876:
heap, with insert and delete-min operations, is linear, as it uses no extra storage other than an array containing a list of the items in the heap. If changes to the priorities of existing items need to be supported, then one must also maintain pointers from the items to their positions in the heap,
304:
comparisons to determine the minimum or maximum of the children and then one more comparison against the parent to determine whether a swap is needed. Therefore, the time to delete the root item, to increase the priority of an item in a min-heap, or to decrease the priority of an item in a max-heap,
213:
To insert a new item into the heap, the item is appended to the end of the array, and then while the heap property is violated it is swapped with its parent, moving it upward in the tree and earlier in the array, until eventually the heap property is satisfied. The same upward-swapping procedure may
270:
swaps. In the upward-swapping procedure, each swap involves a single comparison of an item with its parent, and takes constant time. Therefore, the time to insert a new item into the heap, to decrease the priority of an item in a min-heap, or to increase the priority of an item in a max-heap, is
209:
is swapped with one of its children (the one with the smallest priority in a min-heap, or the one with the largest priority in a max-heap), moving it downward in the tree and later in the array, until eventually the heap property is satisfied. The same downward swapping procedure may be used to
657:
493:
68:
This data structure allows decrease priority operations to be performed more quickly than binary heaps, at the expense of slower delete minimum operations. This tradeoff leads to better running times for algorithms such as
504:
860:
197:
The minimum priority item in a min-heap (or the maximum priority item in a max-heap) may always be found at position 0 of the array. To remove this item from the priority queue, the last item
974:
running time of binary heap versions of these algorithms whenever the number of edges is significantly larger than the number of vertices. An alternative priority queue data structure, the
742:
1122:. Note that Tarjan uses 1-based numbering, not 0-based numbering, so his formulas for the parent and children of a node need to be adjusted when 0-based numbering is used.
1117:
194:, in a min-heap, each item has a priority that is at least as large as its parent; in a max-heap, each item has a priority that is no larger than its parent.
83:
behavior than binary heaps, allowing them to run more quickly in practice despite having a theoretically larger worst-case running time. Like binary heaps,
1336:
384:
1177:
378:
cost for the second swap beyond the cost already counted in the first term, etc. Therefore, the total amount of time to create a heap in this way is
1301:
1279:
1160:
652:{\displaystyle {\frac {d}{d-1}}(n-s_{d}(n))-(d-1-(n{\bmod {d}}))\left(e_{d}\left(\left\lfloor {\frac {n}{d}}\right\rfloor \right)+1\right)}
754:
1228:
Naor, D.; Martel, C. U.; Matloff, N. S. (October 1991), "Performance of priority queue structures in a virtual memory environment",
498:
The exact value of the above (the worst-case number of comparisons during the construction of d-ary heap) is known to be equal to:
1418:
1263:
Algorithms and Data
Structures: 9th International Workshop, WADS 2005, Waterloo, Canada, August 15–17, 2005, Proceedings
886:
941:, the total times for these two types of operations may be balanced against each other, leading to a total time of
1262:
214:
be used to decrease the priority of an item in a min-heap, or to increase the priority of an item in a max-heap.
1005:-ary heap typically runs much faster than a binary heap for heap sizes that exceed the size of the computer's
1186:
117:
items, each of which has a priority associated with it. These items may be viewed as the nodes in a complete
898:
70:
677:
250:
items in it, both the upward-swapping procedure and the downward-swapping procedure may perform as many as
1348:
1001:
4-heaps may perform better than binary heaps in practice, even for delete-min operations. Additionally, a
998:-ary heaps are generally at least as fast, and often faster, than Fibonacci heaps for this application.
910:
201:
in the array is moved into its place, and the length of the array is decreased by one. Then, while item
1260:
Mortensen, C. W.; Pettie, S. (2005), "The complexity of implicit and space efficient priority queues",
1024:-ary heap, each one taking far more time than the extra work incurred by the additional comparisons a
210:
increase the priority of an item in a min-heap, or to decrease the priority of an item in a max-heap.
128:
124:
108:
906:
1353:
1332:
90:
73:
in which decrease priority operations are more common than delete min operations. Additionally,
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and ending at the item at position 0, applying the downward-swapping procedure for each item.
62:
1358:
1313:
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93:
that uses no additional storage beyond that needed to store the array of items in the heap.
223:
items, one may loop over the items in reverse order, starting from the item at position
1014:
975:
38:
35:
1266:, Lecture Notes in Computer Science, vol. 3608, Springer-Verlag, pp. 49–60,
1412:
1105:
1068:
902:
1006:
488:{\displaystyle \sum _{i=1}^{\log _{d}n}\left({\frac {n}{d^{i}}}+1\right)O(d)=O(n).}
80:
334:
items, most of the items are in positions that will eventually hold leaves of the
1302:"Elementary Yet Precise Worst-Case Analysis of Floyd's Heap-Construction Program"
191:
42:
351:
items are non-leaves, and may be swapped downwards at least once, at a cost of
1242:
1017:
1010:
1377:
667:(n) is the sum of all digits of the standard base-d representation of n and e
1116:, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 44,
143:
items are its grandchildren, etc. Thus, the parent of the item at position
1055:(1975), "Priority queues with update and finding minimum spanning trees",
340:-ary tree, and no downward swapping is performed for those items. At most
1317:
1362:
1271:
671:(n) is the exponent of d in the factorization of n. This reduces to
1403:
370:
nodes may be swapped downward two times, incurring an additional
1337:"Shortest paths algorithms: Theory and experimental evaluation"
855:{\displaystyle {\frac {3}{2}}(n-s_{3}(n))-2e_{3}(n)-e_{3}(n-1)}
131:) forms the root of the tree, the items at positions 1 through
51:
children instead of 2. Thus, a binary heap is a 2-heap, and a
582:
1404:
C++ implementation of generalized heap with D-Heap support
283:. In the downward-swapping procedure, each swap involves
205:
and its children do not satisfy the heap property, item
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680:
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387:
55:
is a 3-heap. According to Tarjan and Jensen et al.,
978:, gives an even better theoretical running time of
854:
736:
651:
487:
359:time to find the child to swap them with. At most
1176:Jensen, C.; Katajainen, J.; Vitale, F. (2004),
1118:Society for Industrial and Applied Mathematics
1155:(2nd ed.), Addison-Wesley, p. 216,
127:: the item at position 0 of the array (using
8:
173:and its children are the items at positions
1028:-ary heap makes compared to a binary heap.
962:for the algorithm, an improvement over the
1295:
1293:
1291:
1352:
1241:
923:decrease-priority operations. By using a
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1009:: A binary heap typically requires more
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1037:
1255:
1253:
1212:
1153:Data Structures and Algorithm Analysis
1114:Data Structures and Network Algorithms
1100:
1098:
877:which again uses only linear storage.
217:To create a new heap from an array of
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1041:
919:delete-min operations and as many as
737:{\displaystyle 2n-2s_{2}(n)-e_{2}(n)}
7:
1208:
1206:
1376:Kamp, Poul-Henning (11 June 2010),
913:use a min-heap in which there are
14:
1057:Information Processing Letters
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1:
1335:; Radzik, Tomasz (May 1996),
1179:An extended truth about heaps
125:breadth first traversal order
1069:10.1016/0020-0190(75)90001-0
61:-ary heaps were invented by
1300:Suchenek, Marek A. (2012),
137:are its children, the next
1435:
156:) is the item at position
41:, a generalization of the
107:-ary heap consists of an
1341:Mathematical Programming
330:-ary heap from a set of
45:in which the nodes have
1419:Heaps (data structures)
1378:"You're doing it wrong"
1312:(1), IOS Press: 75–92,
1306:Fundamenta Informaticae
1243:10.1093/comjnl/34.5.428
869:The space usage of the
91:in-place data structure
79:-ary heaps have better
1331:Cherkassky, Boris V.;
1147:Weiss, M. A. (2007), "
911:minimum spanning trees
856:
738:
653:
489:
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289:comparisons and takes
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654:
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123:-ary tree, listed in
899:Dijkstra's algorithm
885:When operating on a
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678:
505:
385:
129:zero-based numbering
71:Dijkstra's algorithm
1333:Goldberg, Andrew V.
1318:10.3233/FI-2012-751
190:. According to the
1363:10.1007/BF02592101
1272:10.1007/11534273_6
994:, but in practice
852:
748:for d = 2, and to
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649:
485:
89:-ary heaps are an
1281:978-3-540-28101-6
1162:978-0-321-37013-6
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628:
524:
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63:Donald B. Johnson
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1230:Computer Journal
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1215:, pp. 77 and 91.
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1191:
1185:, archived from
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1120:, pp. 34–38
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927:-ary heap with
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297:time: it takes
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244:-ary heap with
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1398:External links
1396:
1393:
1392:
1368:
1347:(2): 129–174,
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1287:
1280:
1249:
1236:(5): 428–437,
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1168:
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1124:
1108:(1983), "3.2.
1074:
1053:Johnson, D. B.
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1035:
1033:
1030:
1015:virtual memory
976:Fibonacci heap
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903:shortest paths
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39:data structure
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1213:Tarjan (1983)
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1203:
1192:on 2007-06-09
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1229:
1194:, retrieved
1187:the original
1178:
1171:
1152:
1148:
1113:
1109:
1063:(3): 53–57,
1060:
1056:
1011:cache misses
1007:cache memory
1000:
989:
985:
981:
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81:memory cache
75:
67:
57:
53:ternary heap
52:
47:
28:
26:
19:
17:
15:
1018:page faults
866:for d = 3.
165:− 1)/
43:binary heap
1196:2008-02-05
1032:References
893:edges and
1382:ACM Queue
1349:CiteSeerX
1151:-heaps",
1112:-heaps",
844:−
825:−
800:−
775:−
713:−
688:−
573:−
567:−
558:−
533:−
518:−
415:
390:∑
302:− 1
228:− 1
149:(for any
65:in 1975.
23:-ary heap
1413:Category
631:⌋
618:⌊
234:Analysis
180:through
1020:than a
663:where s
1351:
1278:
1159:
315:/ log
277:/ log
273:O(log
265:/ log
261:= log
154:> 0
1190:(PDF)
1183:(PDF)
889:with
887:graph
238:In a
109:array
34:is a
32:-heap
1276:ISBN
1157:ISBN
1013:and
988:log
968:log
909:for
905:and
901:for
874:-ary
311:log
101:The
16:The
1388:(6)
1359:doi
1314:doi
1310:120
1268:doi
1238:doi
1065:doi
947:log
583:mod
406:log
368:+ 1
349:+ 1
305:is
252:log
178:+ 1
111:of
25:or
1415::
1384:,
1380:,
1357:,
1345:73
1343:,
1339:,
1308:,
1304:,
1290:^
1274:,
1252:^
1234:34
1232:,
1220:^
1205:^
1127:^
1077:^
1059:,
1040:^
984:+
980:O(
964:O(
943:O(
932:=
372:O(
353:O(
321:.
307:O(
291:O(
185:+
183:di
176:di
1390:.
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1361::
1321:.
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1270::
1247:.
1240::
1200:.
1166:.
1149:d
1110:d
1072:.
1067::
1061:4
1026:d
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1003:d
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992:)
990:n
986:n
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970:n
966:m
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958:n
954:n
952:/
950:m
945:m
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936:/
934:m
930:d
925:d
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891:m
872:d
862:,
850:)
847:1
841:n
838:(
833:3
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788:(
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769:(
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626:d
623:n
614:(
608:d
604:e
599:(
595:)
592:)
587:d
579:n
576:(
570:1
564:d
561:(
555:)
552:)
549:n
546:(
541:d
537:s
530:n
527:(
521:1
515:d
511:d
483:.
480:)
477:n
474:(
471:O
468:=
465:)
462:d
459:(
456:O
452:)
448:1
445:+
438:i
434:d
430:n
424:(
418:n
410:d
400:1
397:=
394:i
376:)
374:d
366:d
364:/
362:n
357:)
355:d
347:d
345:/
343:n
337:d
332:n
327:d
319:)
317:d
313:n
309:d
300:d
295:)
293:d
286:d
281:)
279:d
275:n
267:d
263:n
259:n
255:d
247:n
241:d
226:n
220:n
207:x
203:x
199:x
187:d
170:⌋
167:d
163:i
161:(
159:⌊
152:i
146:i
140:d
134:d
120:d
114:n
104:d
86:d
76:d
58:d
48:d
29:d
20:d
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