I know this is a common question and I saw a few threads in Stack Overflow but still couldn't get it.
Here is an accepted answer from Stack overflow:
" Disk seeks are expensive. B-Tree structure is designed specifically to
avoid disk seeks as much as possible. Therefore B-Tree packs much more
keys/pointers into a single node than a binary tree. This property
makes the tree very flat. Usually most B-Trees are only 3 or 4 levels
deep and the root node can be easily cached. This requires only 2-3
seeks to find anything in the tree. Leaves are also "packed" this way,
so iterating a tree (e.g. full scan or range scan) is very efficient,
because you read hundreds/thousands data-rows per single block (seek).
In binary tree of the same capacity, you'd have several tens of levels
and sequential visiting every single value would require at least one
seek. "
I understand that B-Tree has more nodes (Order) than a BST. So it's definitely flat and shallow than a BST.
But these nodes are again stored as linked lists right?
I don't understand when they say that the keys are read as a block thereby minimising the no of I/Os.
Isn't the same argument hold good for BSTs too? Except that the links will be downwards?
Please someone explain it to me?
I understand that B-Tree has more nodes (Order) than a BST. So it's definitely flat and shallow than a BST. I don't understand when they say that the keys are read as a block thereby minimising the no of I/Os.
Isn't the same argument hold good for BSTs too? Except that the links will be downwards?
Basically, the idea behind using a B+tree in file systems is to reduce the number of disk reads. Imagine that all the blocks in a drive are stored as a sequentially allocated array. In order to search for a specific block you would have to do a linear scan and it would take O(n) every time to find a block. Right?
Now, imagine that you got smart and decided to use a BST, great! You would store all your blocks in a BST an that would take roughly O(log(n)) to find a block. Remember that every branch is a disk access, which is highly expensive!
But, we can do better! The problem now is that a BST is really "tall". Because every node only has a fanout (number of children) factor of 2, if we had to store N objects, our tree would be in the order of log(N) tall. So we would have to perform at most log(N) access to find our leaves.
The idea behind the B+tree structure is to increase the fanout factor (number of children), reducing the height of tree and, thus, reducing the number of disk access that we have to make in order to find a leave. Remember that every branch is a disk access. For instance, if you pack X keys in a node of a B+tree every node will point to at most X+1 children.
Also, remember that a B+tree is structured in a way that only the leaves store the actual data. That way, you can pack more keys in the internal nodes in order to fill up one disk block, that, for instance, stores one node of a B+tree. The more keys you pack in a node the more children it will point to and the shorter your tree will be, thus reducing the number of disk access in order to find one leave.
But these nodes are again stored as linked lists right?
Also, in a B+tree structure, sometimes the leaves are stored in a linked list fashion. Remember that only the leaves store the actual data. That way, with the linked list idea, when you have to perform a sequential access after finding one block you would do it faster than having to traverse the tree again in order to find the next block, right? The problem is that you still have to find the first block! And for that, the B+tree is way better than the linked list.
Imagine that if all the accesses were sequential and started in the first block of the disk, an array would be better than the linked list, because in a linked list you still have to deal with the pointers.
But, the majority of disk accesses, according to Tanenbaum, are not sequential and are accesses to files of small sizes (like 4KB or less). Imagine the time it would take if you had to traverse a linked list every time to access one block of 4KB...
This article explains it way better than me and uses pictures as well:
https://loveforprogramming.quora.com/Memory-locality-the-magic-of-B-Trees
A B-tree node is essentially an array, of pairs {key, link}, of a fixed size which is read in one chunk, typically some number of disk blocks. The links are all downwards. At the bottom layer the links point to the associated records (assuming a B+-tree, as in any practical implementation).
I don't know where you got the linked list idea from.
Each node in a B-tree implemented in disk storage consists of a disk block (normally a handful of kilobytes) full of keys and "pointers" that are accessed as an array and not - as you said - a linked list. The block size is normally file-system dependent and chosen to use the file system's read and write operations efficiently. The pointers are not normal memory pointers, but rather disk addresses, again chosen to be easily used by the supporting file system.
The main reason for B-tree is how it behaves on changes. If you have permanent structure, BST is OK, but in that case Hash function is even better. In case of file systems, you want a structure which changes as a whole as little as possible on inserts or deletes, and where you can perform find operation with as little reads as possible - these properties have B-trees.
Related
I have multiple robots, which explore an occupancy grid through some algorithm. I am trying to save the order of explored nodes. But I am not sure, which data structure can be used to save them efficiently.
I first thought of an tree, but the order can be repeatable like 1, 2, 5, 1. So, I feel, it may be too complex to store such an order in tree form. Then, I thought of an array, but it can be too much expensive in terms of memory for large grids.
I am a bit confused now. What data structure would be better(suppose grid is of 10,000 nodes). But the point is the order of explored nodes will be greater than 10,000 in this case as there will be overlap.
Thanks!
A tree makes little sense here with a need to preserve insertion order and the need to allow duplicates. Basically, as I understand it, we want to store the path in which the robot has traveled in the tightest form we can.
A compact, contiguous kind of sequence ends up making the most sense here (array, e.g.). It's cheaper than any linked structure (tree included) since there are no links to store.
There's little we can do to compact memory usage any further.
However, an unrolled list might be helpful here. Since it's not one giant contiguous block and instead a series of smaller blocks (ex: 4 kilobytes each) linked together, you can start, say, off-loading blocks at the front of the list to disk if you want to reduce memory use. The link overhead is trivial since we're only storing a link every N elements, where N could be some large number.
So I see that trees are usually implemented as a list where each node is dynamically allocated and each node contains pointers to two of its children.
But a heap is almost always implemented (or so is recommended in text books) using an array. Why is that? Is there some underlying assumption about the uses of these two data strcutures? For e.g. if you are implementing a priority queue using a min heap then the number of nodes in the queue is constant and so it can be implemented using an array of fixed size. But when you are talking/teaching about a heap in general why recommend implemeting it using an array. Or to flip the question a bit why not recommend learnig about trees with an implementation using arrays?
(I assume by heap you mean binary heap; other heaps are almost always linked nodes.)
A binary heap is always a complete tree, and no operation on it moves whole subtrees around or otherwise alters the topology of the tree in any nontrivial way. This is not an assumption, the first is part of the definition of a heap and the second is immediately obvious from the definition of the operations.
First, since the Ahnentafel layout requires reserving space for every internal node (and all leaf nodes except the rightmost ones), an incomplete tree implemented this way would waste space for nodes that don't exist. Conversely, for a complete tree it's the most efficient layout possible, since all space is actually used for node data, and no space is needed for pointers.
Second, moving a subtree in the array would require copying all child elements to their new positions (since the left child's index is always twice the parent's index, the former changes when the latter changes, recursively down to the leafs). When you have nodes linked via pointers, you only need to move a few pointers around regardless of how large the trees below those pointers are. Moving subtrees is a core component of many algorithms of trees, including all kinds of binary search trees. It needs to be lightning fast for those algorithms to be efficient. Binary heap operations however never need to do this so it's a non-issue.
I'm studying B+trees for indexing and I try to understand more than just memorizing the structure. As far as I understand the inner nodes of a B+tree forms an index on the leaves and the leaves contains pointers to where the data is stored on disk. Correct? Then how are lookups made? If a B+tree is so much better than a binary tree, why don't we use B+trees instead of binary trees everywhere?
I read the wikipedia article on B+ trees and I understand the structure but not how an actual lookup is performed. Could you guide me perhaps with some link to reading material?
What are some other uses of B+ trees besides database indexing?
I'm studying B+trees for indexing and I try to understand more than just memorizing the structure. As far as I understand the inner nodes of a B+tree forms an index on the leaves and the leaves contains pointers to where the data is stored on disk. Correct?
No, the index is formed by the inner nodes (non-leaves). Depending on the implementation the leaves may contain either key/value pairs or key/pointer to value pairs. For example, a database index uses the latter, unless it is an IOT (Index Organized Table) in which case the values are inlined in the leaves. This depends mainly on whether the value is insanely large wrt the key.
Then how are lookups made?
In the general case where the root node is not a leaf (it does happen, at first), the root node contains a sorted array of N keys and N+1 pointers. You binary search for the two keys S0 and S1 such that S0 <= K < S1 (where K is what you are looking for) and this gives you the pointer to the next node.
You repeat the process until you (finally) hit a leaf node, which contains a sorted list of key-values pairs and make a last binary search pass on those.
If a B+tree is so much better than a binary tree, why don't we use B+trees instead of binary trees everywhere?
Binary trees are simpler to implement. One though cookie with B+Trees is to size the number of keys/pointers in inner nodes and the number of key/values pairs in leaves nodes. Another though cookie is to decide on the low and high watermark that leads to grouping two nodes or exploding one.
Binary trees also offer memory stability: an element inserted is not moved, at all, in memory. On the other hand, inserting an element in a B+Tree or removing one is likely to lead to elements shuffling
B+Trees are tailored for small keys/large values cases. They also require that keys can be duplicated (hopefully cheaply).
Could you guide me perhaps with some link to reading material?
I hope the rough algorithm I explained helped out, otherwise feel free to ask in the comments.
What are some other uses of B+ trees besides database indexing?
In the same vein: file-system indexing also benefits.
The idea is always the same: a B+Tree is really great with small keys/large values and caching. The idea is to have all the keys (inner nodes) in your fast memory (CPU Cache >> RAM >> Disk), and the B+Tree achieves that for large collections by pushing keys to the bottom. With all inner nodes in the fast memory, you only have one slow memory access at each search (to fetch the value).
B+ trees are better than binary tree all the dbms use them,
a lookup in B+Tree is LOGF N being F the base of LOG and the fan out. The lookup is performed exactly like in a binary tree but with a bigger fan out and lower height thats why it is way better.
B+Tree are usually known for having the data in the leaf(if they are unclustered probably not), this means you dont have to make another jump to the disk to get the data, you just take it from the leaf.
B+Tree is used almost everywhere, Operating Systems use them, datawarehouse (not so much here but still), lots of applications.
B+Tree are perfect for range queries, and are used whenever you have unique values, like a primary key, or any field with low cardinality.
If you can get this book http://www.amazon.com/Database-Management-Systems-Raghu-Ramakrishnan/dp/0072465638 its one of the best. Its basically the bible for any database guy.
I read this on wikipedia:
In B-trees, internal (non-leaf) nodes can have a variable number of
child nodes within some pre-defined range. When data is inserted or
removed from a node, its number of child nodes changes. In order to
maintain the pre-defined range, internal nodes may be joined or split.
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.
We have to specify this range for B trees. Even when I looked up CLRS (Intro to Algorithms), it seemed to make to use of arrays for keys and children. My question is- is there any way to reduce this wastage in space by defining the keys and children as lists instead of predetermined arrays? Is this too much of a hassle?
Also, for the life of me I'm not able to get a decent psedocode on btreeDeleteNode. Any help here is appreciated too.
When you say "lists", do you mean linked lists?
An array of some kind of element takes up one element's worth of memory per slot, whether that slot is filled or not. A linked list only takes up memory for elements it actually contains, but for each one, it takes up one element's worth of memory, plus the size of one pointer (two if it's a doubly-linked list, unless you can use the xor trick to overlap them).
If you are storing pointers, and using a singly-linked list, then each list link is twice the size of each array slot. That means that unless the list is less than half full, a linked list will use more memory, not less.
If you're using a language whose runtime has per-object overhead (like Java, and like C unless you are handling memory allocation yourself), then you will also have to pay for that overhead on each list link, but only once on an array, and the ratio is even worse.
I would suggest that your balancing algorithm should keep tree nodes at least half full. If you split a node when it is full, you will create two half-full nodes. You then need to merge adjacent nodes when they are less than half full. You can then use an array, safe in the knowledge that it is more efficient than a linked list.
No idea about the details of deletion, sorry!
B-Tree node has an important characteristic, all keys in the node is sorted. When finding a specific key, binary search is used to find the right position. Using binary search keeps the complexity of search algorithm in B-Tree O(logn).
If you replace the preallocated array with some kind of linked list, you lost the ordering. Unless you use some complex data structures, like skip list, to keep the search algorithm with O(logn). But it's totally unnecessary, skip list itself is better.
I have explored the definitions of T-trees and B-/B+ trees. From papers on the web I understand that B-trees perform better in hierarchical memory, such as disk drives and cached memory.
What I can not understand is why T-trees were/are used even for flat memory?
They are advertised as space efficient alternative to AVL trees.
In the worst case, all leaf nodes of a T-tree contain just one element and all internal nodes contain the minimum amount allowed, which is close to full. This means that on average only half of the allocated space is utilized. Unless I am mistaken, this is the same utilization as the worst case of B-trees, when the nodes of a B-tree are half full.
Assuming that both trees store the keys locally in the nodes, but use pointers to refer to the records, the only difference is that B-trees have to store pointers for each of the branches. This would generally cause up to 50% overhead or less (over T-trees), depending on the size of the keys. In fact, this is close to the overhead expected in AVL trees, assuming no parent pointer, records embedded in the nodes, keys embedded in the records. Is this the expected efficiency gain that prevents us from using B-trees instead?
T-trees are usually implemented on top of AVL trees. AVL trees are more balanced than B-trees. Can this be connected with the application of T-trees?
I can give you a personal story that covers half of the answer, that is, why I wrote some Pascal code to program B+ trees some 18 years ago.
my target system was a PC with two disk drives, I had to store an index on non volatile memory and I wanted to understand better what I was learning at university. I was very dissatisfied with the performance of a commercial package, probably DBase III, or some Fox product, I can't remember.
anyhow: I needed these operations:
lookup
insertion
deletion
next item
previous item
maximum size of index was not known
so data had to reside on disk
each access to the support had high cost
reading a whole block cost the same as reading one byte
B+-trees made that small slow PC really fly through the data!
the leafs had two extra pointers so they formed a doubly linked list, for sequential searches.
In reality the difference lies in the system you use. As my tutor in university commented it : if your problem lies in memory shortage, or in hdd shortage will determine which tree and in which implementation you will use. Most probably it will be B+ tree.
Because there are hundreds of implementations, for instance with 2direction queue and one directional queues where you need to loop thought elements, and also there are multiple ways to store the index and retrieve it will determine the real cons and mins of any implementation.