B+Trees are complex disk based trees used to index large amounts of data. They are used in everything from file systems, to relation databases, to new style databases gaining popularity today. Sometimes a domain specific application needs to index a large amount of data, but cannot use a traditional database, or one of the NoSQL databases. In such instances the development team needs to roll their own indices. Here is an introduction to the B+Tree (one of the indexes my team created) and lessons I learned while implementing it.
Introduction to the B+Tree
B+Trees are one of the fundamental index structures used by databases today. This includes new style SQL free databases. The B+Tree popularity stems from their performance approaching optimal performance in terms of disk reads for range queries in a 1 dimensional space. What is a 1 dimensional space when talking about computer data which could be anything (not just numbers)? It is any collection of objects where the user accesses the object using only one attribute at a time.
For example if we have an object which has X, Y, and Z as attributes queries would only take place on X, or Y, or Z, but never on XY, or YZ, or XZ, or XYZ. A collection where multiple attributes are used to access the data elements are known as multidimensional spaces. For these spaces there are many other structures which have better performance than B+Trees.
B+Trees perform particularly well (in comparison to some other indices) when executing range queries. A range query is typically expressed as inequality such as "give me all strings between 'blossom' and 'brunet.'"
When I say their performance is approaching optimal in number of disk reads what do I mean? Why are we not measuring performance in number of instructions executed (like we do when we analyze a binary search)? In memory algorithms and structures like sorted arrays and binary searches are largely bound by the number CPU cycles it takes to execute the algorithm. We usually neglect CPU cache performance and memory locality when analyzing them, arguing these are constant in terms of the asymptotic performance of the algorithm. However, for a disk based structure like B+Trees the time it takes to read (or write) to a disk becomes the dominant term, since disks are extremely slow in comparison to main memory. Therefore for disk structures we analyze their performance in terms of disk reads/writes.
Basic Structure of the B+ Tree
While I will not give a through explanation of the exact structure and properties of B+ Tree (I leave that to algorithm and database textbooks by the likes of Knuth, Sedgewick, and Ullman), I will describe its basic structure.
A B+Tree is best thought of as a key-value store. It is structured as a generalized tree. Instead of having only one key in each node it has N keys in each node, where N is referred to as the degree of the B+Tree. In the B+Tree there are 2 kinds of nodes, interior nodes, and exterior (leaf) nodes. The interior nodes hold keys and pointers to nodes. The exterior nodes hold keys and their associated values. This indicates that the interior nodes have a different (usually higher) degree than the exterior nodes.
The reason the tree is structured this way is because it is rooted in the nature of disk access. Disks to not return 1 byte when you ask for 1 byte instead they return what is called the disk block to which that byte belongs, the operating system then sorts out which byte it is that you need. B+Tree exploit the situation by making their nodes fit exactly into the size of one disk block. Since the degree of the interior nodes is high, this makes the tree extremely wide, which is a good thing since it means fewer disk reads to find the value associated with any one key.
Figure 1. An Example B+Tree
In figure 1 you can see an example B+Tree. For this illustration I neglect showing the values, and have the order of the interior nodes equal to the order of the exterior nodes. In general this will not be the case. One thing to note in this simple example is how the exterior nodes are chained together in order. This is why it is efficient to execute a range query on the B+Tree. One can simply find the first key in the range, and then traverse the leaf nodes until the last key has been found.
Implementing the B+Tree
I made the decision to use TDD (Test Driven Development) for implementing the B+Tree. TDD has a lot of pluses when trying to create a data structure of any kind. When implementing a data structure one typically knows exactly how the structure should function, what it should do, and what it should never do. By writing tests first, you can ensure that when you finish a method, it actually works. This speeds development time especially since you already know how the structure should function. It makes it quicker to find bugs, and to battle test the B+Tree. Since I have released the B+Tree to the rest of my team to use, there has not yet been a bug filled against it.
So knowing that we are using TDD, and knowing what the structure is and how it performs. What is the best way to begin implementing this complex structure? The way I started was to create a general structure called a block file. My block files abstracted the notion of reading and writing blocks (and buffering them). I also created objects to model a block that could contain either keys and pointers, or keys and records (instead of values from here on I will use the term records). Actually my blocks are even more general than that as I intend to reuse them for other disk based index structures like linear hashing in the future.
I also created what I called a ByteSlice. My ByteSlice was an array of bytes of arbitrary length. I use it to represent, keys, records, and pointers; everything in the B+Tree. My ByteSlice implemented a comparator, so it could be sorted, and conversions from integer types of various lengths to the ByteSlice and back again. By implementing this general type my B+Trees can easily deal with any kind of data and perform in exactly the same way.
After the infrastructure was created I began working on my first iteration of the B+Tree. The first iteration was based on the algorithms give by Robert Sedgewick in his excellent book "Algorithms in C++." I managed to get this implementation up, running, and fully tested in a matter of days. However, the version given by Sedgewick which inspired my implementation did not deal gracefully with duplicate keys. Thus, I need to invent my own way of handling duplicate keys.
Approaches to Handling Duplicate Keys in B+Trees
There are several different ways of handling duplicate keys. One way is use an unmodified insert algorithm which allows duplicate keys in blocks but is otherwise unchanged. The issue with a structure such as this is the search algorithm must be modified to take into account several corner cases which arise. For instance one of the invariants of a B+Tree may be violated in this structure. Specifically if there are many duplicate keys, a copy of one of the keys may be in a non-leaf block. However, the key may appear in blocks that which appear logically before the block which is pointed at by the key in the internal block. Thus the search algorithm must be modified to look in the previous blocks to the one suggested by the unmodified search algorithm. This will slow down the common case for search.
There is another issue with this straight forward implementation, if there are many duplicate keys in the index, the index size may be taller than necessary. Consider a situation were for each unique key there are perhaps hundreds of duplicates, the index size will be proportional to the total number of keys in the main file, however, you only need to index an index on the unique keys. One of the files indexed in our program will be indexing has such characteristics to its data. It indexes strings (as the keys) with associated instances where those strings show up in our documents. There can be hundreds to thousands of instances of each unique string.
Therefore the approach I took was to store only the unique keys in the index, and have the duplicates captured in overflow blocks in the main file. An example of such a tree can be seen in figure 2. Consider key 6; there are 5 instances of this key in the tree. The tree is order 3, indicating the keys cannot all fit in one block. To handle this situation an overflow block is chained to the block which is indexed by the tree structure. The overflow block then points to the next relevant block in the tree.
Figure 2. A B+Tree with duplicate keys and overflow blocks.
To create a structure such as this, the insert algorithm had to be modified. Like the previous version these modifications do not come without a cost, in particular the invariant which states all block must be at least half full has been relaxed. This is not true in this B+Tree, some blocks like the one containing key number 7, are not half full. This problem could be partially solved by using key rotations to balance the tree better. However, there are still corner cases where there would be a block which is under-full. One such corner case includes when a key falls between two keys which have overflow blocks. It must then be in a block by itself, since this B+Tree has the invariant which state that if a block is overflowing it can only contain one unique key. In the future we would like to implement key rotations to help partially alleviate the problem of under-full blocks.
The advantage of this approach to B+Trees with duplicate keys is the index size is small no matter how the ratio of duplicate keys to the total number of keys in the file. This property allows our searches to be conducted quicker. Since the overflow blocks are chained into the B+Tree structure we still have the property of being able to fast sequential scans. One consequence is we have defined all queries on our B+Trees to be range queries. This is fine because all of our queries were already range queries. In conclusion we relax the condition that all blocks must be at least half full to gain higher performance during search.
The Lessons Learned
The biggest lessons learned through the journey:
The Value of Test Driven Development The impact TDD had on the development time of the B+Tree vs. other structures in the project cannot be understated. TDD dramatically reduced the time it took to develop the structure (from over a month for some of the other structures to under two weeks for the B+Tree), and has ensured reliability of the structure once it entered production.
The Value of the Iterative Approach By starting simple, testing, and then adding complexity I was able to get a better grasp on the problems posed by the modifications we needed to make to the structure. For instance I before I tried method 2 for duplicate keys, I modeled the data we would be putting into our tree and visualized the resulting structure. I found the structure would perform poorly. However, the same code allowed my to visualize method 2 and see that it would perform well.
Visualizations as Part of Development Writing code allowing you to visualize the structure you are developing can really help you find bugs quicker. The best tool to do this with is graphviz. The pictures used in this blog where generated as part of unit test cases. My building a visualization framework early as part of your unit tests you can further reduce development time. When a bug a appears it can be enormously helpful to visualize the actual structure of the tree at the time the bug manifested.
When the right choice for your project isn't DBMS, but you still need to index large data, don't fear you can write the index structures yourself. By using TDD, iterating, and visualizing as you go you can ensure the index structure you create will perform well, and will never get into an incorrect state. Databases are not a black box, and they are not always the right answer. When required you can create you own system.