3827:) used a 0 to 2 level tree that has similarities to a B-tree. A disk block was 512 36-bit words. If the file fit in a 512 (2) word block, then the file directory would point to that physical disk block. If the file fit in 2 words, then the directory would point to an aux index; the 512 words of that index would either be NULL (the block isn't allocated) or point to the physical address of the block. If the file fit in 2 words, then the directory would point to a block holding an aux-aux index; each entry would either be NULL or point to an aux index. Consequently, the physical disk block for a 2 word file could be located in two disk reads and read on the third.
2621:
splitting operation as long as they can. To maintain this, instead of immediately splitting up a node when it gets full, its keys are shared with a node next to it. This spill operation is less costly to do than split, because it requires only shifting the keys between existing nodes, not allocating memory for a new one. For inserting, first it is checked whether the node has some free space in it, and if so, the new key is just inserted in the node. However, if the node is full (it has
2816:. A disk block might be 16 kilobytes. If each record is 160 bytes, then 100 records could be stored in each block. The disk read time above was actually for an entire block. Once the disk head is in position, one or more disk blocks can be read with little delay. With 100 records per block, the last 6 or so comparisons don't need to do any disk reads—the comparisons are all within the last disk block read.
3896:
3614:
2617:, the internal nodes do not store any pointers to records, thus all pointers to records are stored in the leaf nodes. In addition, a leaf node may include a pointer to the next leaf node to speed up sequential access. Because B+ tree internal nodes have fewer pointers, each node can hold more keys, causing the tree to be shallower, and thus, faster to search.
3265:
2713:
2851:
index which is the root of the tree identifies the relevant block in aux-index in the level below. Reading and searching that aux-index block identifies the relevant block to read, until the final level, known as the leaf level, identifies a record in the main database. Instead of 150 milliseconds, we need only 30 milliseconds to get the record.
2840:). This auxiliary index would be 1% of the size of the original database, but it can be searched quickly. Finding an entry in the auxiliary index would tell us which block to search in the main database; after searching the auxiliary index, we would have to search only that one block of the main database—at a cost of one more disk read.
539:
3812:, the operating system (or disk utility) must sequentially follow the file's linked list in the FAT. Worse, to find a free disk block, it must sequentially scan the FAT. For MS-DOS, that was not a huge penalty because the disks and files were small and the FAT had few entries and relatively short file chains. In the
2832:
another with leaf pages at the lowest level. One page is typically the starting point of the tree, or the "root". This is where the search for a particular key would begin, traversing a path that terminates in a leaf. Most pages in this structure will be leaf pages which refer to specific table rows.
2575:
B-trees have substantial advantages over alternative implementations when the time to access the data of a node greatly exceeds the time spent processing that data, because then the cost of accessing the node may be amortized over multiple operations within the node. This usually occurs when the node
451:
implies that two half-full nodes can be joined to make a legal node, and one full node can be split into two legal nodes (if there's room to push one element up into the parent). These properties make it possible to delete and insert new values into a B-tree and adjust the tree to preserve the B-tree
3528:
with a sibling. The merge causes the parent to lose a separator element, so the parent may become deficient and need rebalancing. The merging and rebalancing may continue all the way to the root. Since the minimum element count doesn't apply to the root, making the root be the only deficient node is
3519:
Rebalancing starts from a leaf and proceeds toward the root until the tree is balanced. If deleting an element from a node has brought it under the minimum size, then some elements must be redistributed to bring all nodes up to the minimum. Usually, the redistribution involves moving an element from
3323:
Searching is similar to searching a binary search tree. Starting at the root, the tree is recursively traversed from top to bottom. At each level, the search reduces its field of view to the child pointer (subtree) whose range includes the search value. A subtree's range is defined by the values, or
3686:
When the input is sorted, all insertions are at the rightmost edge of the tree, and in particular any time a node is split, we are guaranteed that no more insertions will take place in the left half. When bulk loading, we take advantage of this, and instead of splitting overfull nodes evenly, split
2928:
Insertions can be very slow in a sorted sequential file because room for the inserted record must be made. Inserting a record before the first record requires shifting all of the records down one. Such an operation is just too expensive to be practical. One solution is to leave some spaces. Instead
525:
is also inconsistent. Bayer and McCreight (1972) considered the leaf level to be the lowest level of keys, but Knuth considered the leaf level to be one level below the lowest keys. There are many possible implementation choices. In some designs, the leaves may hold the entire data record; in other
3985:
Lehman and Yao showed that all the read locks could be avoided (and thus concurrent access greatly improved) by linking the tree blocks at each level together with a "next" pointer. This results in a tree structure where both insertion and search operations descend from the root to the leaf. Write
2805:
and a rotational delay. The seek time may be 0 to 20 or more milliseconds, and the rotational delay averages about half the rotation period. For a 7200 RPM drive, the rotation period is 8.33 milliseconds. For a drive such as the
Seagate ST3500320NS, the track-to-track seek time is 0.8 milliseconds
2683:
lowest keys stay in the current node, the next (middle) key is inserted in the parent and the rest go to the right sibling. (The newly inserted key might end up in any of the three places.) The situation when right sibling is full, and left isn't is analogous. When both the sibling nodes are full,
3994:
United States Patent 5283894, granted in 1994, appears to show a way to use a 'Meta Access Method' to allow concurrent B+ tree access and modification without locks. The technique accesses the tree 'upwards' for both searches and updates by means of additional in-memory indexes that point at the
3816:
filesystem (used on floppy disks and early hard disks), there were no more than 4,080 entries, and the FAT would usually be resident in memory. As disks got bigger, the FAT architecture began to confront penalties. On a large disk using FAT, it may be necessary to perform disk reads to learn the
2850:
Instead of reading 14 disk blocks to find the desired record, we only need to read 3 blocks. This blocking is the core idea behind the creation of the B-tree, where the disk blocks fill-out a hierarchy of levels to make up the index. Reading and searching the first (and only) block of the aux-aux
2620:
The B tree balances more neighboring internal nodes to keep the internal nodes more densely packed. This variant ensures non-root nodes are at least 2/3 full instead of 1/2. As the most costly part of operation of inserting the node in B-tree is splitting the node, B-trees are created to postpone
2843:
In the above example the index would hold 10,000 entries and would take at most 14 comparisons to return a result. Like the main database, the last six or so comparisons in the auxiliary index would be on the same disk block. The index could be searched in about eight disk reads, and the desired
3500:
Each element in an internal node acts as a separation value for two subtrees, therefore we need to find a replacement for separation. Note that the largest element in the left subtree is still less than the separator. Likewise, the smallest element in the right subtree is still greater than the
2932:
Both insertions and deletions are fast as long as space is available on a block. If an insertion won't fit on the block, then some free space on some nearby block must be found and the auxiliary indices adjusted. The best case is that enough space is available nearby so that the amount of block
2835:
Because each node (or internal page) can have more than two children, a B-tree index will usually have a shorter height (the distance from the root to the farthest leaf) than a Binary Search Tree. In the example above, initial disk reads narrowed the search range by a factor of two. That can be
2831:
can be used to improve performance. A B-tree index creates a multi-level tree structure that breaks a database down into fixed-size blocks or pages. Each level of this tree can be used to link those pages via an address location, allowing one page (known as a node, or internal page) to refer to
3418:
An alternative algorithm supports a single pass down the tree from the root to the node where the insertion will take place, splitting any full nodes encountered on the way pre-emptively. This prevents the need to recall the parent nodes into memory, which may be expensive if the nodes are on
513:
of B-tree as the minimum number of keys in a non-root node. Folk and
Zoellick points out that terminology is ambiguous because the maximum number of keys is not clear. An order 3 B-tree might hold a maximum of 6 keys or a maximum of 7 keys. Knuth (1998) avoids the problem by defining the
2684:
then the two nodes (current node and a sibling) are split into three and one more key is shifted up the tree, to the parent node. If the parent is full, then spill/split operation propagates towards the root node. Deleting nodes is somewhat more complex than inserting however.
3698:
full, to re-establish the normal B-tree rules, combine such nodes with their (guaranteed full) left siblings and divide the keys to produce two nodes at least half full. The only node which lacks a full left sibling is the root, which is permitted to be less than half full.
2588:, the height of the tree decreases and the number of expensive node accesses is reduced. In addition, rebalancing of the tree occurs less often. The maximum number of child nodes depends on the information that must be stored for each child node and the size of a full
2604:
may refer to a specific design or it may refer to a general class of designs. In the narrow sense, a B-tree stores keys in its internal nodes but need not store those keys in the records at the leaves. The general class includes variations such as the
3370:
If the splitting goes all the way up to the root, it creates a new root with a single separator value and two children, which is why the lower bound on the size of internal nodes does not apply to the root. The maximum number of elements per node is
2495:-key siblings and inserting the mid-value key into the parent. Depth only increases when the root is split, maintaining balance. Similarly, a B-tree is kept balanced after deletion by merging or redistributing keys among siblings to maintain the
3788:(FAT). The FAT has an entry for each disk block, and that entry identifies whether its block is used by a file and if so, which block (if any) is the next disk block of the same file. So, the allocation of each file is represented as a
261:, for the purpose of efficiently managing index pages for large random-access files. The basic assumption was that indices would be so voluminous that only small chunks of the tree could fit in main memory. Bayer and McCreight's paper
3678:
data into an initially empty B-tree. While it is quite possible to simply perform a series of successive inserts, inserting sorted data results in a tree composed almost entirely of half-full nodes. Instead, a special "bulk loading"
2515:-key minimum for non-root nodes. A merger reduces the number of keys in the parent potentially forcing it to merge or redistribute keys with its siblings, and so on. The only change in depth occurs when the root has two children, of
3249:
3280:. In particular, the discussion below uses "element", "value", "key", "separator", and "separation value" to mean essentially the same thing. The terms are not clearly defined. There are some subtle issues at the root and leaves.
470:
In Knuth's terminology, the "leaf" nodes are the actual data objects / chunks. The internal nodes that are one level above these leaves are what would be called the "leaves" by other authors: these nodes only store keys (at most
3990:
storage methods. The cost associated with this improvement is that empty pages cannot be removed from the btree during normal operations. (However, see for various strategies to implement node merging, and source code at.)
3363:
The separation value is inserted in the node's parent, which may cause it to be split, and so on. If the node has no parent (i.e., the node was the root), create a new root above this node (increasing the height of the
529:
For simplicity, most authors assume there are a fixed number of keys that fit in a node. The basic assumption is the key size is fixed and the node size is fixed. In practice, variable length keys may be employed.
3734:
Some operating systems require the user to allocate the maximum size of the file when the file is created. The file can then be allocated as contiguous disk blocks. In that case, to convert the file block address
3505:
Choose a new separator (either the largest element in the left subtree or the smallest element in the right subtree), remove it from the leaf node it is in, and replace the element to be deleted with the new
3092:
3599:
While freshly loaded databases tend to have good sequential behaviour, this behaviour becomes increasingly difficult to maintain as a database grows, resulting in more random I/O and performance challenges.
3457:
Do a single pass down the tree, but before entering (visiting) a node, restructure the tree so that once the key to be deleted is encountered, it can be deleted without triggering the need for any further
3509:
The previous step deleted an element (the new separator) from a leaf node. If that leaf node is now deficient (has fewer than the required number of nodes), then rebalance the tree starting from the leaf
3346:
All insertions start at a leaf node. To insert a new element, search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps:
2354:
key nodes and moving the key that would have been in the middle to the parent node. Each split node has the required minimum number of keys. Similarly, if an internal node and its neighbor each have
2921:
Deleting records from a database is relatively easy. The index can stay the same, and the record can just be marked as deleted. The database remains in sorted order. If there are a large number of
2564:
This depth will increase slowly as elements are added to the tree, but an increase in the overall depth is infrequent, and results in all leaf nodes being one more node farther away from the root.
3694:
At the end of bulk loading, the tree is composed almost entirely of completely full nodes; only the rightmost node on each level may be less than full. Because those nodes may also be less than
2847:
Creating an auxiliary index can be repeated to make an auxiliary index to the auxiliary index. That would make an aux-aux index that would need only 100 entries and would fit in one disk block.
2572:
Because a range of child nodes is permitted, B-trees do not need re-balancing as frequently as other self-balancing search trees, but may waste some space, since nodes are not entirely full.
3162:
2632:
is the order of the tree as maximum number of pointers to subtrees from one node), it needs to be checked whether the right sibling exists and has some free space. If the right sibling has
2073:
2918:
does not change, then compiling the index is simple to do, and the index need never be changed. If there are changes, managing the database and its index require additional computation.
498:
Some balanced trees store values only at leaf nodes, and use different kinds of nodes for leaf nodes and internal nodes. B-trees keep values in every node in the tree except leaf nodes.
2592:
or an analogous size in secondary storage. While 2–3 B-trees are easier to explain, practical B-trees using secondary storage need a large number of child nodes to improve performance.
3589:: The rebalancing operations are different for B+ trees (e.g., rotation is different because parent has copy of the key) and B-tree (e.g., three siblings are merged into two siblings).
3351:
If the node contains fewer than the maximum allowed number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered.
2929:
of densely packing all the records in a block, the block can have some free space to allow for subsequent insertions. Those spaces would be marked as if they were "deleted" records.
2801:
Large databases have historically been kept on disk drives. The time to read a record on a disk drive far exceeds the time needed to compare keys once the record is available due to
2031:
1963:
495:, but the cost of search, insert, and delete operations grows with the depth of the tree. As with any balanced tree, the cost grows much more slowly than the number of elements.
2140:
3986:
locks are only required as a tree block is modified. This maximizes access concurrency by multiple users, an important consideration for databases and/or other B-tree-based
4472:
354:
Each internal node's keys act as separation values which divide its subtrees. For example, if an internal node has 3 child nodes (or subtrees) then it must have 2 keys:
3360:
Values less than the median are put in the new left node and values greater than the median are put in the new right node, with the median acting as a separation value.
4351:
4071:
For FAT, what is called a "disk block" here is what the FAT documentation calls a "cluster", which is a fixed-size group of one or more contiguous whole physical disk
407:) are all nodes except for leaf nodes and the root node. They are usually represented as an ordered set of elements and child pointers. Every internal node contains a
1457:
1394:
1181:
1089:
997:
788:
762:
2958:
In addition, a B-tree minimizes waste by making sure the interior nodes are at least half full. A B-tree can handle an arbitrary number of insertions and deletions.
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Replace the separator in the parent with the first element of the right sibling (right sibling loses one node but still has at least the minimum number of elements)
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1024:
932:
674:
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Otherwise, if both immediate siblings have only the minimum number of elements, then merge with a sibling sandwiching their separator taken off from their parent
2559:
2398:
2269:
2100:
1923:
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Replace the separator in the parent with the last element of the left sibling (left sibling loses one node but still has at least the minimum number of elements)
2441:
2303:
2223:
729:
3810:
3773:
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3729:
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2533:
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2418:
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1984:
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694:
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is some associated information. The associated information might be a pointer to a record or records in a random access, but what it was didn't really matter.
3778:
Other operating systems allow a file to grow. The resulting disk blocks may not be contiguous, so mapping logical blocks to physical blocks is more involved.
3172:
6071:
4571:
5161:
3536:
Copy the separator from the parent to the end of the deficient node (the separator moves down; the deficient node now has the minimum number of elements)
2698:
to allow rapid searches for the Nth record in key order, or counting the number of records between any two records, and various other related operations.
2809:
The time to locate one record out of a million in the example above would take 20 disk reads times 10 milliseconds per disk read, which is 0.2 seconds.
3550:
Copy the separator from the parent to the start of the deficient node (the separator moves down; deficient node now has the minimum number of elements)
5747:
5096:
3995:
blocks in each level in the block cache. No reorganization for deletes is needed and there are no 'next' pointers in each block as in Lehman and Yao.
3564:
Copy the separator to the end of the left node (the left node may be the deficient node or it may be the sibling with the minimum number of elements)
3501:
separator. Both of those elements are in leaf nodes, and either one can be the new separator for the two subtrees. Algorithmically described below:
3711:) is also used in filesystems to allow quick random access to an arbitrary block in a particular file. The basic problem is turning the file block
3339:
3691:
as possible: leave the left node completely full and create a right node with zero keys and one child (in violation of the usual B-tree rules).
5183:
458:
The root node's number of children has the same upper limit as internal nodes, but has no lower limit. For example, when there are fewer than
6041:
5472:
4515:
4397:
2374:
keys, then a key may be deleted from the internal node by combining it with its neighbor. Deleting the key would make the internal node have
5679:
3375:−1. When a node is split, one element moves to the parent, but one element is added. So, it must be possible to divide the maximum number
2731:
2723:
3567:
Move all elements from the right node to the left node (the left node now has the maximum number of elements, and the right node – empty)
302:
have been suggested. McCreight has said that "the more you think about what the B in B-trees means, the better you understand B-trees".
3019:
6115:
5223:
5194:
5980:
4997:
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3305:
2749:
226:
2819:
To speed up the search further, the time to do the first 13 to 14 comparisons (which each required a disk access) must be reduced.
3862:
3775:
to the address of the first disk block constituting the file. The scheme is simple, but the file cannot exceed its created size.
5339:
5770:
5179:
4989:
5004:
Section 6.2.4: Multiway Trees, pp. 481–491. Also, pp. 476–477 of section 6.2.3 (Balanced Trees) discusses 2–3 trees.
3573:
If the parent is the root and now has no elements, then free it and make the merged node the new root (tree becomes shallower)
5775:
3921:
3639:
3419:
secondary storage. However, to use this algorithm, we must be able to send one element to the parent and split the remaining
2806:
and the average reading seek time is 8.5 milliseconds. For simplicity, assume reading from disk takes about 10 milliseconds.
423:
children. Thus, the number of elements is always 1 less than the number of child pointers (the number of elements is between
2898:
In practice, if the main database is being frequently searched, the aux-aux index and much of the aux index may reside in a
4658:
3547:
Otherwise, if the deficient node's left sibling exists and has more than the minimum number of elements, then rotate right
5740:
2767:
Sorting and searching algorithms can be characterized by the number of comparison operations that must be performed using
526:
designs, the leaves may only hold pointers to the data record. Those choices are not fundamental to the idea of a B-tree.
4728:
5854:
5837:
4211:
2790:
comparisons. If the table had 1,000,000 records, then a specific record could be located with at most 20 comparisons:
3906:
3624:
3283:
3120:
6053:
5820:
5815:
5304:
5111:
4937:
583:: Maximum number of potential search keys for each node in a B-tree. (this value is constant over the entire tree).
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2035:
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5689:
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4116:
Proceedings of the 1970 ACM SIGFIDET (Now SIGMOD) Workshop on Data
Description, Access and Control - SIGFIDET '70
3342:
A B-tree insertion example with each iteration. The nodes of this B-tree have at most 3 children (Knuth order 3).
3533:
If the deficient node's right sibling exists and has more than the minimum number of elements, then rotate left
5849:
5844:
5803:
5733:
5245:
5150:
3824:
3395:−1 elements, and hence is a legal node, and the other contains one more element, and hence it is legal too. If
3330:
is typically (but not necessarily) used within nodes to find the separation values and child tree of interest.
5139:
5077:
4084:
Two of these were reserved for special purposes, so only 4078 could actually represent disk blocks (clusters).
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improved by creating an auxiliary index that contains the first record in each disk block (sometimes called a
6084:
6061:
3917:
3635:
3327:
222:
49:
1990:
479:/2-1 if they are not the root) and pointers (one for each key) to nodes carrying the data objects / chunks.
6066:
5866:
5216:
4503:
2902:, so they would not incur a disk read. The B-tree remains the standard index implementation in almost all
3520:
a sibling node that has more than the minimum number of nodes. That redistribution operation is called a
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2643:
keys, then keys are redistributed between the two sibling nodes as evenly as possible. For this purpose,
1934:
5992:
5947:
5909:
5383:
5232:
5026:, vol. Mathematical and Information Sciences Report No. 20, Boeing Scientific Research Laboratories
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2974:
404:
54:
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keys plus one more key brought down from the neighbor's parent. The result is an entirely full node of
2111:
4075:. For the purposes of this discussion, a cluster has no significant difference from a physical sector.
5932:
5373:
5328:
5120:
4924:
4459:
4072:
3969:
3879:
3166:
Comer (1979) and Cormen et al. (2001) give the worst case height (the maximum height) of a B-tree as
2695:
2589:
230:
5172:
3570:
Remove the separator from the parent along with its empty right child (the parent loses an element)
3491:
If underflow happens, rebalance the tree as described in section "Rebalancing after deletion" below.
3357:
A single median is chosen from among the leaf's elements and the new element that is being inserted.
5487:
5263:
5019:
4833:
4052:
3956:
A B-tree grows slower with growing data amount, than the linearity of a linked list. Compared to a
2903:
462:−1 elements in the entire tree, the root will be the only node in the tree with no children at all.
254:
210:
76:
5975:
5343:
4042:
4008:
4004:
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Otherwise, if the parent has fewer than the required number of elements, then rebalance the parent
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4754:"Downloads - high-concurrency-btree - High Concurrency B-Tree code in C - GitHub Project Hosting"
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be the minimum number of children an internal (non-root) node must have. For an ordinary B-tree,
2179:
In order to maintain the predefined range of child nodes, internal nodes may be joined or split.
218:
4596:
213:
that maintains sorted data and allows searches, sequential access, insertions, and deletions in
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3324:
keys, contained in its parent node. These limiting values are also known as separation values.
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5793:
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5249:
5209:
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reorganization can be minimized. Alternatively, some out-of-sequence disk blocks may be used.
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27:
A self-balancing, tree-based data structure, that allows read/write access in logarithmic time
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969:
767:
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The auxiliary indices have turned the search problem from a binary search requiring roughly
2276:
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to be the maximum number of children (which is one more than the maximum number of keys).
87:
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2538:
2377:
2248:
2079:
1902:
559:
As with other trees, B-trees can be represented as a collection of three types of nodes:
4803:
3244:{\displaystyle h_{\mathrm {max} }=\left\lfloor \log _{d}{\frac {n+1}{2}}\right\rfloor .}
2423:
2285:
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793:
All leaf nodes have the same number of ancestors (i.e., they are all at the same depth).
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6007:
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5348:
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3714:
3439:−1, which accounts for why some textbooks impose this requirement in defining B-trees.
3100:
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keys, in which case the two siblings and parent are merged, reducing the depth by one.
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keys, then adding a key to that node can be accomplished by splitting the hypothetical
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707:
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91:
5057:
4431:
3960:, both structures have the same performance, but the B-tree scales better for growing
3472:
Deleting an element may put its node under the minimum number of elements and children
2446:
A B-tree is kept balanced after insertion by splitting a would-be overfilled node, of
6099:
6002:
5899:
5884:
5684:
5664:
5507:
5396:
5323:
4873:
4440:
3875:
2922:
2812:
The search time is reduced because individual records are grouped together in a disk
2772:
2585:
229:, the B-tree is well suited for storage systems that read and write relatively large
5258:
4864:
4702:
4131:
4047:
5644:
5608:
5424:
5419:
5401:
5313:
5106:
5032:
5015:
4981:
4959:
4829:
4020:
3755:
into a disk block address, the operating system simply adds the file block address
2650:
keys from the current node, the new key inserted, one key from the parent node and
311:
250:
72:
4911:
5037:
Proceedings of 1971 ACM-SIGFIDET Workshop on Data
Description, Access and Control
4651:
4284:
avoided the issue by saying an index element is a (physically adjacent) pair of (
2279:
of the tree. The factor of 2 will guarantee that nodes can be split or combined.
443:−1; therefore each internal node is at least half full. The relationship between
5997:
5922:
5694:
5659:
5649:
5563:
5497:
5492:
5482:
5391:
5240:
4779:
4753:
3895:
3869:
3789:
3613:
3451:
Locate and delete the item, then restructure the tree to retain its invariants,
3287:
238:
31:
5086:
4531:
4330:(6th ed.). Upper Saddle River, N.J.: Pearson Education. pp. 652–660.
3013:
entries. Hence, the best case height (i.e. the minimum height) of a B-tree is:
5985:
5889:
5704:
5674:
5634:
5477:
5406:
5353:
5273:
5051:
4304:
states, "For this paper the associated information is of no further interest."
3423:−2 elements into two legal nodes, without adding a new element. This requires
2988:
be the maximum number of children a node can have. Each node can have at most
2899:
2581:
4903:
3683:
can be used to produce a more efficient tree with a higher branching factor.
5927:
5874:
5709:
5669:
5516:
5444:
5434:
5115:
4123:
3957:
3680:
2802:
2272:
4219:
2941:
The B-tree uses all of the ideas described above. In particular, a B-tree:
6024:
5598:
5288:
4894:
4877:
4693:
4676:
4032:
2688:
2614:
2606:
35:
5970:
5798:
5714:
5588:
5568:
5541:
5526:
5278:
4474:
Product Manual: Barracuda ES.2 Serial ATA, Rev. F., publication 100468393
3847:
2915:
234:
6019:
5965:
5699:
5603:
5578:
5521:
5368:
5298:
5293:
5268:
5201:
4856:
3866:
3820:
17:
4780:"Lockless concurrent B-tree index meta access method for cached nodes"
3524:. If no sibling can spare an element, then the deficient node must be
6014:
5618:
5593:
5573:
5558:
5467:
5358:
4758:
4425:(3). Institute for System Programming of the RAS (ISP RAS): 203–216.
4415:"SQLite RDBMS Extension for Data Indexing Using B-tree Modifications"
4037:
3965:
3781:
700:
In B-trees, the following properties are maintained for these nodes:
258:
4497:
4814:
4792:
3831:
2998:
It can be shown (by induction for example) that a B-tree of height
5725:
5462:
5363:
5318:
5074:
3851:
3843:
3813:
3529:
not a problem. The algorithm to rebalance the tree is as follows:
3469:
The element in an internal node is a separator for its child nodes
3465:
There are two special cases to consider when deleting an element:
3337:
538:
537:
3379:−1 of elements into two legal nodes. If this number is odd, then
3087:{\displaystyle h_{\mathrm {min} }=\lceil \log _{m}(n+1)\rceil -1}
6036:
5454:
5126:
B-Tree .Net, a modern, virtualized RAM & Disk implementation
4941:(Second ed.). MIT Press and McGraw-Hill. pp. 434–454.
4625:
3987:
3855:
3839:
3835:
509:
Bayer and McCreight (1972), Comer (1979), and others define the
5729:
5205:
3488:
If the value is in a leaf node, simply delete it from the node.
3354:
Otherwise the node is full, evenly split it into two nodes so:
2951:
uses partially full blocks to speed up insertions and deletions
389:, and all values in the rightmost subtree will be greater than
5107:
Dictionary of
Algorithms and Data Structures entry for B*-tree
4419:
Proceedings of the
Institute for System Programming of the RAS
3889:
3792:
in the table. In order to find the disk address of file block
3607:
3415:−1, which is the minimum number of elements allowed per node.
3258:
2948:
uses a hierarchical index to minimize the number of disk reads
2706:
329:
Every node, except for the root and the leaves, has at least ⌈
4265:
4263:
4197:
3447:
There are two popular strategies for deletion from a B-tree.
506:
The literature on B-trees is not uniform in its terminology.
4164:
4162:
4160:
1426:
Points to subtree in which all search keys are greater than
1363:
Points to subtree in which all search keys are greater than
336:
The root node has at least two children unless it is a leaf.
5087:
NIST's
Dictionary of Algorithms and Data Structures: B-tree
2654:
keys from the sibling node are seen as an ordered array of
274:
Bayer and McCreight never explained what, if anything, the
5125:
4547:
4530:
Jan
Jannink. "Implementing Deletion in B+-Trees". Section
2880:
is the blocking factor (the number of entries per block:
1333:
Points to subtree in which all search keys are less than
797:
Each internal node in a B-tree has the following format:
265:
was first circulated in July 1970 and later published in
4677:"Efficient locking for concurrent operations on B-trees"
4118:. Boeing Scientific Research Laboratories. p. 107.
4023:
to reduce lock contention in virtual memory management.
4838:"Organization and Maintenance of Large Ordered Indexes"
4392:. Belgrade, Serbia: Akademska misao. pp. 274–275.
4109:"Organization and maintenance of large ordered indices"
368:. All values in the leftmost subtree will be less than
4718:"An In-Depth Analysis of Concurrent B-tree Algorithms"
4238:
4236:
4102:
4100:
2182:
Usually, the number of keys is chosen to vary between
710:
615:: The pointer to a child node which starts a sub-tree.
5173:"ECS 165B: Database System Implementation: Lecture 6"
5024:
3817:
disk location of a file block to be read or written.
3798:
3761:
3741:
3717:
3175:
3123:
3103:
3022:
2541:
2521:
2501:
2481:
2452:
2426:
2406:
2380:
2360:
2340:
2311:
2288:
2251:
2231:
2208:
2188:
2149:
2114:
2082:
2038:
1993:
1972:
1937:
1905:
1881:
1846:
1774:
1744:
1712:
1682:
1653:
1623:
1561:
Each leaf node in a B-tree has the following format:
1531:
1501:
1468:
1432:
1402:
1369:
1339:
1307:
1277:
1248:
1218:
1189:
1156:
1126:
1097:
1064:
1034:
1005:
972:
942:
913:
883:
770:
737:
682:
655:
623:
591:
263:
Organization and maintenance of large ordered indices
4599:. State University of New York (SUNY) at Stony Brook
3707:
In addition to its use in databases, the B-tree (or
2945:
keeps keys in sorted order for sequential traversing
2925:, then searching and storage become less efficient.
318:
is a tree which satisfies the following properties:
6052:
5946:
5908:
5865:
5784:
5763:
5627:
5506:
5453:
5382:
5239:
5151:"File Organization, ISAM, B+ Tree and Bulk Loading"
4147:
4145:
4143:
4141:
3476:The procedures for these cases are in order below.
2954:
keeps the index balanced with a recursive algorithm
1804:The node bounds are summarized in the table below:
375:, all values in the middle subtree will be between
176:
169:
150:
131:
112:
97:
86:
68:
60:
48:
43:
4958:
4019:A Maple tree is a B-tree developed for use in the
3804:
3767:
3747:
3723:
3674:A common special case is adding a large amount of
3243:
3156:
3109:
3086:
2553:
2527:
2507:
2487:
2467:
2435:
2412:
2392:
2366:
2346:
2326:
2297:
2263:
2237:
2217:
2194:
2155:
2134:
2094:
2067:
2025:
1978:
1957:
1917:
1887:
1852:
1790:
1757:
1725:
1698:
1666:
1639:
1547:
1514:
1484:
1451:
1415:
1388:
1352:
1320:
1293:
1261:
1234:
1202:
1175:
1142:
1110:
1083:
1050:
1018:
991:
958:
926:
899:
782:
756:
723:
688:
668:
639:
607:
4992:. Vol. 3 (Second ed.). Addison-Wesley.
4491:
4489:
4487:
2906:, and many nonrelational databases use them too.
3411:−2 elements in the node. Half of this number is
647:: The pointer to a record which stores the data.
5195:"BULK INSERT (Transact-SQL) in SQL Server 2017"
3157:{\displaystyle d=\left\lceil m/2\right\rceil .}
2665:keys. The array becomes split by half, so that
2584:. By maximizing the number of keys within each
4301:
4281:
4212:"Stanford Center for Professional Development"
4168:
3462:The algorithm below uses the former strategy.
676:: The search key at the zero-based node index
543:
5741:
5217:
5141:A user configurable implementation of B-trees
5112:Open Data Structures - Section 14.2 - B-Trees
5035:(1971). "Binary B-Trees for Virtual Memory".
4815:Introducing the Maple Tree (LWN.net / github)
2779:records, for example, can be done in roughly
8:
4413:Rigin A. M., Shershakov S. A. (2019-09-10).
4350:: CS1 maint: multiple names: authors list (
4326:Navathe, Ramez Elmasri, Shamkant B. (2010).
4313:
4269:
4254:
3286:. There might be a discussion about this on
3075:
3044:
2129:
2115:
2068:{\displaystyle \equiv \lfloor K/2\rfloor +1}
2056:
2042:
2020:
1994:
1952:
1938:
4804:Maple Tree (The Linux Kernel documentation)
4191:
4189:
3924:. Unsourced material may be challenged and
3642:. Unsourced material may be challenged and
5748:
5734:
5726:
5224:
5210:
5202:
4480:. Seagate Technology LLC. 2008. p. 6.
4003:Since B-trees are similar in structure to
2984:be the number of entries in the tree. Let
2844:record could be accessed in 9 disk reads.
225:with more than two children. Unlike other
83:
4957:Folk, Michael J.; Zoellick, Bill (1992).
4893:
4692:
4430:
3944:Learn how and when to remove this message
3797:
3760:
3740:
3716:
3662:Learn how and when to remove this message
3306:Learn how and when to remove this message
3215:
3206:
3181:
3180:
3174:
3138:
3122:
3102:
3051:
3028:
3027:
3021:
3002:with all its nodes completely filled has
2973:be the height of the classic B-tree (see
2750:Learn how and when to remove this message
2540:
2520:
2500:
2480:
2451:
2425:
2405:
2379:
2359:
2339:
2310:
2287:
2250:
2230:
2207:
2187:
2148:
2121:
2113:
2081:
2048:
2037:
2012:
1992:
1971:
1944:
1936:
1904:
1880:
1845:
1782:
1773:
1749:
1743:
1738:Points to a record with a value equal to
1717:
1711:
1690:
1681:
1658:
1652:
1631:
1622:
1539:
1530:
1506:
1500:
1495:Points to a record with a value equal to
1476:
1467:
1437:
1431:
1407:
1401:
1374:
1368:
1344:
1338:
1312:
1306:
1285:
1276:
1253:
1247:
1226:
1217:
1194:
1188:
1161:
1155:
1134:
1125:
1102:
1096:
1069:
1063:
1042:
1033:
1010:
1004:
977:
971:
950:
941:
918:
912:
891:
882:
769:
742:
736:
715:
709:
681:
660:
654:
631:
622:
599:
590:
574:Note the following variable definitions:
491:times as many items as a B-tree of depth
4675:Lehman, Philip L.; Yao, s. Bing (1981).
2975:Tree (data structure) § Terminology
2937:Advantages of B-tree usage for databases
1806:
1611:
1563:
871:
799:
4383:
4381:
4379:
4377:
4375:
4373:
4096:
4064:
4009:parallel algorithms for red-black trees
6111:Computer-related introductions in 1971
5075:B-Trees: Balanced Tree Data Structures
4343:
4107:Bayer, R.; McCreight, E. (July 1970).
3972:systems, is similar but more compact.
1870:Root node when it is an internal node
731:exists in any node in a B+ tree, then
40:
4543:
4499:Designing Data-Intensive Applications
4364:
4242:
4180:
4151:
3882:file system uses a modified B+-tree.
2977:for the tree height definition). Let
2400:keys; joining the neighbor would add
2026:{\displaystyle \lceil (K+1)/2\rceil }
547:
7:
4681:ACM Transactions on Database Systems
3922:adding citations to reliable sources
3640:adding citations to reliable sources
339:All leaves appear on the same level.
5197:. Microsoft Docs. 6 September 2018.
5097:The InfinityDB BTree implementation
4196:Weiner, Peter G. (30 August 2013).
4011:can be applied to B-trees as well.
3731:address into a disk block address.
3387:and one of the new nodes contains (
2245:is the minimum number of keys, and
1958:{\displaystyle \lfloor K/2\rfloor }
5069:B-tree and UB-tree on Scholarpedia
4632:from the original on 13 April 2008
3188:
3185:
3182:
3035:
3032:
3029:
2887:entries per block in our example;
2722:tone or style may not reflect the
227:self-balancing binary search trees
25:
5149:Kaldırım, Semih (28 April 2015).
4793:Introducing maple trees (LWN.net)
2864:disk reads to one requiring only
2135:{\displaystyle \lceil K/2\rceil }
1835:Root node when it is a leaf node
314:'s definition, a B-tree of order
5189:from the original on 2022-10-09.
5167:from the original on 2022-10-09.
5144:(Thesis). Iowa State University.
4965:(2nd ed.). Addison-Wesley.
4664:from the original on 2022-10-09.
4328:Fundamentals of database systems
3894:
3612:
3263:
2962:Best case and worst case heights
2732:guide to writing better articles
2711:
873:Internal node pointer structure
5180:University of California, Davis
4990:The Art of Computer Programming
4577:from the original on 2022-10-09
3485:Search for the value to delete.
4716:Wang, Paul (1 February 1991).
4390:Algorithms and Data Structures
3496:Deletion from an internal node
3072:
3060:
2009:
1997:
403:Internal nodes (also known as
1:
5064:Animated B-Tree visualization
4650:Matthew Dillon (2008-06-21).
4432:10.15514/ispras-2019-31(3)-16
3784:, for example, used a simple
3708:
2691:and B tree features together.
2687:The B tree combines the main
2609:, the B tree and the B tree.
217:. The B-tree generalizes the
5182:. 9 April 2010. p. 23.
4302:Bayer & McCreight (1972)
4282:Bayer & McCreight (1972)
2763:Time to search a sorted file
1613:Leaf node pointer structure
6072:Directed acyclic word graph
5838:Double-ended priority queue
4626:Microsoft Developer Network
4622:"Inside Win2K NTFS, Part 1"
2694:B-trees can be turned into
6132:
5138:Shetty, Soumya B. (2010).
5102:Cache Oblivious B(+)-trees
5054:by David Scot Taylor, SJSU
4938:Introduction to Algorithms
4496:Kleppmann, Martin (2017).
4169:Bayer & McCreight 1972
3515:Rebalancing after deletion
2823:An index speeds the search
764:exists in that node where
544:Bayer & McCreight 1972
502:Differences in terminology
29:
6116:Database index techniques
6080:
4597:"Cache Oblivious B-trees"
4183:, p. 123 footnote 1.
3480:Deletion from a leaf node
2703:B-tree usage in databases
2568:Comparison to other trees
249:B-trees were invented by
82:
5804:Retrieval Data Structure
5680:Left-child right-sibling
5071:Curator: Dr Rudolf Bayer
5039:. San Diego, California.
4388:Tomašević, Milo (2008).
4314:Folk & Zoellick 1992
4270:Folk & Zoellick 1992
4255:Folk & Zoellick 1992
3861:B-trees are used in the
3556:The tree is now balanced
3542:The tree is now balanced
2910:Insertions and deletions
2282:If an internal node has
801:Internal node structure
30:Not to be confused with
6085:List of data structures
6062:Binary decision diagram
5510:data partitioning trees
5468:C-trie (compressed ADT)
4878:"The Ubiquitous B-Tree"
4652:"The HAMMER Filesystem"
4198:"4- Edward M McCreight"
4124:10.1145/1734663.1734671
2775:of a sorted table with
2726:used on Knowledge (XXG)
1452:{\displaystyle k_{i-1}}
1389:{\displaystyle k_{i-1}}
1176:{\displaystyle k_{i-1}}
1084:{\displaystyle k_{i-1}}
992:{\displaystyle k_{i-1}}
783:{\displaystyle i\geq 1}
757:{\displaystyle k_{i-1}}
567:(a.k.a. interior), and
322:Every node has at most
6067:Directed acyclic graph
4561:"Deletion in a B-tree"
4504:Sebastopol, California
4462:, retrieved 2010-01-25
3842:, AIX (jfs2) and some
3806:
3769:
3749:
3725:
3343:
3245:
3158:
3111:
3088:
2730:See Knowledge (XXG)'s
2555:
2529:
2509:
2489:
2469:
2437:
2414:
2394:
2368:
2348:
2328:
2299:
2265:
2239:
2219:
2196:
2175:Insertion and deletion
2157:
2136:
2096:
2069:
2027:
1980:
1959:
1919:
1889:
1854:
1792:
1791:{\displaystyle pr_{i}}
1759:
1727:
1700:
1699:{\displaystyle pr_{i}}
1668:
1641:
1640:{\displaystyle pr_{i}}
1549:
1548:{\displaystyle pr_{i}}
1516:
1486:
1485:{\displaystyle pt_{i}}
1453:
1417:
1390:
1354:
1322:
1295:
1294:{\displaystyle pr_{i}}
1263:
1236:
1235:{\displaystyle pr_{i}}
1204:
1177:
1144:
1143:{\displaystyle pt_{i}}
1112:
1085:
1052:
1051:{\displaystyle pt_{i}}
1020:
993:
960:
959:{\displaystyle pt_{i}}
928:
901:
900:{\displaystyle pt_{0}}
784:
758:
725:
690:
670:
641:
640:{\displaystyle pr_{i}}
609:
608:{\displaystyle pt_{i}}
551:
4986:Sorting and Searching
4895:10.1145/356770.356776
4694:10.1145/319628.319663
3846:filesystems, such as
3807:
3786:File Allocation Table
3770:
3750:
3726:
3341:
3246:
3159:
3112:
3089:
2696:order statistic trees
2556:
2535:and (transitionally)
2530:
2510:
2490:
2470:
2438:
2415:
2395:
2369:
2349:
2329:
2300:
2266:
2240:
2220:
2197:
2158:
2137:
2097:
2070:
2028:
1981:
1960:
1920:
1890:
1855:
1793:
1760:
1758:{\displaystyle k_{i}}
1728:
1726:{\displaystyle k_{i}}
1701:
1669:
1667:{\displaystyle k_{i}}
1642:
1550:
1517:
1515:{\displaystyle k_{i}}
1487:
1454:
1418:
1416:{\displaystyle k_{i}}
1391:
1355:
1353:{\displaystyle k_{0}}
1323:
1321:{\displaystyle k_{i}}
1296:
1264:
1262:{\displaystyle k_{i}}
1237:
1205:
1203:{\displaystyle k_{i}}
1178:
1145:
1113:
1111:{\displaystyle k_{i}}
1086:
1053:
1021:
1019:{\displaystyle k_{i}}
994:
961:
929:
927:{\displaystyle k_{0}}
902:
785:
759:
726:
691:
671:
669:{\displaystyle k_{i}}
642:
610:
541:
342:A non-leaf node with
55:Tree (data structure)
5933:Unrolled linked list
5690:Log-structured merge
5233:Tree data structures
5058:B-Tree visualisation
4953:Chapter 18: B-Trees.
3970:main memory database
3918:improve this section
3796:
3759:
3739:
3715:
3636:improve this section
3604:Initial construction
3276:confusing or unclear
3173:
3121:
3101:
3020:
2904:relational databases
2539:
2519:
2499:
2479:
2468:{\displaystyle 2d+1}
2450:
2424:
2404:
2378:
2358:
2338:
2327:{\displaystyle 2d+1}
2309:
2286:
2249:
2229:
2206:
2186:
2147:
2112:
2080:
2036:
1991:
1970:
1935:
1903:
1879:
1844:
1772:
1742:
1710:
1680:
1651:
1621:
1565:Leaf node structure
1529:
1499:
1466:
1430:
1400:
1367:
1337:
1305:
1275:
1246:
1216:
1187:
1154:
1124:
1095:
1062:
1032:
1003:
970:
940:
911:
881:
768:
735:
708:
680:
653:
621:
589:
534:Informal description
259:Boeing Research Labs
209:is a self-balancing
5981:Self-balancing tree
3999:Parallel algorithms
3830:Apple's filesystem
3284:clarify the article
2554:{\displaystyle d-1}
2393:{\displaystyle d-1}
2264:{\displaystyle d+1}
2095:{\displaystyle K+1}
1918:{\displaystyle K+1}
1614:
1566:
874:
802:
255:Edward M. McCreight
211:tree data structure
77:Edward M. McCreight
5961:Binary search tree
5826:Double-ended queue
5655:Fractal tree index
5250:associative arrays
5158:Bilkent University
5156:. Ankara, Turkey:
5080:2010-03-05 at the
4925:Leiserson, Charles
4857:10.1007/bf00288683
4550:, pp. 439–440
4548:Cormen et al. 2001
4200:– via Vimeo.
3981:Access concurrency
3802:
3765:
3745:
3721:
3407:−1, so there are 2
3344:
3241:
3154:
3107:
3084:
2796:(1,000,000) ⌉ = 20
2551:
2525:
2505:
2485:
2465:
2436:{\displaystyle 2d}
2433:
2410:
2390:
2364:
2344:
2334:key node into two
2324:
2298:{\displaystyle 2d}
2295:
2261:
2235:
2218:{\displaystyle 2d}
2215:
2192:
2153:
2132:
2092:
2065:
2023:
1976:
1955:
1915:
1885:
1850:
1788:
1755:
1723:
1696:
1664:
1637:
1612:
1564:
1545:
1512:
1482:
1449:
1413:
1396:and are less than
1386:
1350:
1318:
1291:
1259:
1232:
1200:
1173:
1140:
1108:
1081:
1048:
1016:
989:
956:
924:
897:
872:
800:
780:
754:
724:{\textstyle k_{i}}
721:
686:
666:
637:
605:
552:
487:+1 can hold about
483:A B-tree of depth
346:children contains
219:binary search tree
6093:
6092:
5895:Hashed array tree
5794:Associative array
5723:
5722:
4882:Computing Surveys
4532:"4 Lazy Deletion"
4517:978-1-449-37332-0
4399:978-86-7466-328-8
4216:scpd.stanford.edu
3954:
3953:
3946:
3805:{\displaystyle i}
3768:{\displaystyle i}
3748:{\displaystyle i}
3724:{\displaystyle i}
3672:
3671:
3664:
3595:Sequential access
3590:
3399:−1 is even, then
3316:
3315:
3308:
3231:
3110:{\displaystyle d}
2876:disk reads where
2760:
2759:
2752:
2724:encyclopedic tone
2578:secondary storage
2528:{\displaystyle d}
2508:{\displaystyle d}
2488:{\displaystyle d}
2413:{\displaystyle d}
2367:{\displaystyle d}
2347:{\displaystyle d}
2238:{\displaystyle d}
2195:{\displaystyle d}
2172:
2171:
2156:{\displaystyle K}
1979:{\displaystyle K}
1888:{\displaystyle K}
1853:{\displaystyle K}
1802:
1801:
1610:
1609:
1559:
1558:
870:
869:
689:{\displaystyle i}
475:-1, and at least
257:while working at
199:
198:
195:
194:
16:(Redirected from
6123:
5918:Association list
5750:
5743:
5736:
5727:
5226:
5219:
5212:
5203:
5198:
5190:
5188:
5177:
5168:
5166:
5160:. pp. 4–6.
5155:
5145:
5040:
5027:
5003:
4976:
4964:
4952:
4915:
4897:
4868:
4845:Acta Informatica
4842:
4817:
4812:
4806:
4801:
4795:
4790:
4784:
4783:
4776:
4770:
4769:
4767:
4766:
4750:
4744:
4743:
4741:
4739:
4733:
4727:. Archived from
4722:
4713:
4707:
4706:
4696:
4672:
4666:
4665:
4663:
4656:
4647:
4641:
4640:
4638:
4637:
4620:(30 June 2006).
4618:Mark Russinovich
4614:
4608:
4607:
4605:
4604:
4593:
4587:
4586:
4584:
4582:
4576:
4565:
4557:
4551:
4541:
4535:
4528:
4522:
4521:
4493:
4482:
4481:
4479:
4469:
4463:
4457:
4451:
4450:
4448:
4447:
4434:
4410:
4404:
4403:
4385:
4368:
4362:
4356:
4355:
4349:
4341:
4323:
4317:
4311:
4305:
4296:is the key, and
4279:
4273:
4267:
4258:
4252:
4246:
4240:
4231:
4230:
4228:
4227:
4218:. Archived from
4208:
4202:
4201:
4193:
4184:
4178:
4172:
4166:
4155:
4149:
4136:
4135:
4113:
4104:
4085:
4082:
4076:
4069:
3949:
3942:
3938:
3935:
3929:
3898:
3890:
3811:
3809:
3808:
3803:
3774:
3772:
3771:
3766:
3754:
3752:
3751:
3746:
3730:
3728:
3727:
3722:
3667:
3660:
3656:
3653:
3647:
3616:
3608:
3585:
3311:
3304:
3300:
3297:
3291:
3267:
3266:
3259:
3250:
3248:
3247:
3242:
3237:
3233:
3232:
3227:
3216:
3211:
3210:
3193:
3192:
3191:
3163:
3161:
3160:
3155:
3150:
3146:
3142:
3116:
3114:
3113:
3108:
3093:
3091:
3090:
3085:
3056:
3055:
3040:
3039:
3038:
3012:
2994:
2983:
2972:
2894:
2886:
2879:
2875:
2863:
2797:
2789:
2778:
2755:
2748:
2744:
2741:
2735:
2734:for suggestions.
2715:
2714:
2707:
2682:
2681:
2669:
2664:
2653:
2649:
2642:
2631:
2627:
2560:
2558:
2557:
2552:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2494:
2492:
2491:
2486:
2474:
2472:
2471:
2466:
2442:
2440:
2439:
2434:
2419:
2417:
2416:
2411:
2399:
2397:
2396:
2391:
2373:
2371:
2370:
2365:
2353:
2351:
2350:
2345:
2333:
2331:
2330:
2325:
2304:
2302:
2301:
2296:
2277:branching factor
2270:
2268:
2267:
2262:
2244:
2242:
2241:
2236:
2224:
2222:
2221:
2216:
2201:
2199:
2198:
2193:
2162:
2160:
2159:
2154:
2141:
2139:
2138:
2133:
2125:
2101:
2099:
2098:
2093:
2074:
2072:
2071:
2066:
2052:
2032:
2030:
2029:
2024:
2016:
1985:
1983:
1982:
1977:
1964:
1962:
1961:
1956:
1948:
1924:
1922:
1921:
1916:
1894:
1892:
1891:
1886:
1859:
1857:
1856:
1851:
1807:
1797:
1795:
1794:
1789:
1787:
1786:
1764:
1762:
1761:
1756:
1754:
1753:
1732:
1730:
1729:
1724:
1722:
1721:
1705:
1703:
1702:
1697:
1695:
1694:
1673:
1671:
1670:
1665:
1663:
1662:
1646:
1644:
1643:
1638:
1636:
1635:
1615:
1567:
1554:
1552:
1551:
1546:
1544:
1543:
1521:
1519:
1518:
1513:
1511:
1510:
1491:
1489:
1488:
1483:
1481:
1480:
1458:
1456:
1455:
1450:
1448:
1447:
1422:
1420:
1419:
1414:
1412:
1411:
1395:
1393:
1392:
1387:
1385:
1384:
1359:
1357:
1356:
1351:
1349:
1348:
1327:
1325:
1324:
1319:
1317:
1316:
1300:
1298:
1297:
1292:
1290:
1289:
1268:
1266:
1265:
1260:
1258:
1257:
1241:
1239:
1238:
1233:
1231:
1230:
1209:
1207:
1206:
1201:
1199:
1198:
1182:
1180:
1179:
1174:
1172:
1171:
1149:
1147:
1146:
1141:
1139:
1138:
1117:
1115:
1114:
1109:
1107:
1106:
1090:
1088:
1087:
1082:
1080:
1079:
1057:
1055:
1054:
1049:
1047:
1046:
1025:
1023:
1022:
1017:
1015:
1014:
998:
996:
995:
990:
988:
987:
965:
963:
962:
957:
955:
954:
933:
931:
930:
925:
923:
922:
906:
904:
903:
898:
896:
895:
875:
803:
789:
787:
786:
781:
763:
761:
760:
755:
753:
752:
730:
728:
727:
722:
720:
719:
695:
693:
692:
687:
675:
673:
672:
667:
665:
664:
646:
644:
643:
638:
636:
635:
614:
612:
611:
606:
604:
603:
581:
435:must be either 2
268:Acta Informatica
215:logarithmic time
203:computer science
171:Space complexity
84:
41:
21:
6131:
6130:
6126:
6125:
6124:
6122:
6121:
6120:
6096:
6095:
6094:
6089:
6076:
6048:
5942:
5938:XOR linked list
5904:
5880:Circular buffer
5861:
5780:
5759:
5757:Data structures
5754:
5724:
5719:
5623:
5502:
5449:
5378:
5374:Weight-balanced
5329:Order statistic
5243:
5235:
5230:
5193:
5186:
5175:
5171:
5164:
5153:
5148:
5137:
5121:Counted B-Trees
5092:B-Tree Tutorial
5082:Wayback Machine
5048:
5031:
5014:
5011:
5009:Original papers
5000:
4980:
4973:
4961:File Structures
4956:
4949:
4933:Stein, Clifford
4919:
4872:
4840:
4828:
4825:
4820:
4813:
4809:
4802:
4798:
4791:
4787:
4778:
4777:
4773:
4764:
4762:
4752:
4751:
4747:
4737:
4735:
4731:
4720:
4715:
4714:
4710:
4674:
4673:
4669:
4661:
4654:
4649:
4648:
4644:
4635:
4633:
4616:
4615:
4611:
4602:
4600:
4595:
4594:
4590:
4580:
4578:
4574:
4563:
4559:
4558:
4554:
4546:, p. 127;
4542:
4538:
4529:
4525:
4518:
4495:
4494:
4485:
4477:
4471:
4470:
4466:
4460:Counted B-Trees
4458:
4454:
4445:
4443:
4412:
4411:
4407:
4400:
4387:
4386:
4371:
4363:
4359:
4342:
4338:
4325:
4324:
4320:
4312:
4308:
4280:
4276:
4268:
4261:
4253:
4249:
4241:
4234:
4225:
4223:
4210:
4209:
4205:
4195:
4194:
4187:
4179:
4175:
4167:
4158:
4150:
4139:
4111:
4106:
4105:
4098:
4094:
4089:
4088:
4083:
4079:
4070:
4066:
4061:
4029:
4017:
4005:red-black trees
4001:
3983:
3978:
3950:
3939:
3933:
3930:
3915:
3899:
3888:
3858:, use B-trees.
3794:
3793:
3757:
3756:
3737:
3736:
3713:
3712:
3709:§ Variants
3705:
3668:
3657:
3651:
3648:
3633:
3617:
3606:
3597:
3517:
3498:
3482:
3445:
3336:
3321:
3312:
3301:
3295:
3292:
3281:
3268:
3264:
3257:
3217:
3202:
3201:
3197:
3176:
3171:
3170:
3134:
3130:
3119:
3118:
3099:
3098:
3047:
3023:
3018:
3017:
3003:
2989:
2978:
2967:
2964:
2939:
2912:
2892:
2888:
2881:
2877:
2871:
2865:
2859:
2855:
2825:
2795:
2791:
2784:
2780:
2776:
2765:
2756:
2745:
2739:
2736:
2729:
2720:This section's
2716:
2712:
2705:
2679:
2667:
2666:
2655:
2651:
2644:
2633:
2629:
2622:
2598:
2570:
2537:
2536:
2517:
2516:
2497:
2496:
2477:
2476:
2475:keys, into two
2448:
2447:
2422:
2421:
2402:
2401:
2376:
2375:
2356:
2355:
2336:
2335:
2307:
2306:
2284:
2283:
2271:is the minimum
2247:
2246:
2227:
2226:
2204:
2203:
2184:
2183:
2177:
2145:
2144:
2110:
2109:
2078:
2077:
2034:
2033:
1989:
1988:
1968:
1967:
1933:
1932:
1901:
1900:
1877:
1876:
1842:
1841:
1830:of child nodes
1829:
1825:of child nodes
1824:
1819:
1814:
1778:
1770:
1769:
1745:
1740:
1739:
1733:does not exist
1713:
1708:
1707:
1686:
1678:
1677:
1654:
1649:
1648:
1627:
1619:
1618:
1606:
1600:
1591:
1585:
1579:
1573:
1535:
1527:
1526:
1502:
1497:
1496:
1472:
1464:
1463:
1433:
1428:
1427:
1403:
1398:
1397:
1370:
1365:
1364:
1340:
1335:
1334:
1328:does not exist
1308:
1303:
1302:
1281:
1273:
1272:
1249:
1244:
1243:
1222:
1214:
1213:
1190:
1185:
1184:
1157:
1152:
1151:
1130:
1122:
1121:
1118:does not exist
1098:
1093:
1092:
1065:
1060:
1059:
1038:
1030:
1029:
1006:
1001:
1000:
973:
968:
967:
946:
938:
937:
914:
909:
908:
887:
879:
878:
866:
860:
854:
845:
839:
833:
827:
821:
815:
809:
766:
765:
738:
733:
732:
711:
706:
705:
678:
677:
656:
651:
650:
627:
619:
618:
595:
587:
586:
579:
557:
536:
504:
415:children and a
395:
388:
381:
374:
367:
360:
308:
247:
221:, allowing for
88:Time complexity
39:
28:
23:
22:
15:
12:
11:
5:
6129:
6127:
6119:
6118:
6113:
6108:
6098:
6097:
6091:
6090:
6088:
6087:
6081:
6078:
6077:
6075:
6074:
6069:
6064:
6058:
6056:
6050:
6049:
6047:
6046:
6045:
6044:
6034:
6033:
6032:
6030:Hilbert R-tree
6027:
6022:
6012:
6011:
6010:
6008:Fibonacci heap
6005:
6000:
5990:
5989:
5988:
5983:
5978:
5976:Red–black tree
5973:
5968:
5958:
5952:
5950:
5944:
5943:
5941:
5940:
5935:
5930:
5925:
5920:
5914:
5912:
5906:
5905:
5903:
5902:
5897:
5892:
5887:
5882:
5877:
5871:
5869:
5863:
5862:
5860:
5859:
5858:
5857:
5852:
5842:
5841:
5840:
5833:Priority queue
5830:
5829:
5828:
5818:
5813:
5808:
5807:
5806:
5801:
5790:
5788:
5782:
5781:
5779:
5778:
5773:
5767:
5765:
5761:
5760:
5755:
5753:
5752:
5745:
5738:
5730:
5721:
5720:
5718:
5717:
5712:
5707:
5702:
5697:
5692:
5687:
5682:
5677:
5672:
5667:
5662:
5657:
5652:
5647:
5642:
5637:
5631:
5629:
5625:
5624:
5622:
5621:
5616:
5611:
5606:
5601:
5596:
5591:
5586:
5581:
5576:
5571:
5566:
5561:
5556:
5539:
5534:
5529:
5524:
5519:
5513:
5511:
5504:
5503:
5501:
5500:
5495:
5490:
5488:Ternary search
5485:
5480:
5475:
5470:
5465:
5459:
5457:
5451:
5450:
5448:
5447:
5442:
5437:
5432:
5427:
5422:
5417:
5412:
5404:
5399:
5394:
5388:
5386:
5380:
5379:
5377:
5376:
5371:
5366:
5361:
5356:
5351:
5346:
5336:
5331:
5326:
5321:
5316:
5311:
5301:
5296:
5291:
5286:
5281:
5276:
5271:
5266:
5261:
5255:
5253:
5237:
5236:
5231:
5229:
5228:
5221:
5214:
5206:
5200:
5199:
5191:
5169:
5146:
5129:
5128:
5123:
5118:
5109:
5104:
5099:
5094:
5089:
5084:
5072:
5066:
5061:
5060:(click "init")
5055:
5052:B-tree lecture
5047:
5046:External links
5044:
5043:
5042:
5029:
5010:
5007:
5006:
5005:
4998:
4978:
4971:
4954:
4947:
4929:Rivest, Ronald
4921:Cormen, Thomas
4917:
4888:(2): 123–137.
4874:Comer, Douglas
4870:
4851:(3): 173–189.
4824:
4821:
4819:
4818:
4807:
4796:
4785:
4771:
4745:
4734:on 4 June 2011
4708:
4687:(4): 650–670.
4667:
4642:
4609:
4588:
4552:
4536:
4523:
4516:
4510:. p. 80.
4508:O'Reilly Media
4483:
4464:
4452:
4405:
4398:
4369:
4367:, p. 488.
4357:
4336:
4318:
4316:, p. 379.
4306:
4274:
4272:, p. 363.
4259:
4257:, p. 362.
4247:
4245:, p. 483.
4232:
4203:
4185:
4173:
4156:
4137:
4095:
4093:
4090:
4087:
4086:
4077:
4063:
4062:
4060:
4057:
4056:
4055:
4050:
4045:
4043:Red–black tree
4040:
4035:
4028:
4025:
4016:
4013:
4000:
3997:
3982:
3979:
3977:
3974:
3952:
3951:
3902:
3900:
3893:
3887:
3884:
3838:, Microsoft's
3823:(and possibly
3801:
3764:
3744:
3720:
3704:
3703:In filesystems
3701:
3670:
3669:
3620:
3618:
3611:
3605:
3602:
3596:
3593:
3592:
3591:
3582:
3581:
3580:
3579:
3578:
3577:
3574:
3568:
3565:
3559:
3558:
3557:
3554:
3551:
3545:
3544:
3543:
3540:
3537:
3516:
3513:
3512:
3511:
3507:
3497:
3494:
3493:
3492:
3489:
3486:
3481:
3478:
3474:
3473:
3470:
3460:
3459:
3455:
3444:
3441:
3368:
3367:
3366:
3365:
3361:
3358:
3352:
3335:
3332:
3320:
3317:
3314:
3313:
3271:
3269:
3262:
3256:
3253:
3252:
3251:
3240:
3236:
3230:
3226:
3223:
3220:
3214:
3209:
3205:
3200:
3196:
3190:
3187:
3184:
3179:
3153:
3149:
3145:
3141:
3137:
3133:
3129:
3126:
3106:
3095:
3094:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3054:
3050:
3046:
3043:
3037:
3034:
3031:
3026:
2963:
2960:
2956:
2955:
2952:
2949:
2946:
2938:
2935:
2923:lazy deletions
2911:
2908:
2890:
2867:
2857:
2824:
2821:
2793:
2782:
2769:order notation
2764:
2761:
2758:
2757:
2719:
2717:
2710:
2704:
2701:
2700:
2699:
2692:
2685:
2618:
2597:
2594:
2569:
2566:
2550:
2547:
2544:
2524:
2504:
2484:
2464:
2461:
2458:
2455:
2432:
2429:
2409:
2389:
2386:
2383:
2363:
2343:
2323:
2320:
2317:
2314:
2294:
2291:
2260:
2257:
2254:
2234:
2214:
2211:
2191:
2176:
2173:
2170:
2169:
2166:
2163:
2152:
2142:
2131:
2128:
2124:
2120:
2117:
2107:
2103:
2102:
2091:
2088:
2085:
2075:
2064:
2061:
2058:
2055:
2051:
2047:
2044:
2041:
2022:
2019:
2015:
2011:
2008:
2005:
2002:
1999:
1996:
1986:
1975:
1965:
1954:
1951:
1947:
1943:
1940:
1930:
1929:Internal node
1926:
1925:
1914:
1911:
1908:
1898:
1895:
1884:
1874:
1871:
1867:
1866:
1863:
1860:
1849:
1839:
1836:
1832:
1831:
1826:
1821:
1816:
1811:
1800:
1799:
1785:
1781:
1777:
1766:
1752:
1748:
1735:
1734:
1720:
1716:
1693:
1689:
1685:
1675:
1661:
1657:
1634:
1630:
1626:
1608:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1557:
1556:
1542:
1538:
1534:
1523:
1509:
1505:
1493:
1479:
1475:
1471:
1460:
1446:
1443:
1440:
1436:
1424:
1410:
1406:
1383:
1380:
1377:
1373:
1361:
1347:
1343:
1330:
1329:
1315:
1311:
1288:
1284:
1280:
1270:
1256:
1252:
1229:
1225:
1221:
1211:
1197:
1193:
1170:
1167:
1164:
1160:
1137:
1133:
1129:
1119:
1105:
1101:
1078:
1075:
1072:
1068:
1045:
1041:
1037:
1027:
1013:
1009:
986:
983:
980:
976:
953:
949:
945:
935:
921:
917:
894:
890:
886:
868:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
795:
794:
791:
779:
776:
773:
751:
748:
745:
741:
718:
714:
698:
697:
685:
663:
659:
648:
634:
630:
626:
616:
602:
598:
594:
584:
556:
555:Node structure
553:
546:) of order 5 (
535:
532:
503:
500:
481:
480:
468:
464:
463:
456:
453:
401:
400:Internal nodes
393:
386:
379:
372:
365:
358:
352:
351:
350:−1 keys.
340:
337:
334:
327:
307:
304:
246:
243:
231:blocks of data
197:
196:
193:
192:
185:
178:
174:
173:
167:
166:
159:
152:
148:
147:
140:
133:
129:
128:
121:
114:
110:
109:
104:
99:
95:
94:
92:big O notation
80:
79:
70:
66:
65:
62:
58:
57:
52:
46:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6128:
6117:
6114:
6112:
6109:
6107:
6104:
6103:
6101:
6086:
6083:
6082:
6079:
6073:
6070:
6068:
6065:
6063:
6060:
6059:
6057:
6055:
6051:
6043:
6040:
6039:
6038:
6035:
6031:
6028:
6026:
6023:
6021:
6018:
6017:
6016:
6013:
6009:
6006:
6004:
6003:Binomial heap
6001:
5999:
5996:
5995:
5994:
5991:
5987:
5984:
5982:
5979:
5977:
5974:
5972:
5969:
5967:
5964:
5963:
5962:
5959:
5957:
5954:
5953:
5951:
5949:
5945:
5939:
5936:
5934:
5931:
5929:
5926:
5924:
5921:
5919:
5916:
5915:
5913:
5911:
5907:
5901:
5900:Sparse matrix
5898:
5896:
5893:
5891:
5888:
5886:
5885:Dynamic array
5883:
5881:
5878:
5876:
5873:
5872:
5870:
5868:
5864:
5856:
5853:
5851:
5848:
5847:
5846:
5843:
5839:
5836:
5835:
5834:
5831:
5827:
5824:
5823:
5822:
5819:
5817:
5814:
5812:
5809:
5805:
5802:
5800:
5797:
5796:
5795:
5792:
5791:
5789:
5787:
5783:
5777:
5774:
5772:
5769:
5768:
5766:
5762:
5758:
5751:
5746:
5744:
5739:
5737:
5732:
5731:
5728:
5716:
5713:
5711:
5708:
5706:
5703:
5701:
5698:
5696:
5693:
5691:
5688:
5686:
5683:
5681:
5678:
5676:
5673:
5671:
5668:
5666:
5665:Hash calendar
5663:
5661:
5658:
5656:
5653:
5651:
5648:
5646:
5643:
5641:
5638:
5636:
5633:
5632:
5630:
5626:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5582:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5554:
5552:
5546:
5544:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5514:
5512:
5509:
5505:
5499:
5496:
5494:
5491:
5489:
5486:
5484:
5481:
5479:
5476:
5474:
5471:
5469:
5466:
5464:
5461:
5460:
5458:
5456:
5452:
5446:
5443:
5441:
5440:van Emde Boas
5438:
5436:
5433:
5431:
5430:Skew binomial
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5409:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5389:
5387:
5385:
5381:
5375:
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5355:
5352:
5350:
5347:
5345:
5341:
5337:
5335:
5332:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5309:Binary search
5306:
5302:
5300:
5297:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5275:
5272:
5270:
5267:
5265:
5262:
5260:
5257:
5256:
5254:
5251:
5247:
5242:
5238:
5234:
5227:
5222:
5220:
5215:
5213:
5208:
5207:
5204:
5196:
5192:
5185:
5181:
5174:
5170:
5163:
5159:
5152:
5147:
5143:
5142:
5136:
5135:
5134:
5133:
5127:
5124:
5122:
5119:
5117:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5079:
5076:
5073:
5070:
5067:
5065:
5062:
5059:
5056:
5053:
5050:
5049:
5045:
5038:
5034:
5033:Bayer, Rudolf
5030:
5025:
5022:(July 1970),
5021:
5020:McCreight, E.
5017:
5016:Bayer, Rudolf
5013:
5012:
5008:
5001:
4999:0-201-89685-0
4995:
4991:
4987:
4983:
4982:Knuth, Donald
4979:
4974:
4972:0-201-55713-4
4968:
4963:
4962:
4955:
4950:
4948:0-262-03293-7
4944:
4940:
4939:
4934:
4930:
4926:
4922:
4918:
4913:
4909:
4905:
4901:
4896:
4891:
4887:
4883:
4879:
4876:(June 1979).
4875:
4871:
4866:
4862:
4858:
4854:
4850:
4846:
4839:
4835:
4834:McCreight, E.
4831:
4827:
4826:
4822:
4816:
4811:
4808:
4805:
4800:
4797:
4794:
4789:
4786:
4781:
4775:
4772:
4761:
4760:
4755:
4749:
4746:
4730:
4726:
4719:
4712:
4709:
4704:
4700:
4695:
4690:
4686:
4682:
4678:
4671:
4668:
4660:
4653:
4646:
4643:
4631:
4627:
4623:
4619:
4613:
4610:
4598:
4592:
4589:
4573:
4569:
4568:cs.rhodes.edu
4562:
4556:
4553:
4549:
4545:
4540:
4537:
4533:
4527:
4524:
4519:
4513:
4509:
4505:
4501:
4500:
4492:
4490:
4488:
4484:
4476:
4475:
4468:
4465:
4461:
4456:
4453:
4442:
4438:
4433:
4428:
4424:
4420:
4416:
4409:
4406:
4401:
4395:
4391:
4384:
4382:
4380:
4378:
4376:
4374:
4370:
4366:
4361:
4358:
4353:
4347:
4339:
4337:9780136086208
4333:
4329:
4322:
4319:
4315:
4310:
4307:
4303:
4299:
4295:
4291:
4287:
4283:
4278:
4275:
4271:
4266:
4264:
4260:
4256:
4251:
4248:
4244:
4239:
4237:
4233:
4222:on 2014-06-04
4221:
4217:
4213:
4207:
4204:
4199:
4192:
4190:
4186:
4182:
4177:
4174:
4170:
4165:
4163:
4161:
4157:
4153:
4148:
4146:
4144:
4142:
4138:
4133:
4129:
4125:
4121:
4117:
4110:
4103:
4101:
4097:
4091:
4081:
4078:
4074:
4068:
4065:
4058:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4030:
4026:
4024:
4022:
4014:
4012:
4010:
4006:
3998:
3996:
3992:
3989:
3980:
3975:
3973:
3971:
3967:
3963:
3959:
3948:
3945:
3937:
3927:
3923:
3919:
3913:
3912:
3908:
3903:This section
3901:
3897:
3892:
3891:
3885:
3883:
3881:
3877:
3876:DragonFly BSD
3873:
3871:
3868:
3864:
3859:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3828:
3826:
3822:
3818:
3815:
3799:
3791:
3787:
3783:
3779:
3776:
3762:
3742:
3732:
3718:
3710:
3702:
3700:
3697:
3692:
3690:
3684:
3682:
3677:
3666:
3663:
3655:
3645:
3641:
3637:
3631:
3630:
3626:
3621:This section
3619:
3615:
3610:
3609:
3603:
3601:
3594:
3588:
3584:
3583:
3575:
3572:
3571:
3569:
3566:
3563:
3562:
3560:
3555:
3552:
3549:
3548:
3546:
3541:
3538:
3535:
3534:
3532:
3531:
3530:
3527:
3523:
3514:
3508:
3504:
3503:
3502:
3495:
3490:
3487:
3484:
3483:
3479:
3477:
3471:
3468:
3467:
3466:
3463:
3458:restructuring
3456:
3454:
3450:
3449:
3448:
3442:
3440:
3438:
3434:
3430:
3426:
3422:
3416:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3362:
3359:
3356:
3355:
3353:
3350:
3349:
3348:
3340:
3333:
3331:
3329:
3328:Binary search
3325:
3318:
3310:
3307:
3299:
3296:February 2012
3289:
3288:the talk page
3285:
3279:
3277:
3272:This article
3270:
3261:
3260:
3254:
3238:
3234:
3228:
3224:
3221:
3218:
3212:
3207:
3203:
3198:
3194:
3177:
3169:
3168:
3167:
3164:
3151:
3147:
3143:
3139:
3135:
3131:
3127:
3124:
3104:
3081:
3078:
3069:
3066:
3063:
3057:
3052:
3048:
3041:
3024:
3016:
3015:
3014:
3010:
3006:
3001:
2996:
2992:
2987:
2981:
2976:
2970:
2961:
2959:
2953:
2950:
2947:
2944:
2943:
2942:
2936:
2934:
2930:
2926:
2924:
2919:
2917:
2909:
2907:
2905:
2901:
2896:
2893:1,000,000 = 3
2884:
2874:
2870:
2862:
2852:
2848:
2845:
2841:
2839:
2833:
2830:
2822:
2820:
2817:
2815:
2810:
2807:
2804:
2799:
2787:
2774:
2773:binary search
2770:
2762:
2754:
2751:
2743:
2733:
2727:
2725:
2718:
2709:
2708:
2702:
2697:
2693:
2690:
2686:
2677:
2673:
2662:
2658:
2647:
2640:
2636:
2625:
2619:
2616:
2612:
2611:
2610:
2608:
2603:
2595:
2593:
2591:
2587:
2586:internal node
2583:
2579:
2573:
2567:
2565:
2562:
2548:
2545:
2542:
2522:
2502:
2482:
2462:
2459:
2456:
2453:
2444:
2430:
2427:
2407:
2387:
2384:
2381:
2361:
2341:
2321:
2318:
2315:
2312:
2292:
2289:
2280:
2278:
2274:
2258:
2255:
2252:
2232:
2212:
2209:
2189:
2180:
2174:
2167:
2164:
2150:
2143:
2126:
2122:
2118:
2108:
2105:
2104:
2089:
2086:
2083:
2076:
2062:
2059:
2053:
2049:
2045:
2039:
2017:
2013:
2006:
2003:
2000:
1987:
1973:
1966:
1949:
1945:
1941:
1931:
1928:
1927:
1912:
1909:
1906:
1899:
1896:
1882:
1875:
1872:
1869:
1868:
1864:
1861:
1847:
1840:
1837:
1834:
1833:
1827:
1822:
1817:
1812:
1809:
1808:
1805:
1783:
1779:
1775:
1767:
1750:
1746:
1737:
1736:
1718:
1714:
1691:
1687:
1683:
1676:
1659:
1655:
1632:
1628:
1624:
1617:
1616:
1602:
1596:
1593:
1587:
1581:
1575:
1569:
1568:
1562:
1540:
1536:
1532:
1524:
1507:
1503:
1494:
1477:
1473:
1469:
1461:
1444:
1441:
1438:
1434:
1425:
1408:
1404:
1381:
1378:
1375:
1371:
1362:
1345:
1341:
1332:
1331:
1313:
1309:
1286:
1282:
1278:
1271:
1254:
1250:
1227:
1223:
1219:
1212:
1210:do not exist
1195:
1191:
1168:
1165:
1162:
1158:
1135:
1131:
1127:
1120:
1103:
1099:
1076:
1073:
1070:
1066:
1043:
1039:
1035:
1028:
1011:
1007:
984:
981:
978:
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850:
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804:
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771:
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657:
649:
632:
628:
624:
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596:
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582:
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575:
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570:
566:
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549:
545:
540:
533:
531:
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524:
519:
517:
512:
507:
501:
499:
496:
494:
490:
486:
478:
474:
469:
466:
465:
461:
457:
455:The root node
454:
450:
446:
442:
438:
434:
430:
426:
422:
418:
414:
410:
406:
402:
399:
398:
397:
392:
385:
378:
371:
364:
357:
349:
345:
341:
338:
335:
333:/2⌉ children.
332:
328:
325:
321:
320:
319:
317:
313:
310:According to
305:
303:
301:
297:
293:
289:
285:
281:
277:
272:
270:
269:
264:
260:
256:
252:
244:
242:
240:
236:
232:
228:
224:
220:
216:
212:
208:
204:
190:
186:
183:
179:
175:
172:
168:
164:
160:
157:
153:
149:
145:
141:
138:
134:
130:
126:
122:
119:
115:
111:
108:
105:
103:
100:
96:
93:
89:
85:
81:
78:
74:
71:
67:
63:
59:
56:
53:
51:
47:
42:
37:
33:
19:
5955:
5855:Disjoint-set
5550:
5542:
5407:
5340:Left-leaning
5283:
5246:dynamic sets
5241:Search trees
5140:
5132:Bulk loading
5131:
5130:
5036:
5023:
4985:
4960:
4936:
4885:
4881:
4848:
4844:
4810:
4799:
4788:
4774:
4763:. Retrieved
4757:
4748:
4736:. Retrieved
4729:the original
4724:
4711:
4684:
4680:
4670:
4645:
4634:. Retrieved
4612:
4601:. Retrieved
4591:
4579:. Retrieved
4567:
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4539:
4526:
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4473:
4467:
4455:
4444:. Retrieved
4422:
4418:
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4389:
4360:
4327:
4321:
4309:
4297:
4293:
4289:
4285:
4277:
4250:
4224:. Retrieved
4220:the original
4215:
4206:
4176:
4115:
4080:
4067:
4021:Linux kernel
4018:
4002:
3993:
3984:
3961:
3955:
3940:
3931:
3916:Please help
3904:
3874:
3870:file systems
3860:
3829:
3819:
3780:
3777:
3733:
3706:
3695:
3693:
3688:
3685:
3675:
3673:
3658:
3649:
3634:Please help
3622:
3598:
3586:
3525:
3521:
3518:
3499:
3475:
3464:
3461:
3452:
3446:
3436:
3432:
3431:rather than
3428:
3424:
3420:
3417:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
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3376:
3372:
3369:
3345:
3326:
3322:
3302:
3293:
3282:Please help
3273:
3165:
3096:
3008:
3004:
2999:
2997:
2990:
2985:
2979:
2968:
2965:
2957:
2940:
2931:
2927:
2920:
2913:
2897:
2882:
2872:
2868:
2860:
2853:
2849:
2846:
2842:
2838:sparse index
2834:
2826:
2818:
2813:
2811:
2808:
2800:
2785:
2766:
2746:
2737:
2721:
2675:
2671:
2660:
2656:
2645:
2638:
2634:
2628:keys, where
2623:
2601:
2599:
2576:data are in
2574:
2571:
2563:
2445:
2281:
2181:
2178:
1803:
1560:
1091:exists, and
796:
699:
578:
573:
568:
564:
560:
558:
528:
522:
520:
515:
510:
508:
505:
497:
492:
488:
484:
482:
476:
472:
459:
448:
444:
440:
436:
432:
428:
424:
420:
416:
412:
408:
390:
383:
376:
369:
362:
355:
353:
347:
343:
330:
323:
315:
309:
299:
295:
291:
287:
283:
279:
278:stands for;
275:
273:
266:
262:
251:Rudolf Bayer
248:
239:file systems
206:
200:
188:
181:
162:
155:
143:
136:
124:
117:
106:
101:
73:Rudolf Bayer
5998:Binary heap
5923:Linked list
5640:Exponential
5628:Other trees
3886:Performance
3790:linked list
2582:disk drives
452:properties.
405:inner nodes
69:Invented by
32:Binary tree
6100:Categories
5986:Splay tree
5890:Hash table
5771:Collection
5584:Priority R
5334:Palindrome
4765:2014-01-27
4738:21 October
4636:2008-04-18
4603:2011-01-17
4544:Comer 1979
4446:2021-08-29
4365:Knuth 1998
4243:Knuth 1998
4226:2011-01-16
4181:Comer 1979
4152:Comer 1979
4092:References
4053:2–3–4 tree
4015:Maple tree
3976:Variations
3676:pre-sorted
3652:April 2018
3506:separator.
3278:to readers
3255:Algorithms
2900:disk cache
2590:disk block
2106:Leaf node
1828:Max number
1823:Min number
1818:Max number
1813:Min number
1810:Node type
1798:is empty.
1555:is empty.
1492:is empty.
548:Knuth 1998
542:A B-tree (
467:Leaf nodes
306:Definition
233:, such as
107:Worst case
6042:Hash tree
5928:Skip list
5875:Bit array
5776:Container
5670:iDistance
5549:implicit
5537:Hilbert R
5532:Cartesian
5415:Fibonacci
5349:Scapegoat
5344:Red–black
5116:Pat Morin
4904:0360-0300
4830:Bayer, R.
4441:203144646
4346:cite book
3958:skip list
3905:does not
3681:algorithm
3623:does not
3334:Insertion
3213:
3079:−
3076:⌉
3058:
3045:⌈
2827:A B-tree
2803:seek time
2600:The term
2546:−
2385:−
2130:⌉
2116:⌈
2057:⌋
2043:⌊
2040:≡
2021:⌉
1995:⌈
1953:⌋
1939:⌊
1442:−
1379:−
1166:−
1074:−
982:−
775:≥
747:−
521:The term
326:children.
235:databases
98:Operation
5971:AVL tree
5850:Multiset
5799:Multimap
5786:Abstract
5685:Link/cut
5397:Binomial
5324:Interval
5184:Archived
5162:Archived
5078:Archived
4984:(1998).
4935:(2001).
4865:29859053
4836:(1972).
4725:dtic.mil
4703:10756181
4659:Archived
4630:Archived
4572:Archived
4292:) where
4132:26930249
4048:2–3 tree
4027:See also
3934:May 2020
3848:Bcachefs
3689:unevenly
3687:them as
3522:rotation
3443:Deletion
3391:−2)/2 =
3235:⌋
3199:⌊
3148:⌉
3132:⌈
2916:database
2895:reads).
2740:May 2022
2596:Variants
2580:such as
2225:, where
1820:of keys
1815:of keys
565:internal
284:balanced
61:Invented
6025:R+ tree
6020:R* tree
5966:AA tree
5645:Fenwick
5609:Segment
5508:Spatial
5425:Pairing
5420:Leftist
5342:)
5314:Dancing
5307:)
5305:Optimal
4823:Sources
4288:,
4073:sectors
4033:B+ tree
3926:removed
3911:sources
3867:Reiser4
3821:TOPS-20
3644:removed
3629:sources
3274:may be
2914:If the
2689:B+ tree
2615:B+ tree
2613:In the
2607:B+ tree
1674:exists
1269:exists
934:exists
427:−1 and
417:minimum
409:maximum
288:between
245:History
102:Average
36:B+ tree
18:B*-tree
6106:B-tree
6054:Graphs
6015:R-tree
5956:B-tree
5910:Linked
5867:Arrays
5695:Merkle
5660:Fusion
5650:Finger
5574:Octree
5564:Metric
5498:Y-fast
5493:X-fast
5483:Suffix
5402:Brodal
5392:Binary
4996:
4969:
4945:
4912:101673
4910:
4902:
4863:
4759:GitHub
4701:
4581:24 May
4514:
4439:
4396:
4334:
4130:
4038:R-tree
3968:, for
3966:T-tree
3880:HAMMER
3782:MS-DOS
3526:merged
3364:tree).
3319:Search
2995:keys.
2678:+ 1)/2
2602:B-tree
2443:keys.
2273:degree
1768:Here,
1525:Here,
1462:Here,
1026:exist
298:, and
280:Boeing
207:B-tree
161:O(log
154:O(log
151:Delete
142:O(log
135:O(log
132:Insert
123:O(log
116:O(log
113:Search
44:B-tree
5948:Trees
5821:Queue
5816:Stack
5764:Types
5705:Range
5675:K-ary
5635:Cover
5478:Radix
5463:Ctrie
5455:Tries
5384:Heaps
5364:Treap
5354:Splay
5319:HTree
5274:(a,b)
5264:2–3–4
5187:(PDF)
5176:(PDF)
5165:(PDF)
5154:(PDF)
4908:S2CID
4861:S2CID
4841:(PDF)
4732:(PDF)
4721:(PDF)
4699:S2CID
4662:(PDF)
4655:(PDF)
4575:(PDF)
4564:(PDF)
4478:(PDF)
4437:S2CID
4128:S2CID
4112:(PDF)
4059:Notes
3852:Btrfs
3844:Linux
3825:TENEX
3814:FAT12
3510:node.
2885:= 100
2829:index
2814:block
2792:⌈ log
2781:⌈ log
2637:<
1706:when
1647:when
1301:when
1242:when
1150:when
1058:when
966:when
907:when
516:order
511:order
431:−1).
312:Knuth
300:Bayer
296:bushy
292:broad
223:nodes
177:Space
6037:Trie
5993:Heap
5811:List
5710:SPQR
5589:Quad
5517:Ball
5473:Hash
5445:Weak
5435:Skew
5410:-ary
4994:ISBN
4967:ISBN
4943:ISBN
4900:ISSN
4740:2022
4583:2022
4512:ISBN
4394:ISBN
4352:link
4332:ISBN
3988:ISAM
3964:. A
3909:any
3907:cite
3865:and
3856:ext4
3854:and
3840:NTFS
3836:APFS
3834:and
3832:HFS+
3696:half
3627:any
3625:cite
3587:Note
3097:Let
2971:≥ –1
2966:Let
2771:. A
2202:and
1594:...
1183:and
999:and
848:...
569:leaf
561:root
523:leaf
447:and
439:or 2
382:and
361:and
253:and
237:and
205:, a
64:1970
50:Type
5845:Set
5715:Top
5569:MVP
5527:BSP
5279:AVL
5259:2–3
4890:doi
4853:doi
4689:doi
4427:doi
4120:doi
3920:by
3878:'s
3863:HFS
3638:by
3435:= 2
3427:= 2
3204:log
3049:log
2982:≥ 0
2891:100
2889:log
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2856:log
2663:+ 1
2648:- 1
2641:− 1
2626:− 1
2275:or
1605:K-1
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3453:OR
3403:=2
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1423:.
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2190:d
2151:K
2127:2
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2084:K
2063:1
2060:+
2054:2
2050:/
2046:K
2018:2
2014:/
2010:)
2007:1
2004:+
2001:K
1998:(
1974:K
1950:2
1946:/
1942:K
1913:1
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1907:K
1883:K
1848:K
1784:i
1780:r
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