Red-Black Trees

algorithmbinary treered-black-tree

I've seen binary trees and binary searching mentioned in several books I've read lately, but as I'm still at the beginning of my studies in Computer Science, I've yet to take a class that's really dealt with algorithms and data structures in a serious way.

I've checked around the typical sources (Wikipedia, Google) and most descriptions of the usefulness and implementation of (in particular) Red-Black trees have come off as dense and difficult to understand. I'm sure for someone with the necessary background, it makes perfect sense, but at the moment it reads like a foreign language almost.

So what makes binary trees useful in some of the common tasks you find yourself doing while programming? Beyond that, which trees do you prefer to use (please include a sample implementation) and why?

Best Answer

Red Black trees are good for creating well-balanced trees. The major problem with binary search trees is that you can make them unbalanced very easily. Imagine your first number is a 15. Then all the numbers after that are increasingly smaller than 15. You'll have a tree that is very heavy on the left side and has nothing on the right side.

Red Black trees solve that by forcing your tree to be balanced whenever you insert or delete. It accomplishes this through a series of rotations between ancestor nodes and child nodes. The algorithm is actually pretty straightforward, although it is a bit long. I'd suggest picking up the CLRS (Cormen, Lieserson, Rivest and Stein) textbook, "Introduction to Algorithms" and reading up on RB Trees.

The implementation is also not really so short so it's probably not really best to include it here. Nevertheless, trees are used extensively for high performance apps that need access to lots of data. They provide a very efficient way of finding nodes, with a relatively small overhead of insertion/deletion. Again, I'd suggest looking at CLRS to read up on how they're used.

While BSTs may not be used explicitly - one example of the use of trees in general are in almost every single modern RDBMS. Similarly, your file system is almost certainly represented as some sort of tree structure, and files are likewise indexed that way. Trees power Google. Trees power just about every website on the internet.

Applications of binary trees

Binary Search Tree - Used in many search applications where data is constantly entering/leaving, such as the map and set objects in many languages' libraries.
Binary Space Partition - Used in almost every 3D video game to determine what objects need to be rendered.
Binary Tries - Used in almost every high-bandwidth router for storing router-tables.
Hash Trees - Used in torrents and specialized image-signatures in which a hash needs to be verified, but the whole file is not available. Also used in blockchains for eg. Bitcoin.
Heaps - Used in implementing efficient priority-queues, which in turn are used for scheduling processes in many operating systems, Quality-of-Service in routers, and A* (path-finding algorithm used in AI applications, including robotics and video games). Also used in heap-sort.
Huffman Coding Tree (Chip Uni) - Used in compression algorithms, such as those used by the .jpeg and .mp3 file-formats.
GGM Trees - Used in cryptographic applications to generate a tree of pseudo-random numbers.
Syntax Tree - Constructed by compilers and (implicitly) calculators to parse expressions.
Treap - Randomized data structure used in wireless networking and memory allocation.
T-tree - Though most databases use some form of B-tree to store data on the drive, databases which keep all (most) their data in memory often use T-trees to do so.

The reason that binary trees are used more often than n-ary trees for searching is that n-ary trees are more complex, but usually provide no real speed advantage.

In a (balanced) binary tree with m nodes, moving from one level to the next requires one comparison, and there are log_2(m) levels, for a total of log_2(m) comparisons.

In contrast, an n-ary tree will require log_2(n) comparisons (using a binary search) to move to the next level. Since there are log_n(m) total levels, the search will require log_2(n)*log_n(m) = log_2(m) comparisons total. So, though n-ary trees are more complex, they provide no advantage in terms of total comparisons necessary.

(However, n-ary trees are still useful in niche-situations. The examples that come immediately to mind are quad-trees and other space-partitioning trees, where divisioning space using only two nodes per level would make the logic unnecessarily complex; and B-trees used in many databases, where the limiting factor is not how many comparisons are done at each level but how many nodes can be loaded from the hard-drive at once)

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Finally fixed by removing /var/lib/nginx/fastcgi directory and then recreating it from root and changing it's owner to www-data.

now it looks like that

drwxr-xr-x 2 www-data root     4096 Sep  5 13:58 fastcgi

Also the problem could be fixed by adjusting fastcgi buffer size using these configuration settings:

    fastcgi_buffers 16 16k;
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What are the applications of binary trees

Applications of binary trees

Php – Nginx Net::ERR_INCOMPLETE_CHUNKED_ENCODING when trying to show large JSON

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