51:
214:. This can be done by copying the sub-arrays into a temporary array, then applying the merge algorithm above. The allocation of a temporary array can be avoided, but at the expense of speed and programming ease. Various in-place merge algorithms have been devised, sometimes sacrificing the linear-time bound to produce an
91:
An example merge sort is given in the illustration. It starts with an unsorted array of 7 integers. The array is divided into 7 partitions; each partition contains 1 element and is sorted. The sorted partitions are then merged to produce larger, sorted, partitions, until 1 partition, the sorted
1062:. In 2018, Saitoh M. et al. introduced MMS for FPGAs, which focused on removing a multi-cycle feedback datapath that prevented efficient pipelining in hardware. Also in 2018, Papaphilippou P. et al. introduced FLiMS that improved the hardware utilization and performance by only requiring
922:
194:
When the inputs are linked lists, this algorithm can be implemented to use only a constant amount of working space; the pointers in the lists' nodes can be reused for bookkeeping and for constructing the final merged list.
1003:
692:, and since the recursive calls are independent of each other, they can be done in parallel. Hybrid approach, where serial algorithm is used for recursion base case has been shown to perform well in practice
814:
54:
A graph exemplifying merge sort. Two red arrows starting from the same node indicates subdivision, while two green arrows ending in the same node corresponds to an execution of the merge algorithm.
662:, whichever is larger, into (nearly) equal halves. It then splits the other array into a part with values smaller than the midpoint of the first, and a part with larger or equal values. (The
807:
1540:
Saitoh, Makoto; Elsayed, Elsayed A.; Chu, Thiem Van; Mashimo, Susumu; Kise, Kenji (April 2018). "A High-Performance and Cost-Effective
Hardware Merge Sorter without Feedback Datapath".
73:
divide the list into sublists of (roughly) equal length, until each sublist contains only one element, or in the case of iterative (bottom up) merge sort, consider a list of
1105:
762:
734:
are in parallel, only the costlier of the two calls needs to be considered. In the worst case, the maximum number of elements in one of the recursive calls is at most
1043:
There are also algorithms that introduce parallelism within a single instance of merging of two sorted lists. These can be used in field-programmable gate arrays (
1035:
will destroy the order, if equal items are spread among both input arrays. As a result, when used for sorting, this algorithm produces a sort that is not stable.
541:
comparisons, i.e., less than half the number used by the heap-based algorithm; in practice, it may be about as fast or slow as the heap-based algorithm.
929:
30:
lists as input and produce a single list as output, containing all the elements of the inputs lists in sorted order. These algorithms are used as
1557:
1375:
1303:
1264:
132:
yields the first element of a list; "dropping" an element means removing it from its list, typically by incrementing a pointer or index.
84:
Repeatedly merge sublists to create a new sorted sublist until the single list contains all elements. The single list is the sorted list.
1289:
1692:
1637:
1462:
1256:
917:{\displaystyle T_{\infty }^{\text{merge}}(n)=T_{\infty }^{\text{merge}}\left({\frac {3}{4}}n\right)+\Theta \left(\log(n)\right)}
1715:
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723:
696:
2023:
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327:
70:
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1414:
1452:
1640:. Pages 158–160 of section 5.2.4: Sorting by Merging. Section 5.3.2: Minimum-Comparison Merging, pp. 197–207.
1499:
Papaphilippou, Philippos; Luk, Wayne; Brooks, Chris (2022). "FLiMS: a Fast
Lightweight 2-way Merger for Sorting".
1730:
1215:
1148:
767:
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1210:
427:
Searching for the next smallest element to be output (find-min) and restoring heap order can now be done in
1936:
1816:
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method which merges another list into itself. The type of the elements merged must support the less-than (
2097:
1685:
1047:), specialized sorting circuits, as well as in modern processors with single-instruction multiple-data (
369:
242:
1182:
C++17 allows for differing execution policies, namely sequential, parallel, and parallel-unsequenced.
2076:
1941:
1796:
1440:
530:
When the input lists to this algorithm are ordered by length, shortest first, it requires fewer than
113:
1542:
2018 IEEE 26th Annual
International Symposium on Field-Programmable Custom Computing Machines (FCCM)
1358:
1065:
558:
2092:
1831:
1333:
1005:, meaning that it takes that much time on an ideal machine with an unbounded number of processors.
727:
1352:. European Symp. Algorithms. Lecture Notes in Computer Science. Vol. 3221. pp. 714–723.
1982:
1957:
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lists to pick off the minimum element each time, and repeat this loop until all lists are empty:
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1987:
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1319:
Katajainen, Jyrki; Pasanen, Tomi; Teuhola, Jukka (1996). "Practical in-place mergesort".
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Existing parallel algorithms are based on modifications of the merge part of either the
2109:
2018:
1885:
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module, that takes multiple sorted iterables, and merges them into a single iterator.
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1992:
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Several solutions to this problem exist. A naive solution is to do a loop over the
81:
sub-lists of size 1. A list containing a single element is, by definition, sorted.
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and linear or constant space (depending on the data access model). The following
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1967:
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1997:
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998:{\displaystyle T_{\infty }^{\text{merge}}(n)=\Theta \left(\log(n)^{2}\right)}
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blocks that fit in memory, sorts these one by one, then merges these blocks.
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2002:
1972:
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170:// By now, either A or B is empty. It remains to empty the other input list.
23:
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since the array with more elements is perfectly split in half. Adding the
2135:
2130:
1844:
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version of the binary merge algorithm can serve as a building block of a
809:
cost of the Binary Search, we obtain this recurrence as an upper bound:
2061:
620:// ensure that A is the larger array: i, j still belong to A; k, ℓ to B
1670:
1393:
Patience is a Virtue: Revisiting Merge and Sort on Modern
Processors
606:
A, B, C : array i, j, k, ℓ, p, q : indices
364:
elements in the lists. It can be improved by storing the lists in a
88:
The merge algorithm is used repeatedly in the merge sort algorithm.
1662:
1144:
315:
Loop over the lists to find the one with the minimum first element.
49:
1048:
1044:
360:
element comparisons to perform its work if there are a total of
66:. Conceptually, the merge sort algorithm consists of two steps:
1674:
703:
elements, i.e., the running time of a serial version of it, is
250:-way merging generalizes binary merging to an arbitrary number
1179:) operator, or it must be provided with a custom comparator.
699:
performed by the algorithm for two arrays holding a total of
1603:"heapq — Heap queue algorithm — Python 3.10.1 documentation"
557:. The following pseudocode demonstrates this algorithm in a
258:-way merging arise in various sorting algorithms, including
164:append head(A) to C drop the head of A
1422:. Proc. 31-st Annual ACM Southeast Conf. pp. 127–135.
108:
demonstrates an algorithm that merges input lists (either
1350:
Stable
Minimum Storage Merging by Symmetric Comparisons
1131:
provide built-in or library support for merging sorted
318:
Output the minimum element and remove it from its list.
168:append head(B) to C drop the head of B
1113:
compare-and-swap units to merge with a parallelism of
740:
16:
Algorithm that combines multiple sorted lists into one
1068:
932:
817:
770:
480:
solution that builds on the binary merge algorithm:
446:
comparisons), and the full problem can be solved in
2118:
2085:
2042:
2011:
1950:
1919:
1878:
1830:
1784:
1708:
1391:Chandramouli, Badrish; Goldstein, Jonathan (2014).
187:append head(B) to C drop the head of B
179:append head(A) to C drop the head of A
1099:
997:
916:
801:
756:
1295:Algorithms and Data Structures: The Basic Toolbox
100:Merging two sorted lists into one can be done in
58:The merge algorithm plays a critical role in the
726:of the algorithm, it is necessary to derive a
1686:
1163:, which merges two consecutive sorted ranges
8:
1015:: if equal items are separated by splitting
1457:(3rd ed.). MIT Press and McGraw-Hill.
1193:'s standard library (since 2.6) also has a
802:{\displaystyle \Theta \left(\log(n)\right)}
654:The algorithm operates by splitting either
1693:
1679:
1671:
688:.) Finally, each pair of halves is merged
387:lists, using the first element as the key.
202:is typically used to merge two sub-arrays
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647:t = p + (r - i) + (s - k) C = A
1632:, Third Edition. Addison-Wesley, 1997.
1237:
476:A third algorithm for the problem is a
254:of sorted input lists. Applications of
1431:
1429:
651:merge(A, B, C) merge(A, B, C)
573:and writes the sorted output to array
266:algorithm that divides its input into
149:list C := new empty list
390:While any of the lists is non-empty:
312:While any of the lists is non-empty:
7:
1348:Kim, Pok-Son; Kutzner, Arne (2004).
1155:, which merges two sorted ranges of
680:; that this always a number between
565:). It operates on two sorted arrays
730:. Since the two recursive calls of
1023:, they will become interleaved in
960:
938:
886:
850:
823:
771:
507:Else, recursively merge the first
198:In the merge sort algorithm, this
153:A is not empty and B is not empty
64:comparison-based sorting algorithm
14:
610:m = j - i, n = ℓ - k
407:Output the first element of list
1653:of Parallel and Serial Merge in
1171:(linked list) class has its own
718:elements need to be copied into
666:subroutine returns the index in
372:) keyed by their first element:
1716:Computational complexity theory
1651:High Performance Implementation
1625:The Art of Computer Programming
1257:Springer Science+Business Media
523:lists, then binary merge these.
494:, output the single input list.
1584:. cppreference.com. 2018-01-08
1501:IEEE Transactions on Computers
1100:{\displaystyle \log _{2}(P)+1}
1088:
1082:
981:
974:
954:
948:
906:
900:
839:
833:
791:
785:
634:// base case, nothing to merge
1:
1416:k-way Merging and k-ary Sorts
1478:Victor J. Duvanenko (2011),
1368:10.1007/978-3-540-30140-0_63
757:{\textstyle {\frac {3}{4}}n}
643:s = binary-search(A, B)
411:and remove it from its list.
1413:Greene, William A. (1993).
1253:The Algorithm Design Manual
1039:Parallel merge of two lists
561:style (adapted from Cormen
559:parallel divide-and-conquer
2173:
2024:Batcher odd–even mergesort
1454:Introduction to Algorithms
330:, this algorithm performs
240:
231:Merge sort § Variants
1216:Join (relational algebra)
1149:Standard Template Library
1119:elements per FPGA cycle.
504:, perform a binary merge.
438:time (more specifically,
2029:Pairwise sorting network
1211:Merge (revision control)
714:. This is optimal since
676:would be, if it were in
2057:Kirkpatrick–Reisch sort
1550:10.1109/FCCM.2018.00038
1509:10.1109/TC.2022.3146509
1937:Oscillating merge sort
1817:Proportion extend sort
1771:Transdichotomous model
1101:
999:
918:
803:
758:
55:
2098:Pre-topological order
1630:Sorting and Searching
1441:Leiserson, Charles E.
1102:
1000:
919:
804:
759:
243:K-way merge algorithm
145:A, B : list
53:
2077:Merge-insertion sort
1942:Polyphase merge sort
1797:Cocktail shaker sort
1544:. pp. 197–204.
1066:
930:
815:
768:
738:
639:r = ⌊(i + j)/2⌋
581:denotes the part of
515:lists and the final
461:time (approximately
2093:Topological sorting
1855:Cartesian tree sort
1321:Nordic J. Computing
1167:. In addition, the
1107:pipeline stages of
1011:The routine is not
947:
859:
832:
728:Recurrence relation
722:. To calculate the
555:parallel merge sort
26:that take multiple
2157:Sorting algorithms
1983:Interpolation sort
1958:American flag sort
1951:Distribution sorts
1932:Cascade merge sort
1702:Sorting algorithms
1484:Dr. Dobb's Journal
1161:std::inplace_merge
1129:computer languages
1097:
1060:odd-even mergesort
995:
933:
914:
845:
818:
799:
754:
478:divide and conquer
210:of a single array
160:head(A) ≤ head(B)
56:
36:sorting algorithms
2144:
2143:
2119:Impractical sorts
1559:978-1-5386-5522-1
1445:Rivest, Ronald L.
1437:Cormen, Thomas H.
1377:978-3-540-23025-0
1305:978-3-540-77978-0
1266:978-1-849-96720-4
1151:has the function
1051:) instructions.
945:
873:
857:
830:
749:
622:swap m and n
379:Build a min-heap
328:In the worst case
305:Input: a list of
96:Merging two lists
2164:
2052:Block merge sort
2012:Concurrent sorts
1911:Patience sorting
1695:
1688:
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1607:
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1589:
1578:
1572:
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1496:
1487:
1486:
1480:"Parallel Merge"
1475:
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1316:
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1282:
1271:
1270:
1255:(2nd ed.).
1245:
1200:
1197:function in the
1196:
1178:
1174:
1170:
1162:
1154:
1123:Language support
1118:
1112:
1106:
1104:
1103:
1098:
1078:
1077:
1034:
1030:
1027:; also swapping
1026:
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1004:
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989:
988:
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943:
941:
926:The solution is
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308:
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290:
288:
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264:external sorting
260:patience sorting
257:
253:
249:
233:for discussion.
228:
213:
209:
205:
131:
127:
124:into a new list
123:
119:
92:array, is left.
38:, most famously
22:are a family of
20:Merge algorithms
2172:
2171:
2167:
2166:
2165:
2163:
2162:
2161:
2147:
2146:
2145:
2140:
2114:
2105:Pancake sorting
2081:
2038:
2007:
1988:Pigeonhole sort
1946:
1915:
1879:Insertion sorts
1874:
1860:Tournament sort
1832:Selection sorts
1826:
1780:
1761:Integer sorting
1756:Sorting network
1746:Comparison sort
1704:
1699:
1657:with source in
1647:
1616:
1614:Further reading
1611:
1610:
1601:
1600:
1596:
1587:
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1539:
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1476:
1472:
1465:
1449:Stein, Clifford
1435:
1434:
1427:
1419:
1412:
1411:
1400:
1390:
1389:
1385:
1378:
1359:10.1.1.102.4612
1347:
1346:
1342:
1318:
1317:
1313:
1306:
1284:
1283:
1274:
1267:
1259:. p. 123.
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1234:
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659:
655:
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599:merge(A, B, C)
590:
586:
582:
578:
577:. The notation
574:
570:
566:
547:
531:
528:
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229:algorithm; see
215:
211:
207:
203:
192:
183:B is not empty
175:A is not empty
129:
128:. The function
125:
121:
117:
98:
48:
17:
12:
11:
5:
2170:
2168:
2160:
2159:
2149:
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2141:
2139:
2138:
2133:
2128:
2122:
2120:
2116:
2115:
2113:
2112:
2110:Spaghetti sort
2107:
2102:
2101:
2100:
2089:
2087:
2083:
2082:
2080:
2079:
2074:
2069:
2064:
2059:
2054:
2048:
2046:
2040:
2039:
2037:
2036:
2031:
2026:
2021:
2019:Bitonic sorter
2015:
2013:
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1923:
1921:
1917:
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1914:
1913:
1908:
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1898:
1893:
1888:
1886:Insertion sort
1882:
1880:
1876:
1875:
1873:
1872:
1870:Weak-heap sort
1867:
1862:
1857:
1852:
1847:
1842:
1840:Selection sort
1836:
1834:
1828:
1827:
1825:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1788:
1786:
1785:Exchange sorts
1782:
1781:
1779:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1738:
1733:
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1721:Big O notation
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1645:External links
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1488:
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1395:. SIGMOD/PODS.
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1334:10.1.1.22.8523
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1272:
1265:
1249:Skiena, Steven
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1096:
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1072:
1056:bitonic sorter
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987:
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649:in parallel do
618:swap A and B
595:
546:
545:Parallel merge
543:
525:
524:
505:
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484:
483:
482:
473:comparisons).
422:
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366:priority queue
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241:Main article:
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2158:
2155:
2154:
2152:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2121:
2117:
2111:
2108:
2106:
2103:
2099:
2096:
2095:
2094:
2091:
2090:
2088:
2084:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2049:
2047:
2045:
2041:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2016:
2014:
2010:
2004:
2001:
1999:
1996:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1978:Counting sort
1976:
1974:
1971:
1969:
1966:
1964:
1961:
1959:
1956:
1955:
1953:
1949:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1924:
1922:
1918:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1883:
1881:
1877:
1871:
1868:
1866:
1863:
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1858:
1856:
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1848:
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1841:
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1835:
1833:
1829:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1802:Odd–even sort
1800:
1798:
1795:
1793:
1790:
1789:
1787:
1783:
1777:
1774:
1772:
1769:
1767:
1766:X + Y sorting
1764:
1762:
1759:
1757:
1754:
1752:
1751:Adaptive sort
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1713:
1711:
1707:
1703:
1696:
1691:
1689:
1684:
1682:
1677:
1676:
1673:
1667:
1664:
1660:
1656:
1652:
1649:
1648:
1644:
1639:
1638:0-201-89685-0
1635:
1631:
1627:
1626:
1621:
1618:
1617:
1613:
1604:
1598:
1595:
1583:
1577:
1574:
1569:
1565:
1561:
1555:
1551:
1547:
1543:
1536:
1533:
1528:
1524:
1519:
1518:10044/1/95271
1514:
1510:
1506:
1502:
1495:
1493:
1489:
1485:
1481:
1474:
1471:
1466:
1464:0-262-03384-4
1460:
1456:
1455:
1450:
1446:
1442:
1438:
1432:
1430:
1426:
1418:
1417:
1409:
1407:
1405:
1403:
1399:
1394:
1387:
1384:
1379:
1373:
1369:
1365:
1360:
1355:
1351:
1344:
1341:
1335:
1330:
1326:
1322:
1315:
1312:
1307:
1301:
1297:
1296:
1291:
1290:Peter Sanders
1287:
1286:Kurt Mehlhorn
1281:
1279:
1277:
1273:
1268:
1262:
1258:
1254:
1250:
1244:
1242:
1238:
1231:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1208:
1204:
1202:
1192:
1185:
1183:
1180:
1166:
1158:
1150:
1146:
1138:
1136:
1134:
1130:
1122:
1120:
1117:
1111:
1094:
1091:
1085:
1079:
1074:
1070:
1061:
1057:
1052:
1050:
1046:
1038:
1036:
1014:
1010:
1006:
991:
985:
977:
971:
968:
964:
957:
951:
934:
924:
910:
903:
897:
894:
890:
883:
879:
875:
870:
867:
861:
846:
842:
836:
819:
810:
795:
788:
782:
779:
775:
751:
746:
743:
733:
729:
725:
711:
707:
698:
693:
691:
674:
665:
664:binary search
650:
646:
642:
638:
635:
632:
629:
625:
621:
617:
613:
609:
605:
602:
598:
594:
593:, exclusive.
564:
560:
556:
552:
544:
542:
538:
534:
520:
512:
506:
501:
496:
491:
486:
485:
481:
479:
474:
470:
466:
458:
454:
450:
443:
435:
431:
413:
406:
401:
397:
392:
391:
389:
378:
377:
373:
371:
367:
347:
339:
335:
329:
317:
314:
313:
311:
304:
303:
299:
292:
284:
270:
265:
261:
244:
237:K-way merging
236:
234:
232:
226:
222:
218:
201:
196:
190:
186:
182:
178:
174:
171:
167:
163:
159:
156:
152:
148:
144:
141:
137:
133:
115:
111:
107:
103:
95:
93:
89:
83:
80:
76:
72:
69:
68:
67:
65:
62:algorithm, a
61:
52:
45:
43:
41:
37:
33:
29:
25:
21:
2044:Hybrid sorts
1993:Proxmap sort
1906:Library sort
1776:Quantum sort
1629:
1628:, Volume 3:
1623:
1620:Donald Knuth
1597:
1586:. Retrieved
1576:
1541:
1535:
1500:
1483:
1473:
1453:
1415:
1392:
1386:
1349:
1343:
1327:(1): 27–40.
1324:
1320:
1314:
1298:. Springer.
1294:
1252:
1189:
1181:
1164:
1142:
1126:
1115:
1109:
1053:
1042:
1008:
1007:
925:
811:
731:
709:
705:
694:
672:
653:
648:
644:
640:
636:
633:
630:
627:
623:
619:
615:
611:
607:
603:
600:
596:
562:
548:
536:
532:
529:
518:
510:
499:
489:
475:
468:
464:
456:
452:
448:
441:
433:
429:
426:
399:
395:
345:
337:
333:
326:
293:
282:
268:
246:
224:
220:
216:
197:
193:
188:
184:
180:
176:
172:
169:
165:
161:
157:
154:
150:
146:
142:
139:
138:merge(A, B)
135:
110:linked lists
99:
90:
87:
78:
77:elements as
74:
57:
19:
18:
2126:Stooge sort
1968:Bucket sort
1920:Merge sorts
1792:Bubble sort
1736:Inplacement
1726:Total order
1582:"std:merge"
1226:Join (Unix)
1133:collections
690:recursively
585:from index
414:Re-heapify
398:= find-min(
102:linear time
71:Recursively
46:Application
34:in various
32:subroutines
2072:Spreadsort
2034:Samplesort
1998:Radix sort
1927:Merge sort
1865:Cycle sort
1850:Smoothsort
1812:Gnome sort
1588:2018-04-28
1232:References
1221:Join (SQL)
1153:std::merge
200:subroutine
106:pseudocode
60:merge sort
40:merge sort
24:algorithms
2067:Introsort
2003:Flashsort
1973:Burstsort
1963:Bead sort
1901:Tree sort
1896:Splaysort
1891:Shellsort
1822:Quicksort
1807:Comb sort
1741:Stability
1527:245669103
1451:(2009) .
1354:CiteSeerX
1329:CiteSeerX
1169:std::list
1157:iterators
1080:
972:
961:Θ
939:∞
898:
887:Θ
851:∞
824:∞
783:
772:Θ
614:m < n
597:algorithm
136:algorithm
2151:Category
2136:Bogosort
2131:Slowsort
1845:Heapsort
1568:52195866
1503:: 1–12.
1292:(2008).
1251:(2010).
1205:See also
1165:in-place
589:through
551:parallel
370:min-heap
2062:Timsort
1661:and in
383:of the
356:
342:
287:
273:
262:and an
147:returns
1709:Theory
1666:GitHub
1659:GitHub
1636:
1566:
1556:
1525:
1461:
1374:
1356:
1331:
1302:
1263:
1191:Python
1186:Python
1159:, and
1013:stable
670:where
631:return
626:m ≤ 0
604:inputs
563:et al.
440:2⌊log
309:lists.
189:return
143:inputs
114:arrays
28:sorted
2086:Other
1731:Lists
1564:S2CID
1523:S2CID
1420:(PDF)
1199:heapq
1195:merge
1173:merge
1143:The
1127:Some
1045:FPGAs
1009:Note:
944:merge
856:merge
829:merge
732:merge
535:⌈log
467:⌊log
432:(log
181:while
173:while
151:while
1634:ISBN
1554:ISBN
1459:ISBN
1372:ISBN
1300:ISBN
1261:ISBN
1177:<
1049:SIMD
1031:and
1019:and
724:span
697:work
695:The
684:and
628:then
616:then
569:and
455:log
393:Let
336:−1)(
223:log
166:else
162:then
130:head
120:and
1663:C++
1546:doi
1513:hdl
1505:doi
1364:doi
1147:'s
1145:C++
1139:C++
1110:P/2
1071:log
1058:or
969:log
895:log
780:log
658:or
645:let
641:let
637:let
608:let
521:/2⌉
513:/2⌋
502:= 2
497:If
492:= 1
487:If
289:− 1
112:or
2153::
1655:C#
1622:.
1562:.
1552:.
1521:.
1511:.
1491:^
1482:,
1447:;
1443:;
1439:;
1428:^
1401:^
1370:.
1362:.
1323:.
1288:;
1275:^
1240:^
1135:.
624:if
612:if
601:is
549:A
271:=
206:,
191:C
185:do
177:do
158:if
155:do
140:is
116:)
42:.
1694:e
1687:t
1680:v
1605:.
1591:.
1570:.
1548::
1529:.
1515::
1507::
1467:.
1380:.
1366::
1337:.
1325:3
1308:.
1269:.
1116:P
1095:1
1092:+
1089:)
1086:P
1083:(
1075:2
1033:B
1029:A
1025:C
1021:B
1017:A
992:)
986:2
982:)
978:n
975:(
965:(
958:=
955:)
952:n
949:(
935:T
911:)
907:)
904:n
901:(
891:(
884:+
880:)
876:n
871:4
868:3
862:(
847:T
843:=
840:)
837:n
834:(
820:T
796:)
792:)
789:n
786:(
776:(
752:n
747:4
744:3
720:C
716:n
712:)
710:n
708:(
706:O
701:n
686:ℓ
682:k
678:B
673:A
668:B
660:B
656:A
591:j
587:i
583:A
579:A
575:C
571:B
567:A
539:⌉
537:k
533:n
519:k
517:⌈
511:k
509:⌊
500:k
490:k
471:⌋
469:k
465:n
463:2
459:)
457:k
453:n
451:(
449:O
444:⌋
442:k
436:)
434:k
430:O
418:.
416:h
409:i
404:.
402:)
400:h
396:i
385:k
381:h
368:(
362:n
358:)
353:2
350:/
346:k
340:−
338:n
334:k
332:(
307:k
296:k
283:M
279:/
276:1
269:k
256:k
252:k
248:k
227:)
225:n
221:n
219:(
217:O
212:A
208:A
204:A
126:C
122:B
118:A
79:n
75:n
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