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The Data-Compression Lexicon, with a History

Like any other scientific or engineering discipline, data compression has a vocabulary that at first seem overwhelmingly strange to an outsider. Terms like Lempel-Ziv compression, arithmetic coding, and statistical modeling get tossed around with reckless abandon.

While the list of buzzwords is long enough to merit a glossary, mastering them is not as daunting a project as it may first seem. With a bit of study and a few notes, any programmer should hold his or her own at a cocktail-party argument over data-compression techniques.

Data-compression techniques can be divided into two major families; lossy and lossless. Lossy data compression concedes a certain loss of accuracy in exchange for greatly increased compression. Lossy compression proves effective when applied to graphics images and digitized voice. By their very nature, these digitized representations of analog phenomena are not perfect to begin with, so the idea of output and input not matching exactly is a little more acceptable. Most lossy compression techniques can be adjusted to different quality levels, gaining higher accuracy in exchange for less effective compression. Until recently, lossy compression has been primarily implemented using dedicated hardware. In the past few years, powerful lossy-compression programs have been moved to desktop CPUs, but even so the field is still dominated by hardware implementations.

Lossless compression consists of those techniques guaranteed to generate an exact duplicate of the input data stream after a compress/expand cycle. This is the type of compression used when storing database records, spreadsheets, or word processing files. In these applications, the loss of even a single bit could be catastrophic. Most techniques discussed in this book will be lossless.

In general, data compression consists of taking a stream of symbols and transforming them into codes. If the compression is effective, the resulting stream of codes will be smaller than the original symbols. The decision to output a certain code for a certain symbol or set of symbols is based on a model. The model is simply a collection of data and rules used to process input symbols and determine which code(s) to output. A program uses the model to accurately define the probabilities for each symbol and the coder to produce an appropriate code based on those probabilities.

Modeling and coding are two distinctly different things. People frequently use the term coding to refer to the entire data-compression process instead of just a single component of that process. You will hear the phrases “Huffman coding” or “Run-Length Encoding,” for example, to describe a data-compression technique, when in fact they are just coding methods used in conjunction with a model to compress data.

Using the example of Huffman coding, a breakdown of the compression process looks something like this:

**Figure 2.1** A Statistical Model with a Huffman Encoder.

In the case of Huffman coding, the actual output of the encoder is determined by a set of probabilities. When using this type of coding, a symbol that has a very high probability of occurrence generates a code with very few bits. A symbol with a low probability generates a code with a larger number of bits.

We think of the model and the program’s coding process as different because of the countless ways to model data, all of which can use the same coding process to produce their output. A simple program using Huffman coding, for example, would use a model that gave the raw probability of each symbol occurring anywhere in the input stream. A more sophisticated program might calculate the probability based on the last 10 symbols in the input stream. Even though both programs use Huffman coding to produce their output, their compression ratios would probably be radically different.

So when the topic of coding methods comes up at your next cocktail party, be alert for statements like “Huffman coding in general doesn’t produce very good compression ratios.” This would be your perfect opportunity to respond with “That’s like saying Converse sneakers don’t go very fast. I always thought the leg power of the runner had a lot to do with it.” If the conversation has already dropped to the point where you are discussing data compression, this might even go over as a real demonstration of wit.

Data compression is perhaps the fundamental expression of Information Theory. Information Theory is a branch of mathematics that had its genesis in the late 1940s with the work of Claude Shannon at Bell Labs. It concerns itself with various questions about information, including different ways of storing and communicating messages.

Data compression enters into the field of Information Theory because of its concern with redundancy. Redundant information in a message takes extra bit to encode, and if we can get rid of that extra information, we will have reduced the size of the message.

Information Theory uses the term entropy as a measure of how much information is encoded in a message. The word entropy was borrowed from thermodynamics, and it has a similar meaning. The higher the entropy of a message, the more information it contains. The entropy of a symbol is defined as the negative logarithm of its probability. To determine the information content of a message in bits, we express the entropy using the base 2 logarithm:

Number of bits = - Log base 2 (probability)

The entropy of an entire message is simply the sum of the entropy of all individual symbols.

Entropy fits with data compression in its determination of how many bits of information are actually present in a message. If the probability of the character ‘e’ appearing in this manuscript is 1/16, for example, the information content of the character is four bits. So the character string “eeeee” has a total content of 20 bits. If we are using standard 8-bit ASCII characters to encode this message, we are actually using 40 bits. The difference between the 20 bits of entropy and the 40 bits used to encode the message is where the potential for data compression arises.

One important fact to note about entropy is that, unlike the thermodynamic measure of entropy, we can use no absolute number for the information content of a given message. The problem is that when we calculate entropy, we use a number that gives us the probability of a given symbol. The probability figure we use is actually the probability for a given model, not an absolute number. If we change the model, the probability will change with it.

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