
266:antisymmetric
263:
259:
255:
247:
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1406:Linial, Nati
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31:
1829:Conjectures
1659:: 327β330,
1019:Saks (1985)
744:that place
726:#P-complete
417:Nati Linial
389:1 − Ξ΄
371:has a pair
248:Definitions
156:. However,
28:conjecture.
1818:Categories
1782:1610.00809
1728:(2): P29,
1611:1811.01500
1317:Kahn, Jeff
1279:Kahn, Jeff
1052:References
871:Sah (2021)
446:semiorders
359:defined Ξ΄(
270:transitive
260:β€ that is
242:semiorders
205:1 −
1807:119631612
1735:1107.5626
1673:189901558
1636:119604901
1531:125593629
1398:120228193
1347:123370506
1309:206793172
1227:119697949
1162:CiteSeerX
1089:118877843
510:Jeff Kahn
450:polytrees
353:Jeff Kahn
351:In 1984,
262:reflexive
191:in which
1707:16157076
1645:(1985),
1494:42034095
1463:42415160
1192:14793475
1120:86860160
1061:(1985),
647:) = 1/3,
508:In 1984
313:earlier
1799:3983482
1760:1979845
1740:Bibcode
1628:4235316
1523:3955809
1432:5149351
1248:1249709
1184:1368815
586:. Here
494:√
475:√
415:and by
403:History
397:) β₯ 1/3
217:be any
77:Example
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268:, and
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1803:S2CID
1777:arXiv
1769:Order
1756:S2CID
1730:arXiv
1703:S2CID
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1669:S2CID
1652:Order
1632:S2CID
1606:arXiv
1527:S2CID
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1459:S2CID
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1223:S2CID
1206:Order
1188:S2CID
1153:Order
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1085:S2CID
1068:Order
756:Notes
438:width
422:Order
379:with
315:than
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355:and
336:and
297:and
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