Can anyone explain why databases tend to use b-tree indexes rather than a linked list of ordered elements.
My thinking is this: On a B+ Tree (used by most databases), the none-leaf nodes are a collection of pointers to other nodes. Each collection (node) is a ordered list. The leaf nodes, which is where all the data pointers are, is a linked list of clusters of data pointers.
The non-leaf nodes are just used to find the correct leaf node in which your target data pointer lives. So as the leaf nodes are just like a linked list, then why not just do away with the tree elements and just have the linked list. Meta data can be provided which gives the minimum and maximum value of each leaf node cluster, so the application can just read the meta data and find the correct leaf where the data pointer lives.
Just to be clear that the most efficent algorithm for searching an random accessed ordered list is an binary search which has a performance of O(log n) which is the same as a b-tree. The benifit of using a linked list rather than a tree is that they don't need to be ballanced.
Is this structure feasible.
After some research and paper reading I found the answer.
In order to cope with large amounts of data such a millions of records, indexes have to be organised into clusters. A cluster is a continuous group of sectors on a disk that can be read into memory quickly. These are usually about 4096 bytes long.
Each one of these clusters can contain a bunch of indexes which can point to other clusters or data on a disk. So if we had a linked list index, each element of the index would be made up of the collection of indexes contained in a single cluster (say 100).
So, when we are looking for a specific record, how do we know which cluster it is on. We perform a binary search to find the cluster in question [O(log n)].
However, to do a binary search we need to know where the range of values in each clusters, so we need meta-data that says the min and max value of each cluster and where that cluster is. This is great. Except if each cluster can contain 100 indexes, and our meta data is also held on a single cluster (for speed) , then our meta data can only point to 100 clusters.
What happens if we want more than 100 clusters. We have to have two meta-data indexes, each pointing to 100 clusters (10 000 records). Well that’s not enough. Lets add another meta-data cluster and we can now access 1 000 000 records. So how do we know which one of the three meta-data clusters we need to query in order to find our target data cluster. We could search one then the other, but that doesn’t scale. So I add another meta-meta-data cluster to indicate which one of the three meta-data clusters I should query to find the target data cluster. Now I have a tree!
So that’s why databases use trees. It’s not the speed it’s the size of the indexes and the need to have indexes referencing other indexes. What I have described above is a B+Tree – child nodes contain references to other child nodes or leaf nodes, and leaf nodes contain references to data on disk.
Phew!
I guess I answered that question in Chapter 1 of my SQL Indexing Tutorial: http://use-the-index-luke.com/sql/anatomy
To summarize the most important parts, with respect to your particular question:
-- from "The Leaf Nodes"
The primary purpose of an index is to provide an ordered
representation of the indexed data. It is, however, not possible to
store the data sequentially because an insert statement would need to
move the following entries to make room for the new one. But moving
large amounts of data is very time-consuming, so that the insert
statement would be very slow. The problem's solution is to establish a
logical order that is independent of physical order in memory.
-- from "The B-Tree":
The index leaf nodes are stored in an arbitrary order—the position on
the disk does not correspond to the logical position according to the
index order. It is like a telephone directory with shuffled pages. If
you search for “Smith” in but open it at “Robinson” in the first
place, it is by no means granted that Smith comes farther back.
Databases need a second structure to quickly find the entry among the
shuffled pages: a balanced search tree—in short: B-Tree.
Linked lists are usually not ordered by key value, but by the moment of insertion: insertion is done at the end of list and each new entry contains a pointer to the previous entry of the list.
They are usually implemented as heap structures.
This has 2 main benefits:
they are very easy to manage (you just need a pointer for each element)
if used in combination with an index you can overcome the problem of sequential access.
If instead you use an ordered list, by key value, you will have ease of access (binary search), but encounter problems each time you edit, delete, insert a new element: you must infact keep your list ordered after performing operation, making algorithms more complex and time consuming.
B+ trees are better structures, having all the properties you stated, and other advantages:
you can make group searches (by intervals of key values) with same cost of a single search: since elements in the leafs result automatically ordered thanks to the insertion algorithm, which is not possible in linked lists cause it would require many linear searches over the list.
cost is logarithmic with number of elements contained and especially since these structures are kept balanced cost of access does not depend on the particulare value you are looking for (very usefull).
these structures are very efficient in update, insert or delete operations.
Related
I think it will give you a better understanding about where I'm coming from by letting you know how I understand how Btree indices work fundamentally. I'm not a DBA and I'm asking this question as a layman with basic understanding of data structures.
The basic idea of an index is that it speeds up searches by skipping significant amount of records when searching through a database.
AFAIK, binary tree data structure, which I presume where Btree indices are based on, helps us to search without scanning the entire database by dividing the data into nodes. For oversimplified example, words that start from A to M are stored in left node, and words that start with N to Z are stored in right node on the first level of the tree. In this case when we search for the word "Jackfruit" it will only search on the left node skipping the right node saving us significant amount of time and IO.
In this sense, how does a bitmap index let us not scan the entire database when searching? If not, how does it speed up searches? Or is it just meant for compression?
Image taken from here
The image above is a conceptual illustration of a bitmap. Using that structure, how does a DB find rows? Does it scan all rows? In binary tree, that fact that you don't have to scan everything is exactly how it helps speed up the search. I can't see any explanation how exactly a DB gets an advantage in searching for rows using bitmap other than the fact that bitmap takes less space.
Btree indexes are good for key searching (duplicates allowed, but mainly distinct values in the column, ie. SSN). Bitmap indexes are better in cases when you have a few distinct values like 'sex', 'state', 'color', and so on.
Oracle bitmap indexes are very different from standard b-tree indexes. In bitmap structures, a two-dimensional array is created with one column for every row in the table being indexed. Each column represents a distinct value within the bitmapped index. This two-dimensional array represents each value within the index multiplied by the number of rows in the table.
Please see http://www.dba-oracle.com/oracle_tips_bitmapped_indexes.htm for a more detailed explanation.
In a B tree and B+tree , If we specify the order as 5 then we can store the 4 keys in a single node and 5 pointers for that node.
It has any limit for setting the order in the above trees (or) its limit is infinite ?
You can design a system with any order you choose from 1 upwards. If you make the order too big, it becomes difficult to find the key in the node, and the tree will be just 1 or 2 levels deep.
For example, if the order is 1,000,000, then you'd need getting on for a trillion records before you split any nodes to the third level in the tree, and you'd probably never get to the fourth level. And you'd have to search through a million keys at each level to find where to go. Even with a binary search, that's up to 20 probes.
If you choose a smaller order, then your searches are smaller. For example, if the order is 32, you have at most 5 searches per level with a binary search to find the key and where to go next. Against this, each time you move down a level, you have to read a new page from disk (if it is a disk-backed B-tree). If it's in-memory, there's very little cost to that.
Often, you design the B-tree with a fixed page size, and tune the order based on the size of the keys and the size of the pointers. Big keys give you a smaller order; small keys give you a bigger order.
I was debating between using BTREE index or HASH index.
Theoretically, what are the advantages of using HASH indexes?
When should they be chosen and more importantly, why?
I have read that hash indexes are good for point queries, but WHY?
I already know that BTREE indexes are best for range queries because you can easily traverse through the leaf nodes by going from left to right.
You don't mention a specific DBMS so this answer is pretty generic.
A properly performing hash index should reach the answer to a point query in a single fetch. A B-Tree will use something like lg_B(n) secondary storage accesses where B is the approximate branch factor and n is the number of entries. Caching and reasonable node sizes will likely keep that to a couple of fetches but still twice that for the hash index. In addition, each B-Tree access has non-trivial computations associated with it in order to traverse the sub-index in each node (something like lg_2(B) data comparison operations per node). The computation time for a hash index is usually very limited (a hash computation and a small number of data comparison operations - hopefully one). The computation time for searching within each node is often significant for B-Tree based indices.
In terms of picking, use a hash index if
you only expect point queries
you don't expect the data to fall into any poorly performing cases for the system hash function (oddball case but thought I should mention it)
B-Tree family are better if you have any kind of range query and/or want sorted results on a pre-determinable set of columns.
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.
Could any one give an explanation on how a DHT works?
Nothing too heavy, just the basics.
Ok, they're fundamentally a pretty simple idea. A DHT gives you a dictionary-like interface, but the nodes are distributed across the network. The trick with DHTs is that the node that gets to store a particular key is found by hashing that key, so in effect your hash-table buckets are now independent nodes in a network.
This gives a lot of fault-tolerance and reliability, and possibly some performance benefit, but it also throws up a lot of headaches. For example, what happens when a node leaves the network, by failing or otherwise? And how do you redistribute keys when a node joins so that the load is roughly balanced. Come to think of it, how do you evenly distribute keys anyhow? And when a node joins, how do you avoid rehashing everything? (Remember you'd have to do this in a normal hash table if you increase the number of buckets).
One example DHT that tackles some of these problems is a logical ring of n nodes, each taking responsibility for 1/n of the keyspace. Once you add a node to the network, it finds a place on the ring to sit between two other nodes, and takes responsibility for some of the keys in its sibling nodes. The beauty of this approach is that none of the other nodes in the ring are affected; only the two sibling nodes have to redistribute keys.
For example, say in a three node ring the first node has keys 0-10, the second 11-20 and the third 21-30. If a fourth node comes along and inserts itself between nodes 3 and 0 (remember, they're in a ring), it can take responsibility for say half of 3's keyspace, so now it deals with 26-30 and node 3 deals with 21-25.
There are many other overlay structures such as this that use content-based routing to find the right node on which to store a key. Locating a key in a ring requires searching round the ring one node at a time (unless you keep a local look-up table, problematic in a DHT of thousands of nodes), which is O(n)-hop routing. Other structures - including augmented rings - guarantee O(log n)-hop routing, and some claim to O(1)-hop routing at the cost of more maintenance.
Read the wikipedia page, and if you really want to know in a bit of depth, check out this coursepage at Harvard which has a pretty comprehensive reading list.
DHTs provide the same type of interface to the user as a normal hashtable (look up a value by key), but the data is distributed over an arbitrary number of connected nodes. Wikipedia has a good basic introduction that I would essentially be regurgitating if I write more -
http://en.wikipedia.org/wiki/Distributed_hash_table
I'd like to add onto HenryR's useful answer as I just had an insight into consistent hashing. A normal/naive hash lookup is a function of two variables, one of which is the number of buckets. The beauty of consistent hashing is that we eliminate the number of buckets "n", from the equation.
In naive hashing, first variable is the key of the object to be stored in the table. We'll call the key "x". The second variable is is the number of buckets, "n". So, to determine which bucket/machine the object is stored in, you have to calculate: hash(x) mod(n). Therefore, when you change the number of buckets, you also change the address at which almost every object is stored.
Compare this to consistent hashing. Let's define "R" as the range of a hash function. R is just some constant. In consistent hashing, the address of an object is located at hash(x)/R. Since our lookup is no longer a function of the number of buckets, we end up with less remapping when we change the number of buckets.
http://michaelnielsen.org/blog/consistent-hashing/
The core of a DHT is a hash table. Key-value pairs are stored in DHT and a value can be looked up with a key. The keys are unique identifiers to values that can range from blocks in a blockchain to addresses and to documents.
What differentiates a DHT from a normal hash table is the fact that storage and lookup on DHT are distributed across multiple (can be millions) nodes or machines. This very characteristic of DHT makes it look like distributed databases used for storage and retrieval. There is no master-slave hierarchy or a centralized control among the participating nodes. All the nodes are treated as peers.
DHT provides freedom to the participating nodes such that the nodes can join or leave the network anytime. Due to this reason, DHTs are widely used in Peer-to-Peer (P2P) networks. In fact, part of the motivation behind the research of DHT stems from its usage in P2P networks.
Characteristics of DHT
Decentralized: Since there is no central authority or coordination
Scalable: The system can easily scale up to millions of nodes
Fault-tolerant: DHT replicates the data storage on all the nodes.
Therefore, even if one node leaves the network, it should not affect other nodes in the network.
Let’s see how lookup happens in a popular DHT protocol like Chord. Consider a circular doubly-linked list of nodes. Each node has a reference pointer to the node previous as well as next to it. The node next to the node in question is called the successor. The node that is previous to the node in question is called the predecessor.
Speaking in terms of a DHT, each node has a unique node ID of k bits and these nodes are arranged in the increasing order of their node IDs.
Assume these nodes are arranged in a ring structure called identifier ring. For each node, the successor has the shortest distance clockwise away. For most nodes, this is the node whose ID is closest to but still greater than the current node’s ID.
To find out the node appropriate for a particular key, first hash the key K and all the nodes to exactly k bits using consistent hashing techniques like SHA-1.
Start at any point in the ring and traverse clockwise till you catch the node whose node ID is closer to the key K, but can be greater than K. This node is the one responsible for storage and lookup for that particular key.
In an iterative style of lookup, each node Q queries its successor node for KV (key-value) pair. If the queried node does not have the target key, it will return a set of nodes S that can be closer to the target. The querying node Q then queries the nodes in S which are closer to itself. This continues until either the target KV pair is returned or when there are no more nodes to query.
This lookup is very suitable for an ideal scenario where all the nodes have a perfect uptime. But how to handle scenarios when nodes leave the network either intentionally or by failure? This calls for the need for a robust join/leave protocol.
Popular DHT protocols and implementations
Chord
Kademlia
Apache Cassandra
Koorde TomP2P
Voldemort
References:
https://en.wikipedia.org/wiki/Distributed_hash_table
https://steffikj19.medium.com/dht-demystified-77dd31727ea7
https://www.linuxjournal.com/article/6797