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treap, if the three numbers 1, 2, and 3 are inserted in the order 1, 3, 2, and then the number 2 is deleted, the remaining two nodes will have the same parent-child relationship that they did prior to the insertion of the middle number. In a randomized binary search tree, the tree after the deletion is equally likely to be either of the two possible trees on its two nodes, independently of what the tree looked like prior to the insertion of the middle number.
1313:
single tree. That is because the subtrees to be joined are on average at depth Î(log n); joining two trees of size n and m needs Î(log(n+m)) random choices on average. If the left or right subtree of the node to be deleted is empty, the join operation is trivial; otherwise, the left or right child of the deleted node is selected as the new subtree root with probability proportional to its number of descendants, and the join proceeds recursively.
361:
1326:) calls per insertion or deletion rather than one call per insertion) and the insertion procedure is slightly more complicated due to the need to update the numbers of descendants per node. A minor technical difference is that, in a treap, there is a small probability of a collision (two keys getting the same priority), and in both cases, there will be statistical differences between a true random number generator and the
743:
888:
29:
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Rather than storing random priorities on each node, the randomized binary search tree stores a small integer at each node, the number of its descendants (counting itself as one); these numbers may be maintained during tree rotation operations at only a constant additional amount of time per rotation.
1227:
Note that if hash values of keys are used as priorities and structurally equal nodes are merged already at construction, then each merged node will be a unique representation of a set of keys. Provided that there can only be one simultaneous root node representing a given set of keys, two sets can be
737:
is larger than this max-value in the first treap and smaller than the min-value in the second treap, assign it the minimum priority, then set its left child to the first heap and its right child to the second heap. Rotate as necessary to fix the heap order. After that, it will be a leaf node, and can
458:
Aragon and Seidel also suggest assigning higher priorities to frequently accessed nodes, for instance by a process that, on each access, chooses a random number and replaces the priority of the node with that number if it is higher than the previous priority. This modification would cause the tree to
400:
An equivalent way of describing the treap is that it could be formed by inserting the nodes highest priority-first into a binary search tree without doing any rebalancing. Therefore, if the priorities are independent random numbers (from a distribution over a large enough space of possible priorities
1334:
Although the treap and the randomized binary search tree both have the same random distribution of tree shapes after each update, the history of modifications to the trees performed by these two data structures over a sequence of insertion and deletion operations may be different. For instance, in a
1312:
The deletion procedure for a randomized binary search tree uses the same information per node as the insertion procedure, but unlike the insertion procedure, it only needs on average O(1) random decisions to join the two subtrees descending from the left and right children of the deleted node into a
1231:
This technique can be used to enhance the merge algorithms to perform fast also when the difference between two sets is small. If input sets are equal, the union and intersection functions could break immediately returning one of the input sets as result, while the difference function should return
2236:
First we will create an additional field F to store the value of the target function for the range represented by that node. we will create a function that calculates the value F based on the values of the L and R children of the node. We will call this target function at the end of all functions
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The randomized binary search tree, introduced by MartĂnez and Roura subsequently to the work of Aragon and Seidel on treaps, stores the same nodes with the same random distribution of tree shape, but maintains different information within the nodes of the tree in order to maintain its randomized
396:
order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement that it be heap-ordered: that is, the priority number for any non-leaf node must be greater than or equal to the priority of its children. Thus, as with
Cartesian trees more
1308:
at the root of a subtree may be performed either as in the treap by inserting it at a leaf and then rotating it upwards, or by an alternative algorithm described by MartĂnez and Roura that splits the subtree into two pieces to be used as the left and right children of the new node.
738:
easily be deleted. The result is one treap merged from the two original treaps. This is effectively "undoing" a split, and costs the same. More generally, the join operation can work on two treaps and a key with arbitrary priority (i.e., not necessary to be the highest).
1438:
Therefore we can quickly calculate the implicit key of the current node as we perform an operation by accumulating the sum of all nodes as we descend the tree. Note that this sum does not change when we visit the left subtree but it will increase by
405:, a search tree formed by inserting the nodes without rebalancing in a randomly chosen insertion order. Because random binary search trees are known to have logarithmic height with high probability, the same is true for treaps. This mirrors the
1303:
within the left or right subtree (depending on whether its key is less than or greater than the root). The numbers of descendants are used by the algorithm to calculate the necessary probabilities for the random choices at each step. Placing
2467:
To show that the subtree of a given node needs to be reversed for each node we will create an extra boolean field R and set its value to true. To propagate this change we will swap the children of the node and set R to true for all of them.
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typically used on digital computers. However, in any case, the differences between the theoretical model of perfect random choices used to design the algorithm and the capabilities of actual random number generators are vanishingly small.
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The information stored per node in the randomized binary tree is simpler than in a treap (a small integer rather than a high-precision random number), but it makes a greater number of calls to the random number generator
1434:
of a node T is the number of nodes less than that node plus one. Note that such nodes can be present not only in its left subtree but also in left subtrees of its ancestors P, if T is in the right subtree of P.
728:
Joining two treaps that are the product of a former split, one can safely assume that the greatest value in the first treap is less than the smallest value in the second treap. Create a new node with value
397:
generally, the root node is the maximum-priority node, and its left and right subtrees are formed in the same manner from the subsequences of the sorted order to the left and right of that node.
344:
among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular,
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The idea behind an implicit treap is to use the array index as a key, but to not store it explicitly. Otherwise, an update (insertion/deletion) would result in changes of the keys in
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We find the element to be deleted and perform a join on its children L and R. We then replace the element to be deleted with the tree that resulted from the join operation.
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which will be used to propagate this change from a node to its children. We will call this function at the beginning of all functions which modify the tree,
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lose its random shape; instead, frequently accessed nodes would be more likely to be near the root of the tree, causing searches for them to be faster.
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to ensure that two nodes are very unlikely to have the same priority) then the shape of a treap has the same probability distribution as the shape of a
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An implicit treap is a simple variation of an ordinary treap which can be viewed as a dynamic array that supports the following operations in
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Split and Union call Join but do not deal with the balancing criteria of treaps directly, such an implementation is usually called the
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in the sorted order, resulting in one of the previous cases. In this final case, the swap may violate the heap-ordering property for
677:
In addition to the single-element insert, delete and lookup operations, several fast "bulk" operations have been defined on treaps:
311:
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is presumed to return two trees: one holding the keys less than its input key, one holding the greater keys. (The algorithm is
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and not compatible with insertion and deletion in the middle. If implicit tree is not possible, explicit tree is used, called
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version of sorting, then Treaps correspond specifically to dynamic quicksort where priorities guide pivot choices.
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of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform.
2866:. Lecture notes from a course by Jeff Erickson at UIUC. Despite the title, this is primarily about treaps and
2448:
We will create an extra field D which will contain the added value for the subtree. We will create a function
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into the treap with maximum priorityâlarger than the priority of any node in the treap. After this insertion,
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in the tree, and create a new node at the leaf position where the binary search determines a node for
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in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the
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be the size of the symmetric difference. The modified merge algorithms will then also be bounded by
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1299:
as the new root of the tree, and otherwise, it calls the insertion procedure recursively to insert
1213:. Moreover, since the recursive calls to union are independent of each other, they can be executed
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2670:
Proceedings of the tenth annual ACM symposium on
Parallel algorithms and architectures - SPAA '98
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2458:, split and join so that after any changes made to the tree the information will not be lost.
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will be found in the right subtreap. This costs as much as a single insertion into the treap.
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has two children, swap its position in the tree with the position of its immediate successor
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Blelloch, Guy E.; Reid-Miller, Margaret (1998), "Fast set operations using treaps",
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helper routine. The complexity of each of union, intersection and difference is
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To build a treap we can simply insert n values in the treap where each takes
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is not the root of the tree and has a larger priority number than its parent
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585:, so additional rotations may need to be performed to restore this property.
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Open Data
Structures - Section 7.2 - Treap: A Randomized Binary Search Tree
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is about computational geometry, and requests like sum are described in
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Second we need to process a query for a given range : We will call the
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tested for equality by pointer comparison, which is constant in time.
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2549:, Washington, D.C.: IEEE Computer Society Press, pp. 540â545,
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after the join is defined as a join of its former right child and
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To split a treap into two smaller treaps, those smaller than key
2710:
Liljenzin, Olle (2013). "Confluently
Persistent Sets and Maps".
2227:
931:, recursive split call is done to either left or right child of
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will be found in the left subtreap, and all values greater than
360:
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2870:; randomized binary search trees are mentioned only briefly.
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The algorithm for intersection is similar, but requires the
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nodes, the insertion algorithm chooses with probability 1/(
2882:. Visual basic 6 implementation of treaps as a COM object.
1388:
Finding sum, minimum or maximum element in a given range.
717:
will be the root node of the treap, all values less than
2546:
30th Annual
Symposium on Foundations of Computer Science
2232:
To perform this calculation we will proceed as follows:
1733:
The split algorithm for an implicit treap is as follows:
1038:. The following recursive algorithm computes the union:
28:
1489:
The join algorithm for an implicit treap is as follows:
1157:, but an in-place destructive version exists as well.)
462:
Naor and Nissim describe an application in maintaining
2753:
2444:
To perform this operation we will proceed as follows:
989:(k > m) (L', b, R') = split(R, k)
981:(k < m) (L', b, R') = split(L, k)
364:
A treap with alphabetic key and numeric max heap order
340:
that maintain a dynamic set of ordered keys and allow
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function. Finally we call the join function to merge
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969:(nil, false, nil) (L, (m, c), R) = expose(T)
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are two closely related forms of binary search tree
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Pure Python and Cython in-memory treap and duptreap
2666:
451:time. If binary search trees are solutions to the
2907:Pure Go persistent treap key-value storage library
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487:To search for a given key value, apply a standard
443:
2874:A high-performance key-value store based on treap
491:in a binary search tree, ignoring the priorities.
3437:
2634:IEEE Journal on Selected Areas in Communications
1273:
526:that reverses the parent-child relation between
2627:"Certificate revocation and certificate update"
2429:. After the query is answered we will call the
1287:is to be inserted into a tree that already has
483:Treaps support the following basic operations:
880:Node(left(L), k(L), join(right(L), k, R))
2927:
2770:
2607:: CS1 maint: DOI inactive as of April 2024 (
2480:for sum, minimum or maximum. The Knowledge's
2435:function twice to restore the original treap.
2228:Find sum, minimum or maximum in a given range
305:
2817:"Treap - Competitive Programming Algorithms"
2577:Seidel, Raimund; Aragon, Cecilia R. (1996),
2576:
2536:Aragon, Cecilia R.; Seidel, Raimund (1989),
2535:
2416:
2401:
2354:
2339:
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545:is a leaf of the tree, simply remove it. If
2771:MartĂnez, Conrado; Roura, Salvador (1997),
2462:
498:into the treap, generate a random priority
2934:
2920:
2673:, New York, NY, USA: ACM, pp. 16â26,
2621:
884:Node(join(L, k, left(R)), k(R), right(R))
312:
298:
27:
2788:
2715:
2709:
2083:we divide the array into two subsections
2941:
2251:function twice and split the treap into
886:
741:
630:time. Therefore a treap can be built in
359:
2886:ActionScript3 implementation of a treap
2844:Collection of treap references and info
689:. These rely on two helper operations,
3438:
2570:
2529:
2915:
1479:{\displaystyle cnt(T\rightarrow L)+1}
1385:Removing an element from any position
2811:
2809:
2807:
1382:Inserting an element in any position
1138:), union(right(t
2902:Pure Go in-memory, immutable treaps
2506:designed to sum "1"'s in every node
1394:Reversing elements in a given range
1391:Addition, painting in a given range
955:The split algorithm is as follows:
589:
478:
13:
2440:Addition/painting in a given range
868:prior(k, k(L)) and prior(k, k(R))
858:The join algorithm is as follows:
672:
348:its height is proportional to the
14:
3472:
2837:
2804:
2760:Confluent Sets and Maps on GitHub
2496:, but Knowledge lacks proper page
2219:
2164:with the new node by calling the
2079:To insert an element at position
2074:
1486:when we visit the right subtree.
1338:
368:The treap was first described by
16:Random search tree data structure
2773:"Randomized binary search trees"
2069:
514:should exist. Then, as long as
2864:Randomized binary search trees
2764:
2703:
2660:
2615:
2095:function and we get two trees
1467:
1461:
1455:
1414:
1408:
1365:
1353:
1328:pseudo-random number generator
655:
640:
617:
605:
561:be the child of the parent of
438:
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355:
1:
3456:Probabilistic data structures
2522:
2510:
1316:
1274:Randomized binary search tree
473:
334:randomized binary search tree
265:Rapidly exploring random tree
2880:VB6 implementation of treaps
985:(L', b, join(R', m, R))
705:, and those larger than key
573:had no parent). Finally, if
7:
2625:; Nissim, K. (April 2000),
2471:
1222:"join-based" implementation
407:binary search tree argument
10:
3477:
2422:{\displaystyle \{B+1..n\}}
2302:{\displaystyle \{1..A-1\}}
661:{\displaystyle O(n\log n)}
464:authorization certificates
444:{\displaystyle O(n\log n)}
3337:
3216:
3163:
3092:
2949:
2579:"Randomized Search Trees"
2538:"Randomized Search Trees"
2502:for indexing. It is like
1371:{\displaystyle O(\log n)}
1295: + 1) to place
993:(join(L, m, L'), b, R'))
746:Join performed on treaps
623:{\displaystyle O(\log n)}
403:random binary search tree
160:
153:
128:
103:
78:
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26:
21:
3390:Left-child right-sibling
2463:Reverse in a given range
2360:{\displaystyle \{A..B\}}
1735:
1491:
996:The union of two treaps
668:time from a list values.
569:the root of the tree if
468:public-key cryptosystems
3446:Heaps (data structures)
3220:data partitioning trees
3178:C-trie (compressed ADT)
2555:10.1109/SFCS.1989.63531
557:from the tree and make
489:binary search algorithm
376:in 1989; its name is a
2741:Cite journal requires
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2237:that modify the tree,
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2790:10.1145/274787.274812
2679:10.1145/277651.277660
2597:(inactive 2024-04-18)
2595:10.1007/s004539900061
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847:{\displaystyle T_{2}}
822:
820:{\displaystyle T_{1}}
795:
793:{\displaystyle T_{2}}
768:
766:{\displaystyle T_{1}}
745:
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346:with high probability
3400:Log-structured merge
2943:Tree data structures
2500:Order statistic tree
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1427:nodes of the tree.
1420:{\displaystyle O(n)}
1402:
1347:
1195:for treaps of sizes
1014:, representing sets
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895:
831:
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506:. Binary search for
494:To insert a new key
417:
281:Randomized algorithm
2726:2013arXiv1301.3388L
549:has a single child
541:from the treap, if
3365:Fractal tree index
2960:associative arrays
2777:Journal of the ACM
2419:
2385:{\displaystyle T3}
2382:
2357:
2325:{\displaystyle T2}
2322:
2299:
2267:{\displaystyle T1}
2264:
2209:{\displaystyle T2}
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2186:{\displaystyle T1}
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2157:{\displaystyle T1}
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2134:{\displaystyle T2}
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2111:{\displaystyle T1}
2108:
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876:prior(k(L), k(R))
872:Node(L, k, R)
864:join(L, k, R)
856:
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255:Random binary tree
43:binary search tree
3433:
3432:
2898:. By Roy Clemmons
2846:by Cecilia Aragon
2821:cp-algorithms.com
2646:10.1109/49.839932
2241:, split and join.
1126:join(union(left(t
1090:) < priority(t
977:(L, true, R)
944:{\displaystyle T}
924:{\displaystyle x}
904:{\displaystyle T}
800:. Right child of
537:To delete a node
413:runs in expected
394:inorder traversal
374:Cecilia R. Aragon
322:
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2141:. Then we merge
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1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1485:
1483:
1482:
1477:
1426:
1424:
1423:
1418:
1377:
1375:
1374:
1369:
1269:
1267:
1265:
1264:
1259:
1256:
1238:
1212:
1202:
1198:
1194:
1192:
1190:
1189:
1184:
1181:
1037:
1028:that represents
1027:
1021:
1017:
1013:
1004:
961:split(T, k)
950:
948:
947:
942:
930:
928:
927:
922:
910:
908:
907:
902:
853:
851:
850:
845:
843:
842:
826:
824:
823:
818:
816:
815:
799:
797:
796:
791:
789:
788:
772:
770:
769:
764:
762:
761:
667:
665:
664:
659:
629:
627:
626:
621:
590:Building a treap
479:Basic operations
450:
448:
447:
442:
326:computer science
314:
307:
300:
227:Countâmin sketch
191:
190:
155:Space complexity
50:
49:
31:
19:
18:
3476:
3475:
3471:
3470:
3469:
3467:
3466:
3465:
3436:
3435:
3434:
3429:
3333:
3212:
3159:
3088:
3084:Weight-balanced
3039:Order statistic
2953:
2945:
2940:
2840:
2835:
2834:
2825:
2823:
2815:
2814:
2805:
2769:
2765:
2758:
2754:
2742:
2740:
2731:
2730:
2708:
2704:
2689:
2665:
2661:
2629:
2620:
2616:
2600:
2599:
2575:
2571:
2565:
2540:
2534:
2530:
2525:
2513:
2474:
2465:
2442:
2399:
2396:
2395:
2394:which contains
2374:
2371:
2370:
2337:
2334:
2333:
2332:which contains
2314:
2311:
2310:
2279:
2276:
2275:
2274:which contains
2256:
2253:
2252:
2230:
2222:
2198:
2195:
2194:
2175:
2172:
2171:
2146:
2143:
2142:
2123:
2120:
2119:
2100:
2097:
2096:
2089:by calling the
2077:
2072:
2065:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1731:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1444:
1441:
1440:
1430:The key value (
1403:
1400:
1399:
1348:
1345:
1344:
1341:
1319:
1276:
1260:
1257:
1252:
1251:
1249:
1240:
1236:
1232:the empty set.
1204:
1200:
1196:
1185:
1182:
1177:
1176:
1174:
1165:
1155:non-destructive
1147:
1145:
1141:
1137:
1133:
1129:
1121:
1117:
1113:
1109:
1105:
1101:
1093:
1089:
1082:
1075:= nil:
1074:
1067:
1060:= nil:
1059:
1051:
1047:
1029:
1023:
1019:
1015:
1012:
1006:
1003:
997:
994:
936:
933:
932:
916:
913:
912:
896:
893:
892:
885:
838:
834:
832:
829:
828:
811:
807:
805:
802:
801:
784:
780:
778:
775:
774:
757:
753:
751:
748:
747:
675:
673:Bulk operations
635:
632:
631:
600:
597:
596:
592:
481:
476:
453:dynamic problem
418:
415:
414:
358:
342:binary searches
338:data structures
318:
232:Quotient filter
208:data structures
206:
54:Time complexity
17:
12:
11:
5:
3474:
3464:
3463:
3458:
3453:
3448:
3431:
3430:
3428:
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3387:
3382:
3377:
3372:
3367:
3362:
3357:
3352:
3347:
3341:
3339:
3335:
3334:
3332:
3331:
3326:
3321:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3249:
3244:
3239:
3234:
3229:
3223:
3221:
3214:
3213:
3211:
3210:
3205:
3200:
3198:Ternary search
3195:
3190:
3185:
3180:
3175:
3169:
3167:
3161:
3160:
3158:
3157:
3152:
3147:
3142:
3137:
3132:
3127:
3122:
3114:
3109:
3104:
3098:
3096:
3090:
3089:
3087:
3086:
3081:
3076:
3071:
3066:
3061:
3056:
3046:
3041:
3036:
3031:
3026:
3021:
3011:
3006:
3001:
2996:
2991:
2986:
2981:
2976:
2971:
2965:
2963:
2947:
2946:
2939:
2938:
2931:
2924:
2916:
2910:
2909:
2904:
2899:
2893:
2888:
2883:
2877:
2871:
2861:
2859:Animated treap
2856:
2847:
2839:
2838:External links
2836:
2833:
2832:
2803:
2783:(2): 288â323,
2763:
2752:
2743:|journal=
2702:
2687:
2659:
2640:(4): 561â570,
2614:
2569:
2563:
2527:
2526:
2524:
2521:
2520:
2519:
2512:
2509:
2508:
2507:
2497:
2473:
2470:
2464:
2461:
2460:
2459:
2441:
2438:
2437:
2436:
2418:
2415:
2412:
2409:
2406:
2403:
2381:
2378:
2356:
2353:
2350:
2347:
2344:
2341:
2321:
2318:
2298:
2295:
2292:
2289:
2286:
2283:
2263:
2260:
2242:
2229:
2226:
2221:
2220:Delete element
2218:
2205:
2202:
2182:
2179:
2153:
2150:
2130:
2127:
2107:
2104:
2076:
2075:Insert element
2073:
2071:
2068:
1882://implicit key
1736:
1492:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1416:
1413:
1410:
1407:
1396:
1395:
1392:
1389:
1386:
1383:
1367:
1364:
1361:
1358:
1355:
1352:
1340:
1339:Implicit treap
1337:
1318:
1315:
1275:
1272:
1143:
1139:
1135:
1131:
1127:
1119:
1115:
1111:
1107:
1103:
1099:
1091:
1087:
1080:
1072:
1065:
1057:
1049:
1045:
1040:
1010:
1001:
957:
940:
920:
900:
860:
841:
837:
814:
810:
787:
783:
760:
756:
740:
739:
726:
687:set difference
674:
671:
670:
669:
657:
654:
651:
648:
645:
642:
639:
619:
616:
613:
610:
607:
604:
591:
588:
587:
586:
535:
492:
480:
477:
475:
472:
440:
437:
434:
431:
428:
425:
422:
390:Cartesian tree
370:Raimund Seidel
357:
354:
320:
319:
317:
316:
309:
302:
294:
291:
290:
289:
288:
283:
275:
274:
270:
269:
268:
267:
262:
257:
249:
248:
242:
241:
240:
239:
234:
229:
224:
219:
211:
210:
200:
199:
187:
186:
183:
182:
172:
162:
158:
157:
151:
150:
140:
130:
126:
125:
115:
105:
101:
100:
90:
80:
76:
75:
70:
65:
61:
60:
58:big O notation
46:
45:
39:
33:
32:
24:
23:
15:
9:
6:
4:
3:
2:
3473:
3462:
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3443:
3441:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3376:
3375:Hash calendar
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3342:
3340:
3336:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3264:
3262:
3256:
3254:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3224:
3222:
3219:
3215:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3170:
3168:
3166:
3162:
3156:
3153:
3151:
3150:van Emde Boas
3148:
3146:
3143:
3141:
3140:Skew binomial
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3119:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3099:
3097:
3095:
3091:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3051:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3019:Binary search
3016:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2966:
2964:
2961:
2957:
2952:
2948:
2944:
2937:
2932:
2930:
2925:
2923:
2918:
2917:
2914:
2908:
2905:
2903:
2900:
2897:
2894:
2892:
2889:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2865:
2862:
2860:
2857:
2855:
2851:
2848:
2845:
2842:
2841:
2822:
2818:
2812:
2810:
2808:
2800:
2796:
2791:
2786:
2782:
2778:
2774:
2767:
2761:
2756:
2748:
2735:
2727:
2723:
2718:
2713:
2706:
2698:
2694:
2690:
2688:0-89791-989-0
2684:
2680:
2676:
2672:
2671:
2663:
2655:
2651:
2647:
2643:
2639:
2635:
2628:
2624:
2618:
2610:
2604:
2596:
2592:
2588:
2584:
2580:
2573:
2566:
2564:0-8186-1982-1
2560:
2556:
2552:
2548:
2547:
2539:
2532:
2528:
2518:
2517:Finger search
2515:
2514:
2505:
2501:
2498:
2495:
2491:
2490:implicit tree
2487:
2483:
2479:
2476:
2475:
2469:
2457:
2453:
2452:
2447:
2446:
2445:
2434:
2433:
2413:
2410:
2407:
2404:
2393:
2392:
2379:
2376:
2351:
2348:
2345:
2342:
2319:
2316:
2293:
2290:
2287:
2284:
2261:
2258:
2250:
2249:
2243:
2240:
2235:
2234:
2233:
2225:
2217:
2203:
2200:
2180:
2177:
2169:
2168:
2151:
2148:
2128:
2125:
2105:
2102:
2094:
2093:
2088:
2085:
2082:
2067:
1734:
1490:
1487:
1473:
1470:
1464:
1458:
1452:
1449:
1446:
1436:
1433:
1432:implicit key)
1428:
1411:
1405:
1393:
1390:
1387:
1384:
1381:
1380:
1379:
1362:
1359:
1356:
1350:
1336:
1332:
1329:
1325:
1314:
1310:
1307:
1302:
1298:
1294:
1290:
1286:
1280:
1271:
1263:
1255:
1247:
1243:
1233:
1229:
1225:
1223:
1218:
1216:
1211:
1207:
1188:
1180:
1172:
1168:
1163:
1158:
1156:
1152:
1125:
1097:
1085:
1078:
1070:
1063:
1055:
1043:
1039:
1036:
1032:
1026:
1009:
1000:
992:
988:
984:
980:
976:
972:
968:
964:
960:
956:
938:
918:
898:
889:
883:
879:
875:
871:
867:
863:
859:
839:
835:
812:
808:
785:
781:
758:
754:
744:
736:
732:
727:
724:
720:
716:
712:
708:
704:
700:
699:
698:
696:
692:
688:
684:
680:
652:
649:
646:
643:
637:
614:
611:
608:
602:
594:
593:
584:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
540:
536:
533:
529:
525:
524:tree rotation
521:
517:
513:
509:
505:
501:
497:
493:
490:
486:
485:
484:
471:
469:
465:
460:
456:
454:
435:
432:
429:
426:
420:
412:
408:
404:
398:
395:
391:
387:
383:
379:
375:
371:
362:
353:
351:
347:
343:
339:
335:
331:
327:
315:
310:
308:
303:
301:
296:
295:
293:
292:
287:
284:
282:
279:
278:
277:
276:
272:
271:
266:
263:
261:
258:
256:
253:
252:
251:
250:
247:
244:
243:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
214:
213:
212:
209:
205:
204:Probabilistic
202:
201:
197:
193:
192:
180:
176:
173:
170:
166:
163:
159:
156:
152:
148:
144:
141:
138:
134:
131:
127:
123:
119:
116:
113:
109:
106:
102:
98:
94:
91:
88:
84:
81:
77:
74:
71:
69:
66:
62:
59:
55:
51:
47:
44:
40:
38:
34:
30:
25:
20:
3461:Search trees
3451:Binary trees
3260:
3252:
3117:
3073:
3050:Left-leaning
2956:dynamic sets
2951:Search trees
2896:Treaps in C#
2876:by Junyi Sun
2824:. Retrieved
2820:
2780:
2776:
2766:
2755:
2734:cite journal
2705:
2669:
2662:
2637:
2633:
2617:
2586:
2583:Algorithmica
2582:
2572:
2545:
2531:
2504:segment tree
2503:
2494:Segment tree
2493:
2486:Fenwick tree
2482:Segment tree
2478:Segment tree
2477:
2466:
2455:
2450:
2449:
2443:
2431:
2430:
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2247:
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1030:
1024:
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734:
733:, such that
730:
722:
718:
714:
710:
706:
702:
694:
690:
683:intersection
676:
582:
578:
574:
570:
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562:
558:
554:
550:
546:
542:
538:
531:
527:
522:, perform a
519:
515:
511:
507:
503:
499:
495:
482:
461:
457:
399:
367:
333:
329:
323:
259:
246:Random trees
222:Count sketch
217:Bloom filter
178:
174:
168:
164:
146:
142:
136:
132:
121:
117:
111:
107:
96:
92:
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72:
67:
3350:Exponential
3338:Other trees
2488:, which is
1283:When a key
1279:structure.
1215:in parallel
1094:):
1022:is a treap
378:portmanteau
356:Description
286:HyperLogLog
41:Randomized
3440:Categories
3294:Priority R
3044:Palindrome
2868:skip lists
2826:2021-11-21
2523:References
2070:Operations
1317:Comparison
1086:priority(t
965:(T = nil)
474:Operations
388:. It is a
73:Worst case
3380:iDistance
3259:implicit
3247:Hilbert R
3242:Cartesian
3125:Fibonacci
3059:Scapegoat
3054:Redâblack
2854:Pat Morin
2717:1301.3388
2291:−
1462:→
1360:
1114:â split t
891:To split
709:, insert
650:
612:
565:(or make
553:, remove
433:
411:quicksort
350:logarithm
237:Skip list
64:Operation
3395:Link/cut
3107:Binomial
3034:Interval
2654:13833836
2623:Naor, M.
2603:citation
2511:See also
2472:See also
1134:), key(t
1118:on key(t
1042:function
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959:function
862:function
332:and the
196:a series
194:Part of
3355:Fenwick
3319:Segment
3218:Spatial
3135:Pairing
3130:Leftist
3052:)
3024:Dancing
3017:)
3015:Optimal
2722:Bibcode
2697:7342709
2050:upd_cnt
1897:cur_key
1852:cur_key
1716:upd_cnt
1266:
1250:
1203:, with
1191:
1175:
1052:):
1044:union(t
273:Related
68:Average
3405:Merkle
3370:Fusion
3360:Finger
3284:Octree
3274:Metric
3208:Y-fast
3203:X-fast
3193:Suffix
3112:Brodal
3102:Binary
2799:714621
2797:
2695:
2685:
2652:
2561:
2367:, and
1822:return
1149:Here,
1124:return
1122:)
1077:return
1062:return
991:return
983:return
975:return
967:return
882:return
878:return
870:return
328:, the
129:Delete
104:Insert
79:Search
3415:Range
3385:K-ary
3345:Cover
3188:Radix
3173:Ctrie
3165:Tries
3094:Heaps
3074:Treap
3064:Splay
3029:HTree
2984:(a,b)
2974:2â3â4
2795:S2CID
2712:arXiv
2693:S2CID
2650:S2CID
2630:(PDF)
2541:(PDF)
2092:split
2029:->
1987:->
1975:->
1966:split
1930:->
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1903:split
1894:<=
1873:->
1771:&
1768:pitem
1759:&
1756:pitem
1747:pitem
1741:split
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1611:prior
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1599:prior
1596:->
1524:pitem
1515:pitem
1506:&
1503:pitem
1151:split
1102:and t
691:split
679:union
409:that
330:treap
260:Treap
161:Space
135:(log
110:(log
85:(log
22:Treap
3420:SPQR
3299:Quad
3227:Ball
3183:Hash
3155:Weak
3145:Skew
3120:-ary
2747:help
2683:ISBN
2609:link
2559:ISBN
2456:i.e.
2451:push
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2248:plit
2239:i.e.
2193:and
2167:join
2118:and
2086:and
1963:else
1825:void
1738:void
1668:join
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1584:else
1497:join
1494:void
1248:log
1235:Let
1199:and
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1096:swap
1018:and
1005:and
773:and
695:join
693:and
685:and
530:and
502:for
386:heap
384:and
382:tree
372:and
37:Type
3425:Top
3279:MVP
3237:BSP
2989:AVL
2969:2â3
2785:doi
2675:doi
2642:doi
2591:doi
2551:doi
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2285:1..
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2035:)),
2020:cnt
2008:add
2002:key
1945:add
1939:key
1891:key
1864:cnt
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1357:log
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1146:))
1110:, t
1048:, t
911:by
647:log
609:log
466:in
430:log
380:of
324:In
56:in
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3309:R+
3289:PH
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3004:B*
2999:B+
2979:AA
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2819:.
2806:^
2793:,
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2779:,
2775:,
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87:n
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