2006:: A heap data structure is useful to merge many already-sorted input streams into a single sorted output stream. Examples of the need for merging include external sorting and streaming results from distributed data such as a log structured merge tree. The inner loop is obtaining the min element, replacing with the next element for the corresponding input stream, then doing a sift-down heap operation. (Alternatively the replace function.) (Using extract-max and insert functions of a priority queue are much less efficient.)
104:, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. In a heap, the highest (or lowest) priority element is always stored at the root. However, a heap is not a sorted structure; it can be regarded as being partially ordered. A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspersed with removals of the root node.
359:
38:
162:
Heaps are typically constructed in-place in the same array where the elements are stored, with their structure being implicit in the access pattern of the operations. Heaps differ in this way from other data structures with similar or in some cases better theoretic bounds such as
370:, in the array, the first index contains the root element. The next two indices of the array contain the root's children. The next four indices contain the four children of the root's two child nodes, and so on. Therefore, given a node at index
2000:: A priority queue is an abstract concept like "a list" or "a map"; just as a list can be implemented with a linked list or an array, a priority queue can be implemented with a heap or a variety of other methods.
2144:. This class implements by default a min-heap; to implement a max-heap, programmer should write a custom comparator. There is no support for the replace, sift-up/sift-down, or decrease/increase-key operations.
451:
Balancing a heap is done by sift-up or sift-down operations (swapping elements which are out of order). As we can build a heap from an array without requiring extra memory (for the nodes, for example),
2199:
package with heap algorithms that operate on an arbitrary type that satisfies a given interface. That package does not support the replace, sift-up/sift-down, or decrease/increase-key operations.
1934:
320:: move a node up in the tree, as long as needed; used to restore heap condition after insertion. Called "sift" because node moves up the tree until it reaches the correct level, as in a
1709:
1548:
1876:
1947:
Brodal queues and strict
Fibonacci heaps achieve optimal worst-case complexities for heaps. They were first described as imperative data structures. The Brodal-Okasaki queue is a
1606:
1980:: A heap allows access to the min or max element in constant time, and other selections (such as median or kth-element) can be done in sub-linear time on data that is in a heap.
159:). The heap relation mentioned above applies only between nodes and their parents, grandparents. The maximum number of children each node can have depends on the type of heap.
237:: pop root and push a new key. This is more efficient than a pop followed by a push, since it only needs to balance once, not twice, and is appropriate for fixed-size heaps.
2959:
2830:
2040:, which wraps these facilities in a container-like class. However, there is no standard support for the replace, sift-up/sift-down, or decrease/increase-key operations.
2218:
has an implementation of a heap in the
Collections-Sequenceable package along with a set of test cases. A heap is used in the implementation of the timer event loop.
459:
After an element is inserted into or deleted from a heap, the heap property may be violated, and the heap must be re-balanced by swapping elements within the array.
436:
403:
2311:
1632:
4055:
2306:
2207:
2595:
2047:
include a heaps library. Unlike the STL, it supports decrease and increase operations, and supports additional types of heap: specifically, it supports
3731:
3109:
469:
Add the new element at the end of the heap, in the first available free space. If this will violate the heap property, sift up the new element (
4025:
3456:
3035:
2578:
2337:
3663:
3207:
479:
Remove the root and insert the last element of the heap in the root. If this will violate the heap property, sift down the new root (
3964:
3092:
2889:
2803:
2719:
2476:
2036:. It treats the iterators as a reference to an array, and uses the array-to-heap conversion. It also provides the container adaptor
1986:: By using heaps as internal traversal data structures, run time will be reduced by polynomial order. Examples of such problems are
4089:
2156:
module that implements a priority queue using a binary heap. The library exposes a heapreplace function to support k-way merging.
3323:
3754:
692:
3759:
3176:
2136:
1608:, a constant factor (half) of these insertions are within a constant factor of the maximum, so asymptotically we can assume
151:
Note that, as shown in the graphic, there is no implied ordering between siblings or cousins and no implied sequence for an
3724:
2147:
1825:). Applied to skew binomial heaps, it yields Brodal-Okasaki queues, persistent heaps with optimal worst-case complexities.
362:
Example of a complete binary max-heap with node keys being integers from 1 to 100 and how it would be stored in an array.
89:, the key of P is less than or equal to the key of C. The node at the "top" of the heap (with no parents) is called the
3838:
3821:
2222:
2131:
31:
1712:
2951:
1881:
4037:
3804:
3799:
3288:
2466:
2270:
2265:
2242:
2141:
193:
3794:
3673:
2510:
2358:
2190:
1948:
1774:
493:
element in the root and sift down. When compared to extraction followed by insertion, this avoids a sift up step.
330:: move a node down in the tree, similar to sift-up; used to restore heap condition after deletion or replacement.
266:): joining two heaps to form a valid new heap containing all the elements of both, preserving the original heaps.
1637:
272:: joining two heaps to form a valid new heap containing all the elements of both, destroying the original heaps.
3833:
3828:
3787:
3717:
3229:
2866:. FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 174–183.
2628:
2078:
2059:
699:
indicates that the given complexity is amortized, otherwise it is a worst-case complexity. For the meaning of "
654:
561:. This is faster than a sequence of consecutive insertions into an originally empty heap, which is log-linear.
3051:
2996:
2420:
Suchenek, Marek A. (2012), "Elementary Yet
Precise Worst-Case Analysis of Floyd's Heap-Construction Program",
1497:
2044:
4068:
4045:
1991:
351:
128:
74:
3128:
2017:
1837:
462:
Although different types of heaps implement the operations differently, the most common way is as follows:
4050:
3850:
3200:
3013:
2968:
2867:
2781:
2610:
2519:
2092:
1553:
3931:
3893:
3216:
2739:
2275:
2260:
2003:
1379:
634:
58:
30:
For the memory heap (in low-level computer programming), which is unrelated to this data structure, see
1265:
2332:. United States of America: The MIT Press Cambridge, Massachusetts London, England. pp. 151–152.
3916:
3357:
3312:
2679:
Brodal, Gerth Stølting; Okasaki, Chris (November 1996), "Optimal purely functional priority queues",
2458:
2032:
algorithms for heaps (usually implemented as binary heaps), which operate on arbitrary random access
340:
3018:
2973:
2822:
2786:
2615:
3471:
3247:
3076:
2872:
2735:
2524:
2226:
1987:
1977:
115:
binary tree (see figure). The heap data structure, specifically the binary heap, was introduced by
3959:
3327:
2100:
3944:
3809:
3769:
3638:
3597:
3423:
3413:
3292:
2771:
2408:
1983:
1014:
629:
156:
97:
619:
217:): returns the node of maximum value from a max heap after removing it from the heap (a.k.a.,
3878:
3777:
3532:
3233:
3193:
3172:
3088:
3031:
2885:
2799:
2715:
2574:
2537:
2472:
2384:
2353:
2333:
2255:
501:-ary) heap out of a given array of elements may be performed in linear time using the classic
152:
116:
2505:
2396:
3901:
3623:
3160:
3118:
3080:
3023:
2978:
2925:
2877:
2839:
2791:
2688:
2620:
2566:
2529:
2454:
2429:
2367:
1974:: One of the best sorting methods being in-place and with no quadratic worst-case scenarios.
191:): find a maximum item of a max-heap, or a minimum item of a min-heap, respectively (a.k.a.
124:
50:
3012:. Proceedings of the 44th symposium on Theory of Computing - STOC '12. pp. 1177–1184.
3921:
3863:
3567:
3317:
2943:
2818:
2203:
412:
379:
2245:
class which uses quaternary (d-ary) min-heap implementation. It is available from .NET 6.
2082:
1611:
131:. When a heap is a complete binary tree, it has the smallest possible height—a heap with
4013:
3991:
3816:
3740:
3520:
3515:
3398:
3332:
2497:
1997:
1322:
716:
609:
101:
62:
17:
2858:
2770:, Lecture Notes in Computer Science, vol. 1851, Springer-Verlag, pp. 63–77,
314:: delete an arbitrary node (followed by moving last node and sifting to maintain heap)
4083:
3986:
3883:
3868:
3668:
3648:
3491:
3380:
3307:
3000:
2947:
2906:
2707:
2624:
2558:
2501:
952:
591:
3242:
1073:
571:
3628:
3592:
3408:
3403:
3385:
3297:
1430:
1204:
889:
669:
644:
639:
624:
596:
3108:
Frederickson, Greg N. (1993), "An
Optimal Algorithm for Selection in a Min-Heap",
2134:
platform (since version 1.5) provides a binary heap implementation with the class
448:. This simple indexing scheme makes it efficient to move "up" or "down" the tree.
167:
in that they require no additional memory beyond that used for storing the keys.
3981:
3906:
3678:
3643:
3633:
3547:
3481:
3476:
3466:
3375:
3224:
2910:
2762:
2758:
748:
586:
367:
112:
108:
42:
3055:
3004:
2088:
3969:
3873:
3688:
3658:
3618:
3461:
3390:
3337:
3257:
2693:
2462:
2315:
2108:
2067:
1751:) time (where both complexities can be amortized). Another algorithm achieves
649:
601:
164:
2541:
2151:
3911:
3858:
3693:
3653:
3500:
3428:
3418:
3027:
2952:"Fibonacci heaps and their uses in improved network optimization algorithms"
2795:
2570:
2122:
1134:
811:
679:
664:
659:
3123:
358:
4008:
3582:
3272:
2843:
2371:
37:
3954:
3782:
3698:
3572:
3552:
3525:
3510:
3262:
3154:
2881:
2194:
2033:
1971:
614:
502:
453:
120:
2982:
2433:
4003:
3949:
3683:
3587:
3562:
3505:
3352:
3282:
3277:
3252:
3185:
2652:
2118:
2929:
3998:
3939:
3602:
3577:
3557:
3542:
3451:
3342:
3267:
2071:
576:
2533:
2176:
has implementations of binary, binomial, and
Fibonacci heaps in the
2177:
2063:
2055:
231:): removing the root node of a max heap (or min heap), respectively
3709:
3446:
3347:
3302:
2776:
2280:
2215:
674:
357:
321:
36:
4020:
3438:
2823:"On the Efficiency of Pairing Heaps and Related Data Structures"
2407:
The Python
Standard Library, 8.4. heapq — Heap queue algorithm,
2395:
The Python
Standard Library, 8.4. heapq — Heap queue algorithm,
2383:
The Python
Standard Library, 8.4. heapq — Heap queue algorithm,
2238:
2183:
2173:
581:
3713:
3189:
123:
sorting algorithm. Heaps are also crucial in several efficient
2596:"Average Case Analysis of Heap Building by Repeated Insertion"
2159:
347:
Each element in the array represents a node of the heap, and
2225:
programming language has a binary max-heap implementation,
2107:
statements and integration with the range-based API of the
542:) is the sum of all digits of the binary representation of
3063:
Proc. 7th Annual ACM-SIAM Symposium on
Discrete Algorithms
2283:, a form of binary search tree based on heap-ordered trees
473:
operation) until the heap property has been reestablished.
96:
The heap is one maximally efficient implementation of an
308:: updating a key within a max- or min-heap, respectively
45:
max-heap with node keys being integers between 1 and 100
1951:
data structure achieving the same optimum, except that
1727:
is the operation of building a heap from a sequence of
2492:
2490:
2488:
505:, with the worst-case number of comparisons equal to 2
85:) of P is greater than or equal to the key of C. In a
1884:
1840:
1640:
1614:
1556:
1500:
557:) is the exponent of 2 in the prime factorization of
415:
382:
291:: return true if the heap is empty, false otherwise.
4036:
3930:
3892:
3849:
3768:
3747:
3611:
3490:
3437:
3366:
3223:
3117:, vol. 104, Academic Press, pp. 197–214,
2764:
2761:(2000), "Improved upper bounds for pairing heaps",
2653:"Binomial Heap | Brilliant Math & Science Wiki"
3171:(2nd ed.). Addison Wesley. pp. 147–162.
2960:Journal of the Association for Computing Machinery
2831:Journal of the Association for Computing Machinery
2051:-ary, binomial, Fibonacci, pairing and skew heaps.
1928:
1870:
1703:
1626:
1600:
1542:
695:of various heap data structures. The abbreviation
430:
397:
2674:
2672:
2553:
2551:
2314:, 14 December 2004. Retrieved on 2017-10-08 from
1769:
1767:
1765:
2170:) as of version 5.3 in the Standard PHP Library.
1781:), a generic transformation reduces the cost of
1967:The heap data structure has many applications.
3083:(2004). "7.3.6. Bottom-Up Heap Construction".
2312:National Institute of Standards and Technology
1929:{\displaystyle O(2^{2{\sqrt {\log \log n}}}).}
256:: create a heap out of given array of elements
3725:
3201:
2300:Black (ed.), Paul E. (2004-12-14). Entry for
8:
2307:Dictionary of Algorithms and Data Structures
1350:
1341:
1293:
1284:
1236:
1223:
1197:
1184:
1175:
1166:
1153:
1107:
1092:
987:
882:
869:
856:
843:
830:
696:
483:operation) to reestablish the heap property.
2471:(1st ed.). MIT Press and McGraw-Hill.
2316:https://xlinux.nist.gov/dads/HTML/heap.html
687:Comparison of theoretic bounds for variants
175:The common operations involving heaps are:
3732:
3718:
3710:
3208:
3194:
3186:
1943:
1941:
1704:{\displaystyle nO(\log n)-O(n)=O(n\log n)}
1494:)) in the existing size of the heap, thus
3122:
3017:
2972:
2871:
2860:Towards a Final Analysis of Pairing Heaps
2785:
2775:
2692:
2614:
2523:
1899:
1895:
1883:
1839:
1639:
1613:
1566:
1555:
1516:
1505:
1499:
719:. Names of operations assume a max-heap.
414:
381:
285:: return the number of items in the heap.
107:A common implementation of a heap is the
2710:(1998). "10.2. Structural Abstraction".
2594:Hayward, Ryan; McDiarmid, Colin (1991).
2103:that allows iteration with D's built-in
2091:. Instances can be constructed from any
1543:{\displaystyle \sum _{k=1}^{n}O(\log k)}
721:
203:: adding a new key to the heap (a.k.a.,
77:C, if P is a parent node of C, then the
2293:
2087:, which is implemented in terms of D's
1483:
456:can be used to sort an array in-place.
3085:Data Structures and Algorithms in Java
3056:"Worst-Case Efficient Priority Queues"
2563:Data Structures and Network Algorithms
2449:
2447:
2445:
2443:
1988:Prim's minimal-spanning-tree algorithm
354:by the elements' indices in the array.
339:Heaps are usually implemented with an
3163:of how the basic heap algorithms work
2905:Haeupler, Bernhard; Sen, Siddhartha;
2074:support. It provides an STL-like API.
1871:{\displaystyle \Omega (\log \log n),}
1731:unsorted elements. It can be done in
1711:. This can also be readily seen from
139:branches for each node always has log
119:in 1964, as a data structure for the
7:
2356:(1964), "Algorithm 232 - Heapsort",
2011:Programming language implementations
1809:(1) time (amortized, if the cost of
1601:{\displaystyle \log n/2=(\log n)-1}
350:The parent / child relationship is
3087:(3rd ed.). pp. 338–341.
2714:(1st ed.). pp. 158–162.
1992:Dijkstra's shortest-path algorithm
1841:
25:
2712:Purely Functional Data Structures
2681:Journal of Functional Programming
2235:module of its standard library.
1793:is the sum of the old costs of
155:(as there would be in, e.g., a
27:Computer science data structure
2561:(1983). "3.3. Leftist heaps".
1920:
1888:
1862:
1844:
1698:
1683:
1674:
1668:
1659:
1647:
1589:
1577:
1537:
1525:
374:, its children are at indices
1:
531:) (for a binary heap), where
497:Construction of a binary (or
440:, and its parent is at index
2625:10.1016/0196-6774(91)90027-v
2077:The standard library of the
489:Remove the root and put the
4056:Directed acyclic word graph
3822:Double-ended priority queue
3167:Bentley, Jon Louis (2000).
3111:Information and Computation
2056:generic heap implementation
1490:Each insertion takes O(log(
32:C dynamic memory allocation
4106:
2468:Introduction to Algorithms
2330:INTRODUCTION TO ALGORITHMS
2328:CORMEN, THOMAS H. (2009).
2271:Queue (abstract data type)
2266:Stack (abstract data type)
2182:distribution available on
2142:Java Collections Framework
29:
4064:
2944:Fredman, Michael Lawrence
2819:Fredman, Michael Lawrence
2694:10.1017/s095679680000201x
2511:SIAM Journal on Computing
2359:Communications of the ACM
111:, in which the tree is a
3788:Retrieval Data Structure
3664:Left-child right-sibling
2084:std.container.BinaryHeap
1789:, while the new cost of
1713:Stirling's approximation
655:Randomized meldable heap
4090:Heaps (data structures)
4069:List of data structures
4046:Binary decision diagram
3494:data partitioning trees
3452:C-trie (compressed ADT)
3028:10.1145/2213977.2214082
2999:; Lagogiannis, George;
2796:10.1007/3-540-44985-X_5
2571:10.1137/1.9781611970265
2498:Sleator, Daniel Dominic
2428:(1), IOS Press: 75–92,
2422:Fundamenta Informaticae
2310:. Online version. U.S.
2137:java.util.PriorityQueue
1634:; formally the time is
18:Heap (computer science)
4051:Directed acyclic graph
3124:10.1006/inco.1993.1030
3006:Strict Fibonacci heaps
2997:Brodal, Gerth Stølting
2506:"Self-Adjusting Heaps"
2079:D programming language
1930:
1872:
1777:heaps (not supporting
1705:
1628:
1602:
1544:
1521:
432:
399:
363:
250:: create an empty heap
46:
2857:Pettie, Seth (2005).
2844:10.1145/320211.320214
2459:Leiserson, Charles E.
2372:10.1145/512274.512284
2276:Tree (data structure)
2261:Search data structure
2101:input range interface
1931:
1873:
1706:
1629:
1603:
1545:
1501:
635:Strict Fibonacci heap
433:
400:
361:
40:
3917:Unrolled linked list
3674:Log-structured merge
3217:Tree data structures
3157:at Wolfram MathWorld
3077:Goodrich, Michael T.
2911:"Rank-pairing heaps"
2882:10.1109/SFCS.2005.75
2502:Tarjan, Robert Endre
2193:language contains a
2018:C++ Standard Library
1978:Selection algorithms
1882:
1838:
1638:
1612:
1554:
1498:
431:{\displaystyle 2i+2}
413:
398:{\displaystyle 2i+1}
380:
129:Dijkstra's algorithm
3965:Self-balancing tree
2983:10.1145/28869.28874
2741:Theory of 2–3 Heaps
2434:10.3233/FI-2012-751
2206:library contains a
2162:has both max-heap (
2093:random-access range
2045:Boost C++ libraries
1759:) for binary heaps.
1627:{\displaystyle k=n}
65:that satisfies the
3945:Binary search tree
3810:Double-ended queue
3639:Fractal tree index
3234:associative arrays
3169:Programming Pearls
2565:. pp. 38–42.
2354:Williams, J. W. J.
1926:
1868:
1701:
1624:
1598:
1540:
630:Skew binomial heap
428:
395:
364:
352:defined implicitly
157:binary search tree
153:in-order traversal
98:abstract data type
47:
4077:
4076:
3879:Hashed array tree
3778:Associative array
3707:
3706:
3081:Tamassia, Roberto
3037:978-1-4503-1245-5
3001:Tarjan, Robert E.
2948:Tarjan, Robert E.
2930:10.1137/100785351
2918:SIAM J. Computing
2909:(November 2011).
2907:Tarjan, Robert E.
2580:978-0-89871-187-5
2504:(February 1986).
2463:Rivest, Ronald L.
2455:Cormen, Thomas H.
2409:heapq.heapreplace
2339:978-0-262-03384-8
2256:Sorting algorithm
1955:is not supported.
1916:
1801:. Here, it makes
1480:
1479:
693:time complexities
117:J. W. J. Williams
16:(Redirected from
4097:
3902:Association list
3734:
3727:
3720:
3711:
3210:
3203:
3196:
3187:
3182:
3142:
3141:
3140:
3139:
3133:
3127:, archived from
3126:
3116:
3105:
3099:
3098:
3073:
3067:
3066:
3065:, pp. 52–58
3060:
3052:Brodal, Gerth S.
3048:
3042:
3041:
3021:
3011:
2993:
2987:
2986:
2976:
2956:
2940:
2934:
2933:
2924:(6): 1463–1485.
2915:
2902:
2896:
2895:
2875:
2865:
2854:
2848:
2847:
2827:
2815:
2809:
2808:
2789:
2779:
2769:
2755:
2749:
2748:
2746:
2732:
2726:
2725:
2704:
2698:
2697:
2696:
2676:
2667:
2666:
2664:
2663:
2649:
2643:
2642:
2640:
2639:
2633:
2627:. Archived from
2618:
2600:
2591:
2585:
2584:
2555:
2546:
2545:
2527:
2494:
2483:
2482:
2451:
2438:
2436:
2417:
2411:
2405:
2399:
2393:
2387:
2381:
2375:
2374:
2350:
2344:
2343:
2325:
2319:
2298:
2234:
2229:
2210:
2197:
2180:
2169:
2166:) and min-heap (
2165:
2154:
2139:
2125:
2111:
2106:
2098:
2085:
2039:
2031:
2027:
2023:
1984:Graph algorithms
1956:
1945:
1936:
1935:
1933:
1932:
1927:
1919:
1918:
1917:
1900:
1877:
1875:
1874:
1869:
1832:
1826:
1771:
1760:
1739:) time whenever
1722:
1716:
1710:
1708:
1707:
1702:
1633:
1631:
1630:
1625:
1607:
1605:
1604:
1599:
1570:
1549:
1547:
1546:
1541:
1520:
1515:
1488:
1380:Strict Fibonacci
1352:
1343:
1295:
1286:
1238:
1225:
1199:
1186:
1177:
1168:
1155:
1109:
1094:
989:
884:
871:
858:
845:
832:
722:
698:
447:
439:
437:
435:
434:
429:
406:
404:
402:
401:
396:
373:
125:graph algorithms
73:, for any given
51:computer science
21:
4105:
4104:
4100:
4099:
4098:
4096:
4095:
4094:
4080:
4079:
4078:
4073:
4060:
4032:
3926:
3922:XOR linked list
3888:
3864:Circular buffer
3845:
3764:
3743:
3741:Data structures
3738:
3708:
3703:
3607:
3486:
3433:
3362:
3358:Weight-balanced
3313:Order statistic
3227:
3219:
3214:
3179:
3166:
3151:
3146:
3145:
3137:
3135:
3131:
3114:
3107:
3106:
3102:
3095:
3075:
3074:
3070:
3058:
3050:
3049:
3045:
3038:
3019:10.1.1.233.1740
3009:
2995:
2994:
2990:
2974:10.1.1.309.8927
2954:
2942:
2941:
2937:
2913:
2904:
2903:
2899:
2892:
2863:
2856:
2855:
2851:
2825:
2817:
2816:
2812:
2806:
2787:10.1.1.748.7812
2767:
2757:
2756:
2752:
2744:
2734:
2733:
2729:
2722:
2706:
2705:
2701:
2678:
2677:
2670:
2661:
2659:
2651:
2650:
2646:
2637:
2635:
2631:
2616:10.1.1.353.7888
2598:
2593:
2592:
2588:
2581:
2557:
2556:
2549:
2534:10.1137/0215004
2496:
2495:
2486:
2479:
2453:
2452:
2441:
2419:
2418:
2414:
2406:
2402:
2394:
2390:
2382:
2378:
2352:
2351:
2347:
2340:
2327:
2326:
2322:
2299:
2295:
2290:
2252:
2232:
2227:
2208:
2204:Core Foundation
2195:
2178:
2167:
2163:
2152:
2135:
2123:
2109:
2104:
2096:
2083:
2037:
2029:
2025:
2021:
2013:
1965:
1960:
1959:
1946:
1939:
1891:
1880:
1879:
1878:upper bound of
1836:
1835:
1834:Lower bound of
1833:
1829:
1772:
1763:
1723:
1719:
1636:
1635:
1610:
1609:
1552:
1551:
1496:
1495:
1489:
1485:
689:
684:
567:
552:
537:
526:
515:
503:Floyd algorithm
441:
411:
410:
408:
378:
377:
375:
371:
337:
173:
144:
35:
28:
23:
22:
15:
12:
11:
5:
4103:
4101:
4093:
4092:
4082:
4081:
4075:
4074:
4072:
4071:
4065:
4062:
4061:
4059:
4058:
4053:
4048:
4042:
4040:
4034:
4033:
4031:
4030:
4029:
4028:
4018:
4017:
4016:
4014:Hilbert R-tree
4011:
4006:
3996:
3995:
3994:
3992:Fibonacci heap
3989:
3984:
3974:
3973:
3972:
3967:
3962:
3960:Red–black tree
3957:
3952:
3942:
3936:
3934:
3928:
3927:
3925:
3924:
3919:
3914:
3909:
3904:
3898:
3896:
3890:
3889:
3887:
3886:
3881:
3876:
3871:
3866:
3861:
3855:
3853:
3847:
3846:
3844:
3843:
3842:
3841:
3836:
3826:
3825:
3824:
3817:Priority queue
3814:
3813:
3812:
3802:
3797:
3792:
3791:
3790:
3785:
3774:
3772:
3766:
3765:
3763:
3762:
3757:
3751:
3749:
3745:
3744:
3739:
3737:
3736:
3729:
3722:
3714:
3705:
3704:
3702:
3701:
3696:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3656:
3651:
3646:
3641:
3636:
3631:
3626:
3621:
3615:
3613:
3609:
3608:
3606:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3523:
3518:
3513:
3508:
3503:
3497:
3495:
3488:
3487:
3485:
3484:
3479:
3474:
3472:Ternary search
3469:
3464:
3459:
3454:
3449:
3443:
3441:
3435:
3434:
3432:
3431:
3426:
3421:
3416:
3411:
3406:
3401:
3396:
3388:
3383:
3378:
3372:
3370:
3364:
3363:
3361:
3360:
3355:
3350:
3345:
3340:
3335:
3330:
3320:
3315:
3310:
3305:
3300:
3295:
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3239:
3237:
3221:
3220:
3215:
3213:
3212:
3205:
3198:
3190:
3184:
3183:
3177:
3164:
3158:
3150:
3149:External links
3147:
3144:
3143:
3100:
3093:
3068:
3043:
3036:
2988:
2967:(3): 596–615.
2935:
2897:
2890:
2873:10.1.1.549.471
2849:
2838:(4): 473–501.
2810:
2804:
2750:
2736:Takaoka, Tadao
2727:
2720:
2708:Okasaki, Chris
2699:
2687:(6): 839–857,
2668:
2644:
2586:
2579:
2559:Tarjan, Robert
2547:
2525:10.1.1.93.6678
2484:
2477:
2439:
2412:
2400:
2388:
2385:heapq.heappush
2376:
2366:(6): 347–348,
2345:
2338:
2320:
2292:
2291:
2289:
2286:
2285:
2284:
2278:
2273:
2268:
2263:
2258:
2251:
2248:
2247:
2246:
2236:
2219:
2213:
2200:
2187:
2171:
2157:
2145:
2128:
2115:
2075:
2052:
2041:
2038:priority_queue
2012:
2009:
2008:
2007:
2001:
1998:Priority queue
1995:
1981:
1975:
1964:
1961:
1958:
1957:
1937:
1925:
1922:
1915:
1912:
1909:
1906:
1903:
1898:
1894:
1890:
1887:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1827:
1817:still runs in
1761:
1717:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1623:
1620:
1617:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1569:
1565:
1562:
1559:
1539:
1536:
1533:
1530:
1527:
1524:
1519:
1514:
1511:
1508:
1504:
1482:
1481:
1478:
1477:
1467:
1461:
1455:
1449:
1439:
1433:
1427:
1426:
1416:
1410:
1404:
1398:
1388:
1382:
1376:
1375:
1365:
1359:
1353:
1344:
1331:
1325:
1319:
1318:
1308:
1302:
1296:
1287:
1274:
1268:
1262:
1261:
1251:
1245:
1239:
1226:
1213:
1207:
1201:
1200:
1187:
1178:
1169:
1156:
1143:
1137:
1135:Bottom-up skew
1131:
1130:
1120:
1110:
1101:
1095:
1082:
1076:
1070:
1069:
1059:
1049:
1043:
1033:
1023:
1017:
1011:
1010:
1000:
990:
981:
971:
961:
955:
949:
948:
938:
928:
918:
908:
898:
892:
886:
885:
872:
859:
846:
833:
820:
814:
808:
807:
797:
787:
777:
767:
757:
751:
745:
744:
741:
738:
735:
732:
729:
726:
717:Big O notation
688:
685:
683:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
610:Fibonacci heap
607:
599:
594:
589:
584:
579:
574:
568:
566:
563:
550:
535:
524:
513:
495:
494:
484:
474:
427:
424:
421:
418:
394:
391:
388:
385:
356:
355:
348:
343:, as follows:
336:
335:Implementation
333:
332:
331:
325:
315:
309:
298:
297:
293:
292:
286:
279:
278:
274:
273:
267:
257:
251:
244:
243:
239:
238:
232:
222:
208:
198:
181:
180:
172:
169:
140:
102:priority queue
63:data structure
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4102:
4091:
4088:
4087:
4085:
4070:
4067:
4066:
4063:
4057:
4054:
4052:
4049:
4047:
4044:
4043:
4041:
4039:
4035:
4027:
4024:
4023:
4022:
4019:
4015:
4012:
4010:
4007:
4005:
4002:
4001:
4000:
3997:
3993:
3990:
3988:
3987:Binomial heap
3985:
3983:
3980:
3979:
3978:
3975:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3947:
3946:
3943:
3941:
3938:
3937:
3935:
3933:
3929:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3899:
3897:
3895:
3891:
3885:
3884:Sparse matrix
3882:
3880:
3877:
3875:
3872:
3870:
3869:Dynamic array
3867:
3865:
3862:
3860:
3857:
3856:
3854:
3852:
3848:
3840:
3837:
3835:
3832:
3831:
3830:
3827:
3823:
3820:
3819:
3818:
3815:
3811:
3808:
3807:
3806:
3803:
3801:
3798:
3796:
3793:
3789:
3786:
3784:
3781:
3780:
3779:
3776:
3775:
3773:
3771:
3767:
3761:
3758:
3756:
3753:
3752:
3750:
3746:
3742:
3735:
3730:
3728:
3723:
3721:
3716:
3715:
3712:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3649:Hash calendar
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3614:
3610:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3538:
3536:
3530:
3528:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3498:
3496:
3493:
3489:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3444:
3442:
3440:
3436:
3430:
3427:
3425:
3424:van Emde Boas
3422:
3420:
3417:
3415:
3414:Skew binomial
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3393:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3373:
3371:
3369:
3365:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3325:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3293:Binary search
3290:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3240:
3238:
3235:
3231:
3226:
3222:
3218:
3211:
3206:
3204:
3199:
3197:
3192:
3191:
3188:
3180:
3174:
3170:
3165:
3162:
3159:
3156:
3153:
3152:
3148:
3134:on 2012-12-03
3130:
3125:
3120:
3113:
3112:
3104:
3101:
3096:
3094:0-471-46983-1
3090:
3086:
3082:
3078:
3072:
3069:
3064:
3057:
3053:
3047:
3044:
3039:
3033:
3029:
3025:
3020:
3015:
3008:
3007:
3002:
2998:
2992:
2989:
2984:
2980:
2975:
2970:
2966:
2962:
2961:
2953:
2950:(July 1987).
2949:
2945:
2939:
2936:
2931:
2927:
2923:
2919:
2912:
2908:
2901:
2898:
2893:
2891:0-7695-2468-0
2887:
2883:
2879:
2874:
2869:
2862:
2861:
2853:
2850:
2845:
2841:
2837:
2833:
2832:
2824:
2821:(July 1999).
2820:
2814:
2811:
2807:
2805:3-540-67690-2
2801:
2797:
2793:
2788:
2783:
2778:
2773:
2766:
2765:
2760:
2754:
2751:
2743:
2742:
2737:
2731:
2728:
2723:
2721:9780521631242
2717:
2713:
2709:
2703:
2700:
2695:
2690:
2686:
2682:
2675:
2673:
2669:
2658:
2657:brilliant.org
2654:
2648:
2645:
2634:on 2016-02-05
2630:
2626:
2622:
2617:
2612:
2608:
2604:
2603:J. Algorithms
2597:
2590:
2587:
2582:
2576:
2572:
2568:
2564:
2560:
2554:
2552:
2548:
2543:
2539:
2535:
2531:
2526:
2521:
2517:
2513:
2512:
2507:
2503:
2499:
2493:
2491:
2489:
2485:
2480:
2478:0-262-03141-8
2474:
2470:
2469:
2464:
2460:
2456:
2450:
2448:
2446:
2444:
2440:
2435:
2431:
2427:
2423:
2416:
2413:
2410:
2404:
2401:
2398:
2397:heapq.heappop
2392:
2389:
2386:
2380:
2377:
2373:
2369:
2365:
2361:
2360:
2355:
2349:
2346:
2341:
2335:
2331:
2324:
2321:
2317:
2313:
2309:
2308:
2303:
2297:
2294:
2287:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2253:
2249:
2244:
2243:PriorityQueue
2240:
2237:
2230:
2224:
2220:
2217:
2214:
2211:
2205:
2201:
2198:
2192:
2188:
2185:
2181:
2175:
2172:
2161:
2158:
2155:
2149:
2146:
2143:
2138:
2133:
2129:
2126:
2121:there is the
2120:
2116:
2113:
2110:std.algorithm
2102:
2094:
2090:
2086:
2080:
2076:
2073:
2069:
2065:
2061:
2057:
2053:
2050:
2046:
2042:
2035:
2020:provides the
2019:
2015:
2014:
2010:
2005:
2002:
1999:
1996:
1993:
1989:
1985:
1982:
1979:
1976:
1973:
1970:
1969:
1968:
1962:
1954:
1950:
1944:
1942:
1938:
1923:
1913:
1910:
1907:
1904:
1901:
1896:
1892:
1885:
1865:
1859:
1856:
1853:
1850:
1847:
1831:
1828:
1824:
1820:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1780:
1776:
1770:
1768:
1766:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1726:
1721:
1718:
1714:
1695:
1692:
1689:
1686:
1680:
1677:
1671:
1665:
1662:
1656:
1653:
1650:
1644:
1641:
1621:
1618:
1615:
1595:
1592:
1586:
1583:
1580:
1574:
1571:
1567:
1563:
1560:
1557:
1534:
1531:
1528:
1522:
1517:
1512:
1509:
1506:
1502:
1493:
1487:
1484:
1475:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1443:
1440:
1437:
1434:
1432:
1429:
1428:
1424:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1392:
1389:
1386:
1383:
1381:
1378:
1377:
1373:
1369:
1366:
1363:
1360:
1357:
1354:
1348:
1345:
1339:
1335:
1332:
1329:
1326:
1324:
1321:
1320:
1316:
1312:
1309:
1306:
1303:
1300:
1297:
1291:
1288:
1282:
1278:
1275:
1272:
1269:
1267:
1264:
1263:
1259:
1255:
1252:
1249:
1246:
1243:
1240:
1234:
1230:
1227:
1221:
1217:
1214:
1211:
1208:
1206:
1203:
1202:
1195:
1191:
1188:
1182:
1179:
1173:
1170:
1164:
1160:
1157:
1151:
1147:
1144:
1141:
1138:
1136:
1133:
1132:
1128:
1124:
1121:
1118:
1114:
1111:
1105:
1102:
1099:
1096:
1090:
1086:
1083:
1080:
1077:
1075:
1072:
1071:
1067:
1063:
1060:
1057:
1053:
1050:
1047:
1044:
1041:
1037:
1034:
1031:
1027:
1024:
1021:
1018:
1016:
1015:Skew binomial
1013:
1012:
1008:
1004:
1001:
998:
994:
991:
985:
982:
979:
975:
972:
969:
965:
962:
959:
956:
954:
951:
950:
946:
942:
939:
936:
932:
929:
926:
922:
919:
916:
912:
909:
906:
902:
899:
896:
893:
891:
888:
887:
880:
876:
873:
867:
863:
860:
854:
850:
847:
841:
837:
834:
828:
824:
821:
818:
815:
813:
810:
809:
805:
801:
798:
795:
791:
788:
785:
781:
778:
775:
771:
768:
765:
761:
758:
755:
752:
750:
747:
746:
742:
739:
736:
734:increase-key
733:
730:
727:
724:
723:
720:
718:
714:
710:
706:
702:
694:
686:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
604:
600:
598:
595:
593:
592:Binomial heap
590:
588:
585:
583:
580:
578:
575:
573:
570:
569:
564:
562:
560:
556:
549:
545:
541:
534:
530:
523:
519:
512:
508:
504:
500:
492:
488:
485:
482:
478:
475:
472:
468:
465:
464:
463:
460:
457:
455:
449:
445:
425:
422:
419:
416:
392:
389:
386:
383:
369:
360:
353:
349:
346:
345:
344:
342:
334:
329:
326:
323:
319:
316:
313:
310:
307:
303:
300:
299:
295:
294:
290:
287:
284:
281:
280:
276:
275:
271:
268:
265:
261:
258:
255:
252:
249:
246:
245:
241:
240:
236:
233:
230:
226:
223:
220:
216:
212:
209:
206:
202:
199:
196:
195:
190:
186:
183:
182:
178:
177:
176:
170:
168:
166:
160:
158:
154:
149:
147:
143:
138:
134:
130:
126:
122:
118:
114:
110:
105:
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
67:heap property
64:
60:
56:
52:
44:
41:Example of a
39:
33:
19:
3976:
3839:Disjoint-set
3534:
3526:
3391:
3367:
3324:Left-leaning
3230:dynamic sets
3225:Search trees
3168:
3136:, retrieved
3129:the original
3110:
3103:
3084:
3071:
3062:
3046:
3005:
2991:
2964:
2958:
2938:
2921:
2917:
2900:
2859:
2852:
2835:
2829:
2813:
2763:
2759:Iacono, John
2753:
2747:, p. 12
2740:
2730:
2711:
2702:
2684:
2680:
2660:. Retrieved
2656:
2647:
2636:. Retrieved
2629:the original
2606:
2602:
2589:
2562:
2518:(1): 52–69.
2515:
2509:
2467:
2425:
2421:
2415:
2403:
2391:
2379:
2363:
2357:
2348:
2329:
2323:
2305:
2301:
2296:
2209:CFBinaryHeap
2048:
1966:
1963:Applications
1953:increase-key
1952:
1830:
1822:
1818:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1779:increase-key
1778:
1756:
1752:
1748:
1744:
1740:
1736:
1732:
1728:
1724:
1720:
1491:
1486:
1473:
1469:
1463:
1457:
1451:
1445:
1441:
1435:
1422:
1418:
1412:
1406:
1400:
1394:
1390:
1384:
1371:
1367:
1361:
1355:
1346:
1337:
1333:
1327:
1314:
1310:
1304:
1298:
1289:
1280:
1276:
1270:
1266:Rank-pairing
1257:
1253:
1247:
1241:
1232:
1228:
1219:
1215:
1209:
1193:
1189:
1180:
1171:
1162:
1158:
1149:
1145:
1139:
1126:
1122:
1116:
1112:
1103:
1097:
1088:
1084:
1078:
1065:
1061:
1055:
1051:
1045:
1039:
1035:
1029:
1025:
1019:
1006:
1002:
996:
992:
983:
977:
973:
967:
963:
957:
944:
940:
934:
930:
924:
920:
914:
910:
904:
900:
894:
878:
874:
865:
861:
852:
848:
839:
835:
826:
822:
816:
803:
799:
793:
789:
783:
779:
773:
769:
763:
759:
753:
712:
708:
704:
700:
690:
670:Ternary heap
645:Pairing heap
640:Min-max heap
625:Leftist heap
602:
597:Brodal queue
558:
554:
547:
543:
539:
532:
528:
521:
517:
510:
506:
498:
496:
490:
487:Replacement:
486:
480:
476:
470:
466:
461:
458:
450:
443:
365:
338:
327:
317:
311:
306:decrease-key
305:
302:increase-key
301:
288:
282:
269:
263:
259:
253:
247:
234:
228:
224:
218:
214:
210:
204:
200:
192:
188:
184:
174:
161:
150:
145:
141:
136:
132:
106:
95:
90:
86:
82:
78:
70:
66:
54:
48:
3982:Binary heap
3907:Linked list
3624:Exponential
3612:Other trees
3161:Explanation
2609:: 126–153.
2233:collections
2099:exposes an
2054:There is a
2004:K-way merge
1785:to that of
731:delete-max
587:Binary heap
477:Extraction:
368:binary heap
248:create-heap
215:extract-min
211:extract-max
165:Radix trees
109:binary heap
3970:Splay tree
3874:Hash table
3755:Collection
3568:Priority R
3318:Palindrome
3178:0201657880
3138:2010-10-31
2662:2019-09-30
2638:2016-01-28
2288:References
2228:BinaryHeap
2212:structure.
2168:SplMinHeap
2164:SplMaxHeap
2097:BinaryHeap
2068:D-ary heap
1949:persistent
1821:(log
1815:delete-max
1813:is) while
1795:delete-max
1791:delete-max
1775:persistent
1747:(log
1444:(log
1393:(log
1336:(log
1279:(log
1231:(log
1218:(log
1161:(log
1148:(log
1115:(log
1087:(log
1054:(log
1038:(log
1028:(log
995:(log
976:(log
966:(log
933:(log
923:(log
913:(log
903:(log
864:(log
851:(log
838:(log
825:(log
782:(log
772:(log
762:(log
743:make-heap
725:Operation
650:Radix heap
467:Insertion:
277:Inspection
229:delete-min
225:delete-max
171:Operations
135:nodes and
4026:Hash tree
3912:Skip list
3859:Bit array
3760:Container
3654:iDistance
3533:implicit
3521:Hilbert R
3516:Cartesian
3399:Fibonacci
3333:Scapegoat
3328:Red–black
3014:CiteSeerX
2969:CiteSeerX
2868:CiteSeerX
2782:CiteSeerX
2777:1110.4428
2611:CiteSeerX
2542:0097-5397
2520:CiteSeerX
2231:, in the
2124:Data.Heap
2081:includes
2034:iterators
2026:push_heap
2022:make_heap
1911:
1905:
1857:
1851:
1842:Ω
1725:make-heap
1693:
1663:−
1654:
1593:−
1584:
1561:
1532:
1503:∑
1323:Fibonacci
728:find-max
691:Here are
680:Weak heap
665:Soft heap
660:Skew heap
620:Leaf heap
605:-ary heap
328:sift-down
100:called a
4084:Category
3955:AVL tree
3834:Multiset
3783:Multimap
3770:Abstract
3669:Link/cut
3381:Binomial
3308:Interval
3054:(1996),
3003:(2012).
2738:(1999),
2465:(1990).
2250:See also
2202:Apple's
2030:pop_heap
1972:Heapsort
1743:runs in
1550:. Since
1074:2–3 heap
953:Binomial
707:)" and "
615:K-D Heap
572:2–3 heap
565:Variants
454:heapsort
296:Internal
289:is-empty
242:Creation
189:find-min
185:find-max
148:height.
127:such as
121:heapsort
113:complete
87:min heap
71:max heap
4009:R+ tree
4004:R* tree
3950:AA tree
3629:Fenwick
3593:Segment
3492:Spatial
3409:Pairing
3404:Leftist
3326:)
3298:Dancing
3291:)
3289:Optimal
2140:in the
2127:module.
2119:Haskell
2112:package
2105:foreach
1805:run in
1205:Pairing
890:Leftist
737:insert
715:)" see
438:
409:
405:
376:
318:sift-up
254:heapify
235:replace
69:: In a
61:-based
4038:Graphs
3999:R-tree
3940:B-tree
3894:Linked
3851:Arrays
3679:Merkle
3644:Fusion
3634:Finger
3558:Octree
3548:Metric
3482:Y-fast
3477:X-fast
3467:Suffix
3386:Brodal
3376:Binary
3175:
3091:
3034:
3016:
2971:
2888:
2870:
2802:
2784:
2718:
2613:
2577:
2540:
2522:
2475:
2336:
2150:has a
2148:Python
2089:ranges
2072:B-heap
1811:insert
1787:insert
1431:Brodal
749:Binary
577:B-heap
446:−1)/2⌋
366:For a
312:delete
201:insert
93:node.
43:binary
3932:Trees
3805:Queue
3800:Stack
3748:Types
3689:Range
3659:K-ary
3619:Cover
3462:Radix
3447:Ctrie
3439:Tries
3368:Heaps
3348:Treap
3338:Splay
3303:HTree
3258:(a,b)
3248:2–3–4
3132:(PDF)
3115:(PDF)
3059:(PDF)
3010:(PDF)
2955:(PDF)
2914:(PDF)
2864:(PDF)
2826:(PDF)
2772:arXiv
2768:(PDF)
2745:(PDF)
2632:(PDF)
2599:(PDF)
2281:Treap
2216:Pharo
2153:heapq
2066:with
740:meld
675:Treap
341:array
322:sieve
264:union
260:merge
179:Basic
83:value
81:(the
57:is a
4021:Trie
3977:Heap
3795:List
3694:SPQR
3573:Quad
3501:Ball
3457:Hash
3429:Weak
3419:Skew
3394:-ary
3173:ISBN
3155:Heap
3089:ISBN
3032:ISBN
2886:ISBN
2800:ISBN
2716:ISBN
2575:ISBN
2538:ISSN
2473:ISBN
2334:ISBN
2302:heap
2241:has
2239:.NET
2223:Rust
2221:The
2196:heap
2189:The
2184:CPAN
2179:Heap
2174:Perl
2132:Java
2130:The
2117:For
2070:and
2062:and
2058:for
2043:The
2028:and
2016:The
1990:and
1803:meld
1799:meld
1797:and
1783:meld
1773:For
1741:meld
1466:(1)
1460:(1)
1454:(1)
1438:(1)
1415:(1)
1409:(1)
1403:(1)
1387:(1)
1364:(1)
1358:(1)
1349:(1)
1330:(1)
1307:(1)
1301:(1)
1292:(1)
1273:(1)
1250:(1)
1244:(1)
1212:(1)
1183:(1)
1174:(1)
1142:(1)
1106:(1)
1100:(1)
1081:(1)
1048:(1)
1022:(1)
986:(1)
960:(1)
897:(1)
819:(1)
812:Skew
756:(1)
582:Beap
546:and
520:) −
481:sink
471:swim
407:and
283:size
270:meld
227:(or
213:(or
205:push
194:peek
187:(or
91:root
75:node
59:tree
55:heap
53:, a
3829:Set
3699:Top
3553:MVP
3511:BSP
3263:AVL
3243:2–3
3119:doi
3024:doi
2979:doi
2926:doi
2878:doi
2840:doi
2792:doi
2689:doi
2621:doi
2567:doi
2530:doi
2430:doi
2426:120
2368:doi
2304:in
2160:PHP
2064:C++
1908:log
1902:log
1854:log
1848:log
1690:log
1651:log
1581:log
1558:log
1529:log
1351:am.
1342:am.
1294:am.
1285:am.
1237:am.
1224:am.
1198:am.
1185:am.
1176:am.
1167:am.
1154:am.
1108:am.
1093:am.
988:am.
883:am.
870:am.
857:am.
844:am.
831:am.
697:am.
509:− 2
491:new
304:or
219:pop
79:key
49:In
4086::
3684:PQ
3598:VP
3588:R*
3583:R+
3563:PH
3537:-d
3529:-d
3506:BK
3353:UB
3278:B*
3273:B+
3253:AA
3079:;
3061:,
3030:.
3022:.
2977:.
2965:34
2963:.
2957:.
2946:;
2922:40
2920:.
2916:.
2884:.
2876:.
2836:46
2834:.
2828:.
2798:,
2790:,
2780:,
2683:,
2671:^
2655:.
2619:.
2607:12
2605:.
2601:.
2573:.
2550:^
2536:.
2528:.
2516:15
2514:.
2508:.
2500:;
2487:^
2461:;
2457:;
2442:^
2424:,
2362:,
2191:Go
2095:.
2024:,
1940:^
1764:^
1476:)
1448:)
1425:)
1397:)
1374:)
1340:)
1317:)
1283:)
1260:)
1235:)
1222:)
1196:)
1165:)
1152:)
1129:)
1119:)
1091:)
1068:)
1058:)
1042:)
1032:)
1009:)
999:)
980:)
970:)
947:)
937:)
927:)
917:)
907:)
881:)
868:)
855:)
842:)
829:)
806:)
796:)
786:)
776:)
766:)
442:⌊(
3733:e
3726:t
3719:v
3603:X
3578:R
3543:M
3539:)
3535:k
3531:(
3527:k
3392:d
3343:T
3322:(
3287:(
3283:B
3268:B
3236:)
3232:/
3228:(
3209:e
3202:t
3195:v
3181:.
3121::
3097:.
3040:.
3026::
2985:.
2981::
2932:.
2928::
2894:.
2880::
2846:.
2842::
2794::
2774::
2724:.
2691::
2685:6
2665:.
2641:.
2623::
2583:.
2569::
2544:.
2532::
2481:.
2437:.
2432::
2370::
2364:7
2342:.
2318:.
2186:.
2114:.
2060:C
2049:d
1994:.
1924:.
1921:)
1914:n
1897:2
1893:2
1889:(
1886:O
1866:,
1863:)
1860:n
1845:(
1823:n
1819:O
1807:Θ
1757:n
1755:(
1753:Θ
1749:n
1745:O
1737:n
1735:(
1733:Θ
1729:n
1715:.
1699:)
1696:n
1687:n
1684:(
1681:O
1678:=
1675:)
1672:n
1669:(
1666:O
1660:)
1657:n
1648:(
1645:O
1642:n
1622:n
1619:=
1616:k
1596:1
1590:)
1587:n
1578:(
1575:=
1572:2
1568:/
1564:n
1538:)
1535:k
1526:(
1523:O
1518:n
1513:1
1510:=
1507:k
1492:k
1474:n
1472:(
1470:Θ
1464:Θ
1458:Θ
1452:Θ
1446:n
1442:Θ
1436:Θ
1423:n
1421:(
1419:Θ
1413:Θ
1407:Θ
1401:Θ
1395:n
1391:Θ
1385:Θ
1372:n
1370:(
1368:Θ
1362:Θ
1356:Θ
1347:Θ
1338:n
1334:O
1328:Θ
1315:n
1313:(
1311:Θ
1305:Θ
1299:Θ
1290:Θ
1281:n
1277:O
1271:Θ
1258:n
1256:(
1254:Θ
1248:Θ
1242:Θ
1233:n
1229:o
1220:n
1216:O
1210:Θ
1194:n
1192:(
1190:Θ
1181:Θ
1172:Θ
1163:n
1159:O
1150:n
1146:O
1140:Θ
1127:n
1125:(
1123:Θ
1117:n
1113:O
1104:Θ
1098:Θ
1089:n
1085:O
1079:Θ
1066:n
1064:(
1062:Θ
1056:n
1052:Θ
1046:Θ
1040:n
1036:Θ
1030:n
1026:Θ
1020:Θ
1007:n
1005:(
1003:Θ
997:n
993:Θ
984:Θ
978:n
974:Θ
968:n
964:Θ
958:Θ
945:n
943:(
941:Θ
935:n
931:Θ
925:n
921:Θ
915:n
911:Θ
905:n
901:Θ
895:Θ
879:n
877:(
875:Θ
866:n
862:O
853:n
849:O
840:n
836:O
827:n
823:O
817:Θ
804:n
802:(
800:Θ
794:n
792:(
790:Θ
784:n
780:Θ
774:n
770:Θ
764:n
760:Θ
754:Θ
713:f
711:(
709:Θ
705:f
703:(
701:O
603:d
559:N
555:N
553:(
551:2
548:e
544:N
540:N
538:(
536:2
533:s
529:N
527:(
525:2
522:e
518:N
516:(
514:2
511:s
507:N
499:d
444:i
426:2
423:+
420:i
417:2
393:1
390:+
387:i
384:2
372:i
324:.
262:(
221:)
207:)
197:)
146:N
142:a
137:a
133:N
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
