(This is a follow-up to my previous blog post.)

Any run-length encoding requires you to store the number of repetitions. In my example, AAABBBBBZWWK becomes 3A-5B-1Z-2W-1K, we must store 5 counters (3,5,1,2,1) and 5 characters.

Storing counters using a fixed number of bits

Most programming languages represent integers using a fixed number of bits in binary format. For example, Java represents integers (int) using 32 bits. However, in my example (3A-5B-1Z-2W-1K) storing the counters using 32 bits and the characters using 8 bits means that I use 25 bytes which is more than twice the length of the original string (AAABBBBBZWWK).

Thus, we have a simple optimization problem: determine the best number of bits.  In practice, it might be better to store the data in a byte-aligned way. That is, you should be using 8, 16, 32 or 64 bits. Indeed, reading numbers represented using an arbitrary number of bits may involve a CPU processing overhead.

If you use too few bits, some long runs will have to count as several small runs. If you use too many bits, you are wasting storage. Unfortunately, determining on a case-by-case basis the best number of bits requires multiple scans of the data. It also implies added software complexity.

But you don’t have to use the binary format!

You can still use  a fixed number of bits for your counters, but with quantized codes instead of the binary format. For example, using 3 bits, you could only allow the counter values 1,2,16, 24, 32,128,256,1024. In this example, the sequence of bits 000 is interpreted as the value 1, the sequence of bits 001 as the value 2, the sequence 010 as 16, and so on. Determining the best codes implies that you must scan the data, compute the histogram of the counters, and then apply some optimization algorithm (such as dynamic programming). The decoding speed might be slight slower as you need to look-up the codes from a table.

Using variable-length counters for optimal compression

If you are willing to sacrifice coding and decoding speed, then you can represent the counters using universal codes. Thus, instead of using a fixed number of bits and optimizing the representation (as in the quantized coding idea), you seek an optimal variable-length representation of the counters. With this added freedom, you can be much more efficient.

The illustrate the idea behind variable-length codes, we consider Gamma codes: the code 1 is 1, the code 01 is 2, the code 001 is 3, the code 0001 is 4, and so on. Thus, we use x bits to represent the number x.

Fortunately, we can do much better than Gamma codes and represent the number x using roughly 2 log x bits with delta codes. Firstly, write x as x=2^N +(x modulo 2^N) where N is the logarithm. Then we can represent the number N-1 as a Gamma code using N-1 bits, and then store (x modulo 2^N) in binary format (using N-1 bits). Thus, we can represent all numbers up to 2^N-1 using 2N-2 bits.

Unfortunately, the current breed of microprocessors are not kind to variable-length representations so the added compression is at the expense decoding speed.

References and further reading: Holloway et al., How to Barter Bits for Chronons, 2007. See also the slides of my recent talk Compressing column-oriented indexes.

Run-length encoding (RLE) is probably the most important and fundamental string compression technique. Countless multimedia formats and protocols use one form or RLE compression or another.

RLE is also deceptively simple. It represents repeated values as a counter and a character. Thus, the string AAABBBBBZWWK becomes 3A-5B-1Z-2W-1K.

If that is all there was to RLE, then the wikipedia page on run-length encoding would be just fine. Yet, I think it needs help.

Why do we use RLE?

• You can read and write RLE data in one pass, using almost no memory.
• Given a vector V compressed with RLE, you can apply any scalar function f to its component in time O(|V |) where |V | is the compressed size of the vector.
• Given two vectors V and V‘ compressed with RLE, you can do arithmetic (e.g. V+V‘) in time O(|V |+|V’|).

(Some RLE formats have worse complexity bounds.)

Any downsides to RLE?

• Random access is slower. Sometimes, only sequential read (from the beginning) is possible. Updating an RLE-compressed array can be difficult.
• You need long runs of identical values.
• Some RLE formats negatively affect CPU vectorization. Thus, if the compression rates are modest, it could actually take longer to process an RLE-compressed array.

What is the RLE format?

There is no unique RLE format. How you use the RLE idea depends on your goals such as (1) maximize the compression rate (2) maximize the processing speed.

Here are some common variations:

• Instead of using a counter for each run of characters, you only add a counter after a value has been repeated twice. For example, the string AAABBBBBZWWK becomes AA1-BB3-Z-WW-K. Thus, if many characters are not repeated, you will rarely use an unnecessary counter.
• You can use a single bit to decide whether a counter is used. For example, the string AAABBBBZWWK becomes A-True-3, B-True-5, Z-False, W-True-2, K-False. Again, this may avoid many unnecessary counters if values are often not repeated.
• Instead of a counter, you may store the location of the run in the array. For example, the string AAABBBBBZWWK becomes 1A-4B-9Z-10W-11K. The benefit of this approach is to allow random access in logarithmic time using binary search. However, it is also incompatible with some techniques to avoid unnecessary counters. So, it is a compression-speed trade-off. For even more speed, you can store both the location of the run and its length (thus avoiding a subtraction).
• To help vectorization, you can group the characters into blocks of k characters. For example, using blocks of two characters, the string AAABBBBBZWWK becomes 1AA-1AB-2BB-1ZW-1WK. Again, this is a compression-speed trade-off because there will be fewer runs to compress after grouping the characters into long blocks.

Some References (to my own work):

• Daniel Lemire, Compressing column-oriented indexes (slides)
• Daniel Lemire, Owen Kaser, Kamel Aouiche, Sorting improves word-aligned bitmap indexes. Data & Knowledge Engineering 69 (1), pages 3-28, 2010.
• Daniel Lemire, Owen Kaser, Reordering Columns for Smaller Indexes, working paper.

Current practical database compression techniques stress speed over compression:

• Vectorwise is using Super-scalar RAM-CPU cache compression which includes a carefully implemented dictionary coder.
• C-store—and presumably Vertica—is using similar compression techniques as well as simple run-length encoding of projection indexes.
• Oracle is compressing its bitmaps using Byte-aligned bitmap compression—a variation on run-length encoding.
• Wu et al.’s Fastbit as well as my very own Lemur Bitmap Index C++ Library use less aggressive compression techniques, for faster results. In fact, one my recent empirical results is that on a two-CPU dualcore machine, using 64-bit words instead of 32-bit words in word-aligned compression—which nearly halves the compression—can make the processing faster.
• LucidDB similarly compresses its bitmap indexes with a simple variation on run-length encoding.

In a comment to my previous blog post, Rasmus Pagh asks more or less this question:

Given that we have more and more CPU cores—and generally more powerful CPUs—shouldn’t we compress the data more aggressively?

As the CPUs get more powerful, shouldn’t all database become I/O bound? That is, the main difficulty is to bring enough data into the CPUs?

Apparently not.

• As we have more CPU cores, we also have more bandwidth to bring data to the the cores. Otherwise, CPU cores would be constantly data-starved in most multimedia and business applications.
• Multicore CPUs are not the only game in town: RAM and storage have also been revolutionized. In 2009, it is not unpractical to run entirely database applications from RAM. After all, many business databases fit in 16 GB for RAM. And even when we do not have enough RAM, we have SSDs.
• Lightweight compression techniques often favor vectorization which is also getting more and more important and powerful.

Hence, for most database applications fast decompression remains preferable to aggressive compression.

Morteza Zaker sent me pointer to their work comparing bitmap indexes and B-trees in the Oracle database. They examine the folklore surrounding bitmap indexes—which are often thought to be mostly useful over low cardinality columns (columns having few distinct values, such as gender). Their conclusion is that the Bitmap index is the conclusive choice for data warehouse design for columns with high or low cardinality. Their claim is backed by experiments using columns with millions of distinct values. This confirms the observation made in my earlier post: The mythical bitmap index.

They make an interesting distinction between full cardinality columns, columns where each value appears only once, and high cardinality columns where at least a few values are often repeated (think about storing last names). A bitmap index is inadequate for a full cardinality column because there is no compression possible, and this is probably how the folklore around bitmap indexes came about. Yet, unlike transaction processing, data warehousing is usually not concerned with full cardinality columns.

Reference: Zaker et al., An Adequate Design for Large Data Warehouse Systems: Bitmap Index Versus B-Tree Index, IJCC 2 (2), 2008.

Further reading: Trading compression for speed with vectorization, To improve your indexes: sort your tables! and my research paper Sorting improves word-aligned bitmap indexes.

Bitmap indexes are used by search engines (such as Apache Lucene), they are available in DBMSes such as Oracle and PostgreSQL. They are used in column stores such as the Open Source engines Eigenbase and C-Store, as well as by many commercial solutions such as Vertica.

Bitmap indexes are silly data structures. Map each value to an array of booleans. Hence, if you have n rows in your table, and k distinct values, you get an n by k matrix containing booleans. Thus, some people falsely assume that bitmap indexes are only adequate when there are few distinct values (e.g., the gender column, male and female being the only two options). However—using techniques based on run-length encoding—the total size of your bitmaps is proportional to the size of the original table, irrespective of the number of distinct values!

Bitmap indexes are fast because they benefit from vectorization. Indeed, let the predicate “sex=male” is satisfied on rows 1, 5, 32, 45, 54 and 63. I can determine which rows satisfy the extended predicate “(sex=male) AND (city=Montreal)” using a single instruction! The secret? A bitwise AND between the bitmaps “sex=male” and “city=Montreal”. You can compute unions, differences and intersections between sets of integers in [1,N] using only N/64 operations. All microprocessors have built-in parallelism because they operate on several bits at once.

To benefit from vectorization, you need to store the data in a word-aligned manner: that is, you store consecutive segments of bits uncompressed. The longer the words, the less compression. Roughly speaking, 64-bit bitmap indexes are nearly twice as large as 32-bit bitmap indexes. What is the effect on the processing speed? We found that despite being much larger, 64-bit bitmap indexes were faster. That is right: it was faster to load twice as much data from disk!

Yet, we often equate concise data structures with more speed. This assumption can be misguided. Given a choice between more compression, or more vectorization, I would choose more vectorization.

References:

• Daniel Lemire, Owen Kaser, Kamel Aouiche, Sorting improves word-aligned bitmap indexes, to appear in Data & Knoweldge Engineering. (View it on scribdb.)
• Lemur Bitmap Index C++ Library (open source software library)

Further reading: See my posts Compressed bitmaps in Java, To improve your indexes: sort your tables!, and The mythical bitmap index.

Most database users know row-oriented databases such as Oracle or MySQL. In such engines, the data is organized by rows. Database researcher and guru Michael Stonebraker has been advocating column-oriented databases. The idea is quite simple: by organizing the data into columns, we can compress it more efficiently (using simple ideas like run-length encoding). He even founded a company, Vertica, to sell this idea.

Daniel Tunkelang is back from SIGMOD: he reports that column-oriented databases have grabbed much mindshare. While I did not attend SIGMOD, I am not surprised. Daniel Abadi was awarded the 2008 SIGMOD Jim Gray Doctoral Dissertation Award for his excellent thesis on Column-Oriented Database Systems. Such great work supported by influential people such as Stonebraker is likely to get people talking.

But are column-oriented databases the next big thing? No.

• Column stores have been around for a long time in the form of bitmap and projection indexes. Conceptually, there is little difference. (See my own work on bitmap indexes.)
• While it is trivial to change or delete a row in a row-oriented database, it is harder in column-oriented databases. Hence, applications are limited to data warehousing.
• Column-oriented databases are faster for some applications. Sometimes faster by two orders of magnitude, especially on low selectivity queries. Yet, part of these gains are due to the recent evolution in our hardware. Hardware configurations where reading data sequentially is very cheap favor sequential organization of the data such as column stores. What might happen in the world of storage and microprocessors in the next ten years?

I believe Nicolas Bruno said it best in Teaching an Old Elephant New Tricks:

(…) some C-store proponents argue that C-stores are fundamentally different from traditional engines, and therefore their benefits cannot be incorporated into a relational engine short of a complete rewrite (…) we (…) show that many of the benefits of C-stores can indeed be simulated in traditional engines with no changes whatsoever.  Finally, we predict that traditional relational engines will eventually leverage most of the benefits of C-stores natively, as is currently happening in other domains such as XML data.

That is not to say that you should avoid Vertica’s products or do research on column-oriented databases. However, do not bet your career on them. The hype will not last.

(For a contrarian point of view, read Adabi and Madden’s blog post on why column stores are fundamentally superior.)