I heard some statements like, consider the height of the AVL tree and the maximum keys that an AVL tree node can contain, the search of AVL tree will be time-consuming because of the disk io.
However, imagine that an index file contains the whole AVL tree structure, and then the size of the index file is less than a fan size, we can just read the whole AVL tree in only once disk io.
It seems like using AVL tree does not bring about extra disk io, how do you explain B tree is better?
Databases use balanced binary trees(plus) avl is only a special case of these balanced trees, so there is no need for it
we can just read the whole AVL tree in only once disk io
Yes, it could work like that. Essentially, the whole data structure would be brought into memory. IO would no longer be a concern.
Some databases use this strategy. For example, SQL Server In-Memory "Hekaton" does this and delivers ~100x the normal throughput for OLTP.
Hekaton uses two index data structues: hash tables and trees. I think the trees are called cw-trees and are similar to b-trees.
For general purpose database workloads it is very desirable to not need everything in memory. B-trees are a great design tradeoff in those cases.
Its coz B-Trees usually have larger number of keys in single node and hence reducing the depth of the search, in record indexing the link traversal time is longer if the depth is more, hence for cache locality and making the tree wider than deeper, multiple keys are stored in array of a node which improves cache performance and quick lookup comparatively.
I know that indexes are implemented using B-Tree. I have read the Microsoft documentation on spatial indexes. It seems that they implement spatial data using B tree as well.
But why is a grid required or how does grid hierarchy work or how does SQL Server search using spatial data values? All that stuff is still not clear to me.
It would be really helpful if anyone please explain it.
Thanks :-)
As you’re probably aware, the standard index in SQL Server uses a B+ tree structure, which is a variation of the B-tree index. B-tree is nothing but a data structure that keeps data sorted to support search operations, sequential access, and data modifications such as inserts and deletes.
A B-tree index contains at least two levels: the root and the leaf. The root is the top most node and can have child nodes. If there are no child nodes, then the tree is called a Null tree. If there are child nodes, they can be either leaf nodes or intermediate nodes. A leaf node is the bottom part of the tree. Intermediate levels can exist between the root and leaf levels. The difference between a B-tree index and a B+ tree index is that all records are stored only at the leaf level for B+ tree, whereas in a B-tree we can store both keys and data in the intermediate nodes.
SQL Server spatial indexes are built on top of the B+ tree structure, which allows the indexes to use that structure and its access methods. The spatial indexes also use the fundamental principles of XML indexing. XML indexing was introduced in SQL Server 2005 and supports two basic types of indexes: primary and secondary. The primary XML index is a B+ tree that essentially contains one row for each node in the XML instance.
So how does SQL Server implement the spatial index? As already mentioned, SQL Server starts with a B+ tree structure, which organizes data into a linear fashion. Because of this, the indexes must have a way to represent the two-dimensional spatial information as linear data. For this, SQL Server uses a process referred to as the hierarchical uniform decomposition of space. When the index is created, the database engine decomposes, or refactors, the space into a collection of axes aligned along a four-level grid hierarchy. Figure 1 provides an overview of what this process looks like.
Taken from
https://www.red-gate.com/simple-talk/sql/t-sql-programming/sql-server-spatial-indexes/
As per my understanding,
Binary heap(data structure) is used to represent Priority queue ADT. It is a complete binary tree satisfying heap property.
Heap property - If A is a parent node of B then the key (the value) of node A is ordered with respect to the key of node B with the same ordering applying across the heap.
Firstly, it helps me remember term heap, if there is a reason behind terming this data structure as heap. Because, we also use the term heap memory.
Dictionary meaning of heap - an untidy collection of things piled up haphazardly.
Question,
After learning Reb-Black tree & AVL tree data structure,
Why do we think of new data structure(Binary heap)?
Does Binary Heap solve set of problems that Red-Black or AVL tree does not fit into?
The major difference between a binary heap and a red-black tree is the performance on certain operations.
Binary Heap
Pros
It makes an ideal priority queue, since the min/max element (depending on your implementation) is always O(1) access time, so no need to search for it.
It's also very fast for insertion of new values (O(1) on average, O(log(n)) worst case.
Cons
Slow searches for random elements.
RB Tree
Pros
Better searching and insertion performance.
Cons
Slower min/max searches.
More overhead in general.
It should be noted that RB trees can make good schedulers too, such as the Completely Fair Scheduler introduced in Linux kernel v2.6.
I am trying to figure out how to insert an item into a B+ tree using locks and don't really understand the theory behind it.
So for searching, my view is that I put a lock on the root node, and then decide which child node I should go to and lock it, at this point I can release the parent node and continue this operation until I reach the leaf node.
But inserting is a lot more complicated because I can't allow any other threads to interfere with the insertion. My idea is put a lock on each node along the path to the leaf node but putting that many locks is quite expensive, and then the question I have is what happens when the leaf node splits because it is too large?
Does anyone know how to properly insert an item into a B+ tree using locks?
There are many different strategies for dealing with locking in B-Trees in general; most of these actually deal with B+Trees and its variations since they have been dominating the field for decades. Summarising these strategies would be tantamount to summarising the progress of four decades; it's virtually impossible. Here are some highlights.
One strategy for minimising the amount of locking during initial descent is to lock not the whole path starting from the root, but only the sub-path beginning at the last 'stable' node (i.e. a node that won't split or merge as a result of the currently planned operation).
Another strategy is to assume that no split or merge will happen, which is true most of the time anyway. This means the descent can be done by locking only the current node and the child node one will descend into next, then release the lock on the previously 'current' node and so on. If it turns out that a split or merge is necessary after all then re-descend from the root under a heavier locking regime (i.e. path rooted at last stable node).
Another staple in the bag of tricks is to ensure that each node 'descended through' is stable by preventative splitting/merging; that is, when the current node would split or merge under a change bubbling up from below then it gets split/merged right away before continuing the descent. This can simplify operations (including locking) and it is somewhat popular in reinventions of the wheel - homework assignments and 'me too' implementations, rather than sophisticated production-grade systems.
Some strategies allow most normal operations to be performed without any locking at all but usually they require that the standard B+Tree structure be slightly modified; see B-link trees for example. This means that different concurrent threads operating on the tree can 'see' different physical views of this tree - depending on when they got where and followed which link - but they all see the same logical view.
Seminal papers and good overviews:
Efficient Locking for Concurrent Operations on B-Trees (Lehman/Yao 1981)
Concurrent Operations on B*-Trees with Overtaking (Sagiv 1986)
A survey of B-tree locking techniques (Graefe 2010)
B+Tree Locking (slides from Stanford U, including Blink trees)
A Blink Tree method and latch protocol for synchronous deletion in a high concurreny environment (Malbrain 2010)
A Lock-Free B+Tree (Braginsky/Petrank 2012)
In a b-tree you can store both keys and data in the internal and leaf nodes, but in a b+ tree you have to store the data in the leaf nodes only.
Is there any advantage of doing the above in a b+ tree?
Why not use b-trees instead of b+ trees everywhere, as intuitively they seem much faster?
I mean, why do you need to replicate the key (data) in a b+ tree?
The image below helps show the differences between B+ trees and B trees.
Advantages of B+ trees:
Because B+ trees don't have data associated with interior nodes, more keys can fit on a page of memory. Therefore, it will require fewer cache misses in order to access data that is on a leaf node.
The leaf nodes of B+ trees are linked, so doing a full scan of all objects in a tree requires just one linear pass through all the leaf nodes. A B tree, on the other hand, would require a traversal of every level in the tree. This full-tree traversal will likely involve more cache misses than the linear traversal of B+ leaves.
Advantage of B trees:
Because B trees contain data with each key, frequently accessed nodes can lie closer to the root, and therefore can be accessed more quickly.
The principal advantage of B+ trees over B trees is they allow you to pack in more pointers to other nodes by removing pointers to data, thus increasing the fanout and potentially decreasing the depth of the tree.
The disadvantage is that there are no early outs when you might have found a match in an internal node. But since both data structures have huge fanouts, the vast majority of your matches will be on leaf nodes anyway, making on average the B+ tree more efficient.
B+Trees are much easier and higher performing to do a full scan, as in look at every piece of data that the tree indexes, since the terminal nodes form a linked list. To do a full scan with a B-Tree you need to do a full tree traversal to find all the data.
B-Trees on the other hand can be faster when you do a seek (looking for a specific piece of data by key) especially when the tree resides in RAM or other non-block storage. Since you can elevate commonly used nodes in the tree there are less comparisons required to get to the data.
In a B tree search keys and data are stored in internal or leaf nodes. But in a B+-tree data is stored only in leaf nodes.
Full scan of a B+ tree is very easy because all data are found in leaf nodes. Full scan of a B tree requires a full traversal.
In a B tree, data may be found in leaf nodes or internal nodes. Deletion of internal nodes is very complicated. In a B+ tree, data is only found in leaf nodes. Deletion of leaf nodes is easy.
Insertion in B tree is more complicated than B+ tree.
B+ trees store redundant search keys but B tree has no redundant value.
In a B+ tree, leaf node data is ordered as a sequential linked list but in a B tree the leaf node cannot be stored using a linked list. Many database systems' implementations prefer the structural simplicity of a B+ tree.
Example from Database system concepts 5th
B+-tree
corresponding B-tree
Adegoke A, Amit
I guess one crucial point you people are missing is difference between data and pointers as explained in this section.
Pointer : pointer to other nodes.
Data :- In context of database indexes, data is just another pointer to real data (row) which reside somewhere else.
Hence in case of B tree each node has three information keys, pointers to data associated with the keys and pointer to child nodes.
In B+ tree internal node keep keys and pointers to child node while leaf node keep keys and pointers to associated data. This allows more number of key for a given size of node. Size of node is determined mainly by block size.
Advantage of having more key per node is explained well above so I will save my typing effort.
B+ Trees are especially good in block-based storage (eg: hard disk). with this in mind, you get several advantages, for example (from the top of my head):
high fanout / low depth: that means you have to get less blocks to get to the data. with data intermingled with the pointers, each read gets less pointers, so you need more seeks to get to the data
simple and consistent block storage: an inner node has N pointers, nothing else, a leaf node has data, nothing else. that makes it easy to parse, debug and even reconstruct.
high key density means the top nodes are almost certainly on cache, in many cases all inner nodes get quickly cached, so only the data access has to go to disk.
Define "much faster". Asymptotically they're about the same. The differences lie in how they make use of secondary storage. The Wikipedia articles on B-trees and B+trees look pretty trustworthy.
In B+ Tree, since only pointers are stored in the internal nodes, their size becomes significantly smaller than the internal nodes of B tree (which store both data+key).
Hence, the indexes of the B+ tree can be fetched from the external storage in a single disk read, processed to find the location of the target. If it has been a B tree, a disk read is required for each and every decision making process. Hope I made my point clear! :)
**
The major drawback of B-Tree is the difficulty of Traversing the keys
sequentially. The B+ Tree retains the rapid random access property of
the B-Tree while also allowing rapid sequential access
**
ref: Data Structures Using C// Author: Aaro M Tenenbaum
http://books.google.co.in/books?id=X0Cd1Pr2W0gC&pg=PA456&lpg=PA456&dq=drawback+of+B-Tree+is+the+difficulty+of+Traversing+the+keys+sequentially&source=bl&ots=pGcPQSEJMS&sig=F9MY7zEXYAMVKl_Sg4W-0LTRor8&hl=en&sa=X&ei=nD5AUbeeH4zwrQe12oCYAQ&ved=0CDsQ6AEwAg#v=onepage&q=drawback%20of%20B-Tree%20is%20the%20difficulty%20of%20Traversing%20the%20keys%20sequentially&f=false
The primary distinction between B-tree and B+tree is that B-tree eliminates the redundant storage of search key values.Since search keys are not repeated in the B-tree,we may not be able to store the index using fewer tree nodes than in corresponding B+tree index.However,since search key that appear in non-leaf nodes appear nowhere else in B-tree,we are forced to include an additional pointer field for each search key in a non-leaf node.
Their are space advantages for B-tree, as repetition does not occur and can be used for large indices.
Take one example - you have a table with huge data per row. That means every instance of the object is Big.
If you use B tree here then most of the time is spent scanning the pages with data - which is of no use. In databases that is the reason of using B+ Trees to avoid scanning object data.
B+ Trees separate keys from data.
But if your data size is less then you can store them with key which is what B tree does.
A B+tree is a balanced tree in which every path from the root of the tree to a leaf is of the same length, and each nonleaf node of the tree has between [n/2] and [n] children, where n is fixed for a particular tree. It contains index pages and data pages.
Binary trees only have two children per parent node, B+ trees can have a variable number of children for each parent node
One possible use of B+ trees is that it is suitable for situations
where the tree grows so large that it does not fit into available
memory. Thus, you'd generally expect to be doing multiple I/O's.
It does often happen that a B+ tree is used even when it in fact fits into
memory, and then your cache manager might keep it there permanently. But
this is a special case, not the general one, and caching policy is a
separate from B+ tree maintenance as such.
Also, in a B+ tree, the leaf pages are linked together in
a linked list (or doubly-linked list), which optimizes traversals
(for range searches, sorting, etc.). So the number of pointers is
a function of the specific algorithm that is used.