510:
before and after these locations are already in their correct place and do not need to be merged. Then, the smaller of these shrunk runs is copied into temporary memory, and the copied elements are merged with the larger shrunk run into the now free space. If the leftmost shrunk run is smaller, the merge proceeds from left to right. If the rightmost shrunk run is smaller, merging proceeds from right to left (i.e. beginning with elements at the ends of the temporary space and leftmost run, and filling the free space from its end). This optimization reduces the number of required element movements, the running time and the temporary space overhead in the general case.
327:
567:
494:
612:
531:
514:
element of the first run is 10 and it would have to be added at the fifth position of the second run in order to preserve its order. Therefore, and are already in their final positions and the runs in which elements movements are required are and . With this knowledge, we only need to allocate a temporary buffer of size 2 instead of 4.
264:. The algorithm finds subsequences of the data that are already ordered (runs) and uses them to sort the remainder more efficiently. This is done by merging runs until certain criteria are fulfilled. Timsort was Python's standard sorting algorithm from version 2.3 to version 3.10, and is used to sort arrays of non-primitive type in
311:, they are merged. This goes on until all data is traversed; then, all runs are merged two at a time and only one sorted run remains. The advantage of merging ordered runs instead of merging fixed size sub-lists (as done by traditional mergesort) is that it decreases the total number of comparisons needed to sort the entire list.
578:. According to benchmarks done by the developer, galloping is beneficial only when the initial element of one run is not one of the first seven elements of the other run. This implies an initial threshold of 7. To avoid the drawbacks of galloping mode, two actions are taken: (1) When galloping is found to be less efficient than
546:), Timsort considers that it is likely that many consecutive elements may still be selected from that run and switches into galloping mode. Let us assume that R1 is responsible for triggering it. In this mode, the algorithm performs a two-stage search for the place in the run R1 where the next element
501:
The original merge sort implementation is not in-place and it has a space overhead of N (data size). In-place merge sort implementations exist, but have a high time overhead. In order to achieve a middle term, Timsort performs a merge sort with a small time overhead and smaller space overhead than N.
513:
Example: two runs and must be merged. Note that both runs are already sorted individually. The smallest element of the second run is 4 and it would have to be added at the fourth position of the first run in order to preserve its order (assuming that the first position of a run is 1). The largest
602:
In order to also take advantage of data sorted in descending order, Timsort reverses strictly descending runs when it finds them and adds them to the stack of runs. Since descending runs are later blindly reversed, excluding runs with equal elements maintains the algorithm's stability; i.e., equal
509:
to find the location where the first element of the second run would be inserted in the first ordered run, keeping it ordered. Then, it performs the same algorithm to find the location where the last element of the first run would be inserted in the second ordered run, keeping it ordered. Elements
797:
also has good aspects: It's stable (items that compare equal retain their relative order, so, e.g., if you sort first on zip code, and a second time on name, people with the same name still appear in order of increasing zip code; this is important in apps that, e.g., refine the results of queries
480:
and the invariants are checked again. Once the invariants hold, the search for a new run in the data can start. These invariants maintain merges as being approximately balanced while maintaining a compromise between delaying merging for balance, exploiting fresh occurrence of runs in
409:
In order to achieve sorting stability, only consecutive runs are merged. Between two non-consecutive runs, there can be an element with the same key inside the runs. Merging those two runs would change the order of equal keys. Example of this situation ( are ordered runs): 1 4 2
734:
of Java software, the researchers found that this check is not sufficient, and they were able to find run lengths (and inputs which generated those run lengths) which would result in the invariants being violated deeper in the stack after the top of the stack was merged.
738:
As a consequence, for certain inputs the allocated size is not sufficient to hold all unmerged runs. In Java, this generates for those inputs an array-out-of-bound exception. The smallest input that triggers this exception in Java and
Android v7 is of size
718:
Specifically, the invariants on stacked run sizes ensure a tight upper bound on the maximum size of the required stack. The implementation preallocated a stack sufficient to sort 2 bytes of input, and avoided further overflow checks.
759:
The Java implementation was corrected by increasing the size of the preallocated stack based on an updated worst-case analysis. The article also showed by formal methods how to establish the intended invariant by checking that the
1106:
The current algorithm is an adaptive, iterative merge sort inspired by timsort. It is designed to be very fast in cases where the slice is nearly sorted, or consists of two or more sorted sequences concatenated one after
619:
Because merging is most efficient when the number of runs is equal to, or slightly less than, a power of two, and notably less efficient when the number of runs is slightly more than a power of two, Timsort chooses
634:, is equal to, or slightly less than, a power of two. The final algorithm takes the six most significant bits of the size of the array, adds one if any of the remaining bits are set, and uses that result as the
558:
such that R1 < x <= R1, i.e. a region of uncertainty comprising 2 − 1 consecutive elements of R1. The second stage performs a straight binary search of this region to find the exact location in R1 for
590:
is reduced by one, thus encouraging the return to galloping mode. Otherwise, the value is incremented by one, thus discouraging a return to galloping mode. In the case of random data, the value of
707:
for sorting object references or pointers because these require expensive memory indirection to access data and perform comparisons and
Quicksort's cache coherence benefits are greatly reduced.
497:
To merge, Timsort copies the elements of the smaller array (X in this illustration) to temporary memory, then sorts and fills elements in final order into the combined space of X and Y.
1224:
de Gouw, Stijn; Rot, Jurriaan; de Boer, Frank S.; Bubel, Richard; Hähnle, Reiner (July 2015). "OpenJDK's Java.utils.Collection.sort() is Broken: The Good, the Bad and the Worst Case".
980:
764:
topmost runs in the stack satisfy the two rules above. This approach was initially adopted by Python until it switched to
Powersort in 2022 with the release of Python 3.11.
715:
In 2015, Dutch and German researchers in the EU FP7 ENVISAGE project found a bug in the standard implementation of
Timsort. It was fixed in 2015 in Python, Java and Android.
690:
179:
90:
406:
Timsort is a stable sorting algorithm (order of elements with same key is kept) and strives to perform balanced merges (a merge thus merges runs of similar sizes).
314:
Each run has a minimum size, which is based on the size of the input and is defined at the start of the algorithm. If a run is smaller than this minimum run size,
219:
130:
1027:
Code stolen in large part from Python's, listobject.c, which itself had no license header. However, thanks to Tim Peters for the parts of the code I ripped-off.
307:. It iterates over the data collecting elements into runs and simultaneously putting those runs in a stack. Whenever the runs on the top of the stack match a
853:
TimSort is an intriguing sorting algorithm designed in 2002 for Python, whose worst-case complexity was announced, but not proved until our recent preprint.
1263:
988:
1151:
534:
Elements (pointed to by blue arrow) are compared and the smaller element is moved to its final position (pointed to by red arrow).
538:
An individual merge of runs R1 and R2 keeps the count of consecutive elements selected from a run. When this number reaches the
1246:
696:
elements. In the best case, which occurs when the input is already sorted, it runs in linear time, meaning that it is an
1010:
1359:
1130:
930:
831:
638:. This algorithm works for all arrays, including those smaller than 64; for arrays of size 63 or less, this sets
563:. Galloping mode is an attempt to adapt the merge algorithm to the pattern of intervals between elements in runs.
1382:
1690:
257:
1174:
291:
It uses techniques from Peter McIlroy's 1993 paper "Optimistic
Sorting and Information Theoretic Complexity".
261:
1318:
956:"[#JDK-6804124] (coll) Replace "modified mergesort" in java.util.Arrays.sort with timsort"
651:
281:
185:
136:
96:
47:
522:
Merging can be done in both directions: left-to-right, as in the traditional mergesort, or right-to-left.
1723:
285:
574:
Galloping is not always efficient. In some cases galloping mode requires more comparisons than a simple
570:
All red elements are smaller than blue (here, 21). Thus they can be moved in a chunk to the final array.
269:
1521:
726:
group of three consecutive runs, but the implementation only checked it for the top three. Using the
326:
1823:
1397:
1264:"Proving that Android's, Java's and Python's sorting algorithm is broken (and showing how to fix it)"
1196:
1225:
781:
1695:
1090:
579:
885:
Nearly-Optimal
Mergesorts: Fast, Practical Sorting Methods That Optimally Adapt to Existing Runs
657:
146:
57:
1603:
1483:
1437:
1407:
1064:
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based on user input). ... It has no bad cases (O(N log N) is worst case; N−1 compares is best).
364:
are merged and replaced on the stack. In this way, merging is continued until all runs satisfy
331:
277:
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1828:
1764:
1352:
242:
615:
Timsort algorithm searches for minimum-size ordered sequences, minruns, to perform its sort.
1743:
1608:
1463:
40:
1299:
1119:
McIlroy, Peter (January 1993). "Optimistic
Sorting and Information Theoretic Complexity".
195:
106:
8:
1759:
1498:
749:(2). (Older Android versions already triggered this exception for certain inputs of size
731:
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is chosen from the range 32 to 64 inclusive, such that the size of the data, divided by
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1624:
1598:
1402:
908:
888:
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551:
1311:
586:. If the selected element is from the same array that returned an element previously,
582:, galloping mode is exited. (2) The success or failure of galloping is used to adjust
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1242:
1126:
827:
811:
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In pursuit of balanced merges, Timsort considers three runs on the top of the stack,
245:
30:
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912:
841:
1710:
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1234:
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256:, designed to perform well on many kinds of real-world data. It was implemented by
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is used to add more elements to the run until the minimum run size is reached.
315:
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822:
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1644:
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of consecutive ordered elements that already exist in most real-world data,
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1572:
1442:
1122:
Proceedings of the Fourth Annual ACM-SIAM Symposium on
Discrete Algorithms
530:
1792:
1634:
1458:
1392:
1284:
Python Issue
Tracker – Issue 23515: Bad logic in timsort's merge_collapse
1738:
1718:
1700:
1664:
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1531:
1516:
1478:
1233:. Lecture Notes in Computer Science. Vol. 9206. pp. 273–289.
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Auger, Nicolas; Jugé, Vincent; Nicaud, Cyril; Pivoteau, Carine (2018).
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1639:
1629:
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1557:
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1473:
1337:
704:
482:
1283:
870:
Patience is a Virtue: Revisiting Merge and Sort on Modern
Processors
1802:
1797:
1511:
893:
550:
of the run R2 would be inserted. In the first stage it performs an
642:
equal to the array size and Timsort reduces to an insertion sort.
1039:
887:. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. p. 12.
265:
16:
Hybrid sorting algorithm based on insertion sort and merge sort
722:
However, the guarantee requires the invariants to apply to
727:
1298:
Auger, Nicolas; Nicaud, Cyril; Pivoteau, Carine (2015).
809:
1223:
867:
660:
198:
149:
109:
60:
554:, also known as a galloping search, until finding a
1297:
1175:"Understanding timsort, Part 1: Adaptive Mergesort"
868:Chandramouli, Badrish; Goldstein, Jonathan (2014).
594:becomes so large that galloping mode never recurs.
684:
213:
173:
124:
84:
1146:
1144:
1142:
1815:
1300:"Merge Strategies: from Merge Sort to TimSort"
1139:
863:
861:
525:
485:and making merge decisions relatively simple.
1353:
284:, and inspired the sorting algorithm used in
924:
922:
858:
1360:
1346:
1017:. Lines 23-25 of the initial comment block
1015:Mercurial repository of Octave source code
299:Timsort was designed to take advantage of
1312:"On the Worst-Case Complexity of TimSort"
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902:
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882:
821:
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610:
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468:If any of these invariants is violated,
325:
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1118:
883:Munro, J. Ian; Wild, Sebastian (2018).
624:to try to ensure the former condition.
488:
1816:
1367:
1319:"Strategies for Stable Merge Sorting."
1310:Auger, Jugé, Nicaud, Pivoteau (2018).
1214:
779:
710:
1341:
1173:MacIver, David R. (11 January 2010).
1334:– original explanation by Tim Peters
1262:de Gouw, Stijn (24 February 2015).
981:"Class: java.util.TimSort<T>"
606:
308:
13:
1291:
1194:
1040:"Getting things sorted in V8 · V8"
597:
517:
14:
1840:
1325:
985:Android Gingerbread Documentation
928:
321:
692:comparisons to sort an array of
425:, and maintains the invariants:
1383:Computational complexity theory
1277:
1255:
1188:
1112:
1083:
1057:
1065:"Is sort() stable in Swift 5?"
1032:
1003:
973:
948:
876:
803:
782:"[Python-Dev] Sorting"
773:
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472:is merged with the smaller of
208:
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119:
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79:
64:
1:
1158:. 18 May 2022. Archived from
786:Python Developers Mailinglist
767:
1239:10.1007/978-3-319-21690-4_16
1011:"liboctave/util/oct-sort.cc"
780:Peters, Tim (20 July 2002).
603:elements won't be reversed.
294:
7:
1227:Computer Aided Verification
931:"Python Now Uses Powersort"
645:
540:minimum galloping threshold
526:Galloping mode during merge
330:The runs are inserted in a
262:Python programming language
10:
1845:
1691:Batcher odd–even mergesort
1317:Sam Buss, Alexander Knop.
904:10.4230/lipics.esa.2018.63
685:{\displaystyle O(n\log n)}
505:First, Timsort performs a
174:{\displaystyle O(n\log n)}
85:{\displaystyle O(n\log n)}
1785:
1752:
1709:
1678:
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823:10.4230/LIPIcs.ESA.2018.4
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184:
135:
95:
46:
36:
26:
1696:Pairwise sorting network
1724:Kirkpatrick–Reisch sort
260:in 2002 for use in the
1604:Oscillating merge sort
1484:Proportion extend sort
1438:Transdichotomous model
1201:CPython git repository
686:
616:
571:
535:
498:
403:
215:
175:
126:
86:
1765:Pre-topological order
687:
614:
569:
533:
496:
329:
216:
176:
127:
87:
1744:Merge-insertion sort
1609:Polyphase merge sort
1464:Cocktail shaker sort
1125:. pp. 467–474.
658:
489:Merge space overhead
214:{\displaystyle O(n)}
196:
147:
125:{\displaystyle O(n)}
107:
58:
1760:Topological sorting
1522:Cartesian tree sort
1162:on 28 January 2016.
732:formal verification
711:Formal verification
23:
1650:Interpolation sort
1625:American flag sort
1618:Distribution sorts
1599:Cascade merge sort
1369:Sorting algorithms
1156:Python source code
703:It is superior to
682:
617:
572:
552:exponential search
536:
499:
457:| > |
436:| > |
404:
394:| > |
372:| > |
211:
171:
122:
82:
21:
1811:
1810:
1786:Impractical sorts
1248:978-3-319-21689-8
1095:doc.rust-lang.org
246:sorting algorithm
232:
231:
31:Sorting algorithm
1836:
1824:Comparison sorts
1719:Block merge sort
1679:Concurrent sorts
1578:Patience sorting
1362:
1355:
1348:
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1307:
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987:. Archived from
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777:
755:
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748:
747:
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698:adaptive sorting
695:
691:
689:
688:
683:
654:, Timsort takes
607:Minimum run size
462:
456:
447:
441:
435:
401:
399:
393:
385:
383:
377:
371:
355:
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270:Android platform
220:
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189:space complexity
180:
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131:
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123:
91:
89:
88:
83:
24:
20:
1844:
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1839:
1838:
1837:
1835:
1834:
1833:
1814:
1813:
1812:
1807:
1781:
1772:Pancake sorting
1748:
1705:
1674:
1655:Pigeonhole sort
1613:
1582:
1546:Insertion sorts
1541:
1527:Tournament sort
1499:Selection sorts
1493:
1447:
1428:Integer sorting
1423:Sorting network
1413:Comparison sort
1371:
1366:
1328:
1294:
1292:Further reading
1289:
1282:
1278:
1268:
1266:
1260:
1256:
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992:
991:on 16 July 2015
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978:
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953:
949:
939:
937:
927:
920:
881:
877:
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859:
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834:
813:[DROPS]
808:
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598:Descending runs
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518:Merge direction
491:
466:
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443:
442:| + |
437:
431:
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379:
378:| + |
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367:
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349:
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343:
342:| ≤ |
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309:merge criterion
297:
248:, derived from
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5:
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1780:
1779:
1777:Spaghetti sort
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1756:
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1721:
1715:
1713:
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1706:
1704:
1703:
1698:
1693:
1688:
1686:Bitonic sorter
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1553:Insertion sort
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1542:
1540:
1539:
1537:Weak-heap sort
1534:
1529:
1524:
1519:
1514:
1509:
1507:Selection sort
1503:
1501:
1495:
1494:
1492:
1491:
1486:
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1476:
1471:
1466:
1461:
1455:
1453:
1452:Exchange sorts
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1415:
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1405:
1400:
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1388:Big O notation
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1326:External links
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1197:"listsort.txt"
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1152:"listsort.txt"
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1111:
1091:"slice - Rust"
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1031:
1002:
972:
960:JDK Bug System
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918:
875:
872:. SIGMOD/PODS.
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322:Merge criteria
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1645:Counting sort
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1433:X + Y sorting
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1418:Adaptive sort
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1829:Stable sorts
1728:
1711:Hybrid sorts
1660:Proxmap sort
1573:Library sort
1443:Quantum sort
1304:hal-01212839
1303:
1279:
1267:. Retrieved
1257:
1226:
1204:. Retrieved
1200:
1190:
1178:. Retrieved
1168:
1160:the original
1155:
1121:
1114:
1105:
1098:. Retrieved
1094:
1085:
1073:. Retrieved
1069:Swift Forums
1068:
1059:
1047:. Retrieved
1043:
1034:
1026:
1019:. Retrieved
1014:
1005:
993:. Retrieved
989:the original
984:
975:
963:. Retrieved
959:
950:
938:. Retrieved
935:I Programmer
934:
884:
878:
869:
852:
845:. Retrieved
812:
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789:. Retrieved
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483:cache memory
477:
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305:natural runs
304:
300:
298:
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234:
233:
18:
1793:Stooge sort
1635:Bucket sort
1587:Merge sorts
1459:Bubble sort
1403:Inplacement
1393:Total order
1332:timsort.txt
1314:. ESA 2018.
1049:21 December
1021:18 February
995:24 February
847:1 September
791:24 February
700:algorithm.
140:performance
100:performance
51:performance
1818:Categories
1739:Spreadsort
1701:Samplesort
1665:Radix sort
1594:Merge sort
1532:Cycle sort
1517:Smoothsort
1479:Gnome sort
1321:SODA 2019.
1206:5 December
1180:5 December
1100:8 December
894:1805.04154
768:References
652:worst case
592:min_gallop
588:min_gallop
584:min_gallop
544:min_gallop
388:ii. |
274:GNU Octave
258:Tim Peters
250:merge sort
186:Worst-case
48:Worst-case
1734:Introsort
1670:Flashsort
1640:Burstsort
1630:Bead sort
1568:Tree sort
1563:Splaysort
1558:Shellsort
1489:Quicksort
1474:Comb sort
1408:Stability
730:tool for
705:Quicksort
674:
366:i. |
295:Operation
268:, on the
266:Java SE 7
163:
97:Best-case
74:
1803:Bogosort
1798:Slowsort
1512:Heapsort
1107:another.
913:21678081
842:44091254
646:Analysis
1729:Timsort
965:11 June
940:21 June
650:In the
356:, then
235:Timsort
225:Optimal
137:Average
22:Timsort
1376:Theory
1245:
1129:
1075:4 July
1044:v8.dev
911:
840:
830:
640:minrun
636:minrun
632:minrun
628:Minrun
622:minrun
463:|
451:|
448:|
430:|
400:|
384:|
354:|
336:|
243:stable
239:hybrid
1753:Other
1398:Lists
1269:6 May
1231:(PDF)
909:S2CID
889:arXiv
838:S2CID
756:(2))
724:every
334:. If
332:stack
282:Swift
276:, on
272:, in
237:is a
41:Array
27:Class
1271:2017
1243:ISBN
1208:2019
1182:2015
1127:ISBN
1102:2022
1077:2019
1051:2018
1023:2013
997:2011
967:2014
942:2024
849:2018
828:ISBN
793:2011
762:four
386:and
360:and
301:runs
286:Rust
252:and
1235:doi
899:doi
818:doi
753:536
746:864
743:108
728:KeY
671:log
476:or
160:log
71:log
1820::
1302:.
1241:.
1216:^
1199:.
1154:.
1141:^
1104:.
1093:.
1067:.
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1013:.
983:.
958:.
933:.
921:^
907:.
897:.
860:^
851:.
836:.
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816:.
795:.
784:.
751:65
741:67
421:,
417:,
288:.
280:,
278:V8
241:,
228:No
1361:e
1354:t
1347:v
1306:.
1273:.
1251:.
1237::
1210:.
1184:.
1135:.
1079:.
1053:.
999:.
969:.
944:.
915:.
901::
891::
820::
694:n
680:)
677:n
668:n
665:(
662:O
561:x
556:k
548:x
542:(
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470:Y
460:X
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439:Y
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423:Z
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415:X
402:.
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381:X
375:Y
369:Z
362:Y
358:X
351:X
345:Y
339:Z
209:)
206:n
203:(
200:O
169:)
166:n
157:n
154:(
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120:)
117:n
114:(
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80:)
77:n
68:n
65:(
62:O
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