I have a basic understanding of both red black trees and 2-3-4 trees and how they maintain the height balance to make sure that the worst case operations are O(n logn).
But, I am not able to understand this text from Wikipedia
2-3-4 trees are an isometry of red-black trees, meaning that they are equivalent data structures. In other words, for every 2-3-4 tree, there exists at least one red-black tree with data elements in the same order. Moreover, insertion and deletion operations on 2-3-4 trees that cause node expansions, splits and merges are equivalent to the color-flipping and rotations in red-black trees.
I don't see how the operations are equivalent. Is this quote on Wikipedia accurate? How can one see that the operations are equivalent?
rb-tree(red-black-tree) is not isomorphic to 2-3-4-tree. Because the 3-node in 2-3-4-tree can be lean left or right if we try to map this 3-node to a rb-tree. But llrb-tree(Left-leaning red-black tree) does.
Words from Robert Sedgewick(In Introduction section):
In particular, the paper describes a way to maintain
a correspondence between red-black trees and 2-3-4 trees,
by interpreting red links as internal links in 3-nodes and
4-nodes. Since red links can lean either way in 3-nodes
(and, for some implementations in 4-nodes), the correspondence is not necessarily 1-1
Also Page29 and Page30 of presentation from Robert Sedgewick. This a presentation about LLRB tree.
And "Analogy to B-trees of order 4" section of "Red-black Tree" in the wikipedia, it contains a good graph.
Related
I am a bit confused a term "lookup algorithm of avl trees". When I have searched this in google, I see so many website with related about b-tree not avl tree.
So, Is b-tree algorithm equal lookup algorithm of avl tree ?
If not, what is "lookup algorithm of avl tree" ? Moreover, what is the meaning of "lookup algorithm"? Please give me a link, of course if possible.
b-tree is a data structure - a generalized binary tree.
A lookup algorithm is an algorithm used to lookup values in the data structure. It is how you decide to find items in the data structure.
An avl tree is a type of b-tree (in the abstract).
The lookup algorithm is just the way that you look through the nodes in the tree to find a specific value.
An AVL tree is a self-balancing binary search tree, so the lookup algorithm of an AVL tree is the exact same as for a binary tree.
A B-tree is not the same thing as a binary tree, so it has a different lookup algorithm. The difference is that in a B-tree each node can have several values and more than two children, so the lookup algorithm follows the same basic principle as for a binary tree, but it's a bit more complex.
AVL tree is kind of balancing in the binary tree. B-tree is abbreviation for "Bayer-tree" - a kind of multinode (exceeding 2) tree. So these algorithms are different, since look-up in B-tree takes also look-up over particular page
I've read some Q&As about self-balancing binary trees, but I'm not quite familiar with all of them.
The first one of them I got to know is AVL, the second is Red-Black tree.
There are something I don't quite understand: according to some books and articles, AVL can perform searching a little bit faster than Red-Black tree, well, this is understandable.
Then what's Red-Black tree's edge over AVL?
In AVL, probably after each insertion, we have to check for balance, but in Red-Black tree we don't have to do something like that frequently, right?
PS:
I search SO for something similar, but I didn't get satisfying answer.
Hope some friends can give me a detailed comparison of self-balancing trees.
An AVL tree has the following property: from each node, the difference in height of the left and the right subtree is at most 2.
In a red-black tree, on the other hand, the height of the left or right subtree of any node is at most twice the height of the other tree. That is, they differ at most by a factor of 2.
This shows intuitively that lookup is indeed faster in an AVL tree on average.
However, when inserting or deleting a node, we have to rebalance the AVL tree more often, to preserve the much stricter height invariant (on the other hand, rebalancing in a red-black tree is algorithmically much more complicated). This means that in practice, a red-black tree may perform much better than an AVL tree, in particular when it’s often changed.
I have read it in a couple of places that avl tree search faster, but not able to understand. As I understand :
max height of red-black tree = 2*log(N+1)
height of AVL tree = 1.44*logo(N+1)
Is it because AVL is shorter?
Yes.
The number of steps required to find an item depends on the distance between the item and the root.
Since the AVL tree is packed tighter (i.e. it has a lower max height) it means more items are closer to the root than in the red-black case.
The extra tight packing also means the AVL tree requires more work when inserting elements.
The best choice for any app depends on whether it is insert intensive or search intensive...
AVL tree is better than red-black tree if the input key is almost ascending/descending because then we would have to do single rotation(left-left or right-right case) to add this element. Also, since the tree would be tightly balanced, the search would also be faster.
But for randomly selected input key, RBTree are better since they require less rotation for insertion in comparison to AVL.
Overall, it depends on the input sequence, which would decide how tilted our tree is, and the operation performed.For insert-intensive use Red-Black Tree and for search-intensive use AVL.
AVL tree and RBTree do have respective advantages as well as disadvantages. You'll perceive that better if you've already learned how they work.
AVL is slightly faster than RBTree in insert operation because there would be at most one rotation involved in insertion, while there may be two for RBTree.
RBTree only require at most three rotations in deletion, but this is not guaranteed in AVL. So it can delete nodes faster than AVL.
However, above all, they both have strict logarithmic tree height.
Pick up any subtree, the property that makes AVL "balanced" guarantees that the difference of height between two child subtrees is at most one, which is to say, intuitively, the whole tree is rigidly balanced.
But when it comes to an RBTree, the rule becomes likely "looser", since property of RBTree can only guarantee the depth of a tree is not larger than twice as the logarithm of the total number of nodes.
Here're some facts that may be more precise:
An AVL tree's height is strictly less than: 1.44log(n+2)-0.328
(approximately)
A red-black tree's height is at most 2log(n+1)
See https://en.wikipedia.org/wiki/AVL_tree#Comparison_to_other_structures for detailed information.
There are lots of questions around about red-black trees but none of them answer how they work. Why is it called red-black? How does this keep the tree balanced (thus increasing performance over an unbalanced normal binary search tree)? I'm just looking for an overview of how and why it works.
For searches and traversals, it's the same as any binary tree.
For inserts and deletes, more sophisticated algorithms are applied which aim to ensure that the tree cannot be too unbalanced. These guarantee that all single-item operations will always run in at worst O(log n) time, whereas in a simple binary tree the binary tree can become so unbalanced that it's effectively a linked list, giving O(n) worst case performance for each single-item operation.
The basic idea of the red-black tree is to imitate a B-tree with up to 3 keys and 4 children per node. B-trees (or variations such as B+ trees) are mainly used for database indexes and for data stored on hard disk.
Each binary tree node has a "colour" - red or black. Each black node is, in the B-tree analogy, the subtree root for the subtree that fits within that B-tree node. If this node has red children, they are also considered part of the same B-tree node. So it is possible (though not done in practice) to convert a red-black tree to a B-tree and back, with (most) structure preserved. The only possible anomoly is that when a B-tree node has two keys and three children, you have a choice of which key to goes in the black node in the equivalent red-black tree.
For example, with red-black trees, every line from root to leaf has the same number of black nodes. This rule is derived from the B-tree rule that all leaf nodes are at the same depth.
Although this is the basic idea from which red-black trees are derived, the algorithms used in practice for inserts and deletes are modified to enforce all the B-tree rules (there might be a minor exception - I forget) during updates, but are tailored for the binary tree form. This means that doing a red-black tree insert or delete may give a different structure for the result than that you'd expect comparing with doing the B-tree insert or delete.
For more detail, follow the Wikipedia link that MigDus already supplied.
A red-black tree is an ordered binary tree where each vertex is coloured red or black. The intuition is that a red vertex should be seen as being at the same height as its parent (i.e., an edge to a red vertex is thought of as "horizontal" rather than "descending").
[I don't believe the Wikipedia entry makes this point clear.]
The usual rules for red-black trees require that a red vertex never point to another red vertex. This means that the possible vertex arrangements for any subtree rooted with a black vertex (bbb, bbr, rbb, rbr -- for [left child][root][right child]) correspond to 234 trees.
Searching a red-black tree is just the same as searching an ordinary binary tree. Insertion and deletion are similar, except that a "fix-up" rotation may be required at some point to preserve the red-black invariant.
Cheers!
What are the applications of 2-3-4 trees? Are they widely used in applications for providing better performance of the applications ?
Edit : Which algorithms make the best use of 2-3-4 trees ?
2-3-4 trees are self-balancing and usually are usually very efficient for finding, adding and deleting elements, so like all trees they can be used for storing and retrieving elements in non-linear order. Unfortunately they tend to use more memory than other trees, because even nodes with only 2 data items still need to have enough memory to possibly store 4 of them.
This is why 2-3-4 trees are used as models for red-black trees, which are like standard BSTs except that nodes can be either red or black, and various rules exist about how to choose which colour a node is.
The key is that algorithms for searching/adding/deleting in a 2-3-4 tree are VERY similar to the ones for a red-black tree, so usually 2-3-4 trees are studied as a way of understanding red-black trees. The red-black trees themselves are quite widely used - I believe the standard Java Collections Framework tree is a red-black tree.
Answer to the applications of 2-3-4 Trees are:
• Linux Kernel.
• Completely Fair Scheduler
• To keep track of Virtual Memory Segments of a Process.
Because it is similar of Red Black Trees and as #Adam said we use it in Java Collections Framework itself.