# An Explanation of the Deflate Algorithm

## mail_to("feldspar","netcom.com")Antaeus Feldspar

It's important before trying to understand DEFLATE to understand the other two compression strategies that make it up -- Huffman coding and LZ77 compression.

### Huffman coding

Huffman coding is a form of prefix coding, which you may not think you know. But you've almost certainly used a prefix code -- when using the phone. Starting from the dial tone, you press a sequence of what may be five, seven, eight, eleven, twelve, or some other number of keys -- and each sequence of keys reaches another specific phone line.

Now suppose you're in an office setting with an internal switchboard, as many large companies do. All other phones within the bank only require five numbers to dial, instead of seven -- that's because it's expected that you'll be calling those numbers more often. You may still need to call other numbers, though -- so all of those numbers have a `9' added to the front.

That's a prefix code. Each element that you could want to specify has a code made up of numbers, and because no code for one element begins with the code for any other element, you can type in that code and there will be no ambiguity about that being the one you mean.

A Huffman code is a prefix code prepared by a special algorithm. Here, instead of each code being a series of numbers between 0 and 9, each code is a series of bits, either 0 or 1. Instead of each code representing a phone, each code represents an element in a specific ``alphabet'' (such as the set of ASCII characters, which is the primary but not the only use of Huffman coding in DEFLATE).

A Huffman algorithm starts by assembling the elements of the ``alphabet,'' each one being assigned a ``weight'' -- a number that represents its relative frequency within the data to be compressed. These weights may be guessed at beforehand, or they may be measured exactly from passes through the data, or some combination of the two. In any case, the elements are selected two at a time, the elements with the lowest weights being chosen. The two elements are made to be leaf nodes of a node with two branches (I really hope you know nodes and trees...) Anyway, suppose we had a set of elements and weights that looked like this:

```    A    16
B    32
C    32
D     8
E     8
```

We would pick D and E first, and make them branches of a single node -- one being the `0' branch, and one the `1' branch.

```     ( )
0 /   \ 1
D     E
```

At this point, no element has been given its complete code yet, but we now know that the codes for D and E will be exactly the same, except for the last binary digit: D will end in 0 and E in 1.

The combined node D-and-E is placed back with the other (as yet) uncombined elements, and given a weight equal to the sum of its leaf nodes: in this case, 8 + 8 = 16. Again, we take the two nodes with lowest weight, which are A, and D-and-E, and join them into a larger node.

```       ( )
0 /   \ 1
( )    A
0 /   \ 1
D     E
```

This time, when the node A-D-E is put back into the list of elements, all remaining elements have the same weight, 32. Which two of the three are selected to be combined first is not important, at least not in the classical Huffman algorithm.

When all nodes have been recombined into a single ``Huffman tree,'' then by starting at the root and selecting 0 or 1 at each step, you can reach any element in the tree. Each element now has a Huffman code, which is the sequence of 0's and 1's that represents that path through the tree.

Now, it should be fairly easy to see how such a tree, and such a set of codes, could be used for compression. If compressing ordinary text, for example, probably more than half of the ASCII character set could be left out of the tree altogether. Frequently used characters, like `E' and `T' and `A,' will probably get much shorter codes, and even if some codes are actually made longer, they will be the ones that are used less often.

However, there is also the question: how do you pass the tree along with the encoded data? It turns out that there is a fairly simple way, if you modify slightly the algorithm used to generate the tree.

In the classic Huffman algorithm, a single set of elements and weights could generate multiple trees. In the variation used by the Deflate standard, there are two additional rules: elements that have shorter codes are placed to the left of those with longer codes. (In our previous example, D and E wind up with the longest codes, and so they would be all the way to the right.) Among elements with codes of the same length, those that come first in the element set are placed to the left. (If D and E end up being the only elements with codes of that length, then D will get the 0 branch and E the 1 branch, as D comes before E.)

It turns out that when these two restrictions are placed upon the trees, there is at most one possible tree for every set of elements and their respective codelengths. The codelengths are all that we need to reconstruct the tree, and therefore all that we need to transmit.

### LZ77 compression

LZ77 compression works by finding sequences of data that are repeated. The term ``sliding window'' is used; all it really means is that at any given point in the data, there is a record of what characters went before. A 32K sliding window means that the compressor (and decompressor) have a record of what the last 32768 (32 * 1024) characters were. When the next sequence of characters to be compressed is identical to one that can be found within the sliding window, the sequence of characters is replaced by two numbers: a distance, representing how far back into the window the sequence starts, and a length, representing the number of characters for which the sequence is identical.

I realize this is a lot easier to see than to just be told. Let's look at some highly compressible data:

```        Blah blah blah blah blah!
```

Our datastream starts by receiving the following characters: `B,' `l,' `a,' `h,' ` ,' and `b.' However, look at the next five characters:

```         vvvvv
Blah blah blah blah blah!
^^^^^
```

There is an exact match for those five characters in the characters that have already gone into the datastream, and it starts exactly five characters behind the point where we are now. This being the case, we can output special characters to the stream that represent a number for length, and a number for distance.

The data so far:

```        Blah blah b
```

The compressed form of the data so far:

```        Blah b[D=5,L=5]
```

The compression can still be increased, though to take full advantage of it requires a bit of cleverness on the part of the compressor. Look at the two strings that we decided were identical. Compare the character that follows each of them. In both cases, it's `l' -- so we can make the length 6, and not just five. But if we continue checking, we find the next characters, and the next characters, and the next characters, are still identical -- even if the so-called ``previous'' string is overlapping the string we're trying to represent in the compressed data!

It turns out that the 18 characters that start at the second character are identical to the 18 characters that start at the seventh character. It's true that when we're decompressing, and read the length, distance pair that describes this relationship, we don't know what all those 18 characters will be yet -- but if we put in place the ones that we know, we will know more, which will allow us to put down more... or, knowing that any length-and-distance pair where length > distance is going to be repeating (distance) characters again and again, we can set up the decompressor to do just that.

It turns out our highly compressible data can be compressed down to just this:

```        Blah b[D=5, L=18]!
```

### Putting it all together

The deflate compressor is given a great deal of flexibility as to how to compress the data. The programmer must deal with the problem of designing smart algorithms to make the right choices, but the compressor does have choices about how to compress data.

There are three modes of compression that the compressor has available:

1. Not compressed at all. This is an intelligent choice for, say, data that's already been compressed. Data stored in this mode will expand slightly, but not by as much as it would if it were already compressed and one of the other compression methods was tried upon it.
2. Compression, first with LZ77 and then with Huffman coding. The trees that are used to compress in this mode are defined by the Deflate specification itself, and so no extra space needs to be taken to store those trees.
3. Compression, first with LZ77 and then with Huffman coding with trees that the compressor creates and stores along with the data.

The data is broken up in ``blocks,'' and each block uses a single mode of compression. If the compressor wants to switch from non-compressed storage to compression with the trees defined by the specification, or to compression with specified Huffman trees, or to compression with a different pair of Huffman trees, the current block must be ended and a new one begun.

The details of how the LZ77 and Huffman work together need some closer examination. Once the raw data has been turned into a string of characters and special length, distance pairs, these elements must be represented with Huffman codes.

Though this is NOT, repeat, NOT standard terminology, call the point where we start reading in bits a ``dial tone.'' After all, in our analogy, the dial tone is where you can start specifying a series of numbers that will end up mapping to a specific phone. So call the very beginning a ``dial tone.'' At that dial tone, one of three things could follow: a character, a length-distance pair, or the end of the block. Since we must be able to tell which it is, all the possible characters (``literals''), elements that indicate ranges of possible lengths (``lengths''), and a special end-of-block indicator are all merged into a single alphabet. That alphabet then becomes the basis of a Huffman tree. Distances don't need to be included in this alphabet, since they can only appear directly after lengths. Once the literal has been decoded, or the length-distance pair decoded, we are at another ``dial-tone'' point and we start reading again. If we got the end-of-block symbol, of course, we're either at the beginning of another block or at the end of the compressed data.

Length codes or distance codes may actually be a code that represents a base value, followed by extra bits that form an integer to be added to the base value.

Probably the trickiest part of the DEFLATE specification to understand is the way trees are encoded to go along with the data, when that data is compressed with specialized trees.

The trees are transmitted by their codelengths, as previously discussed. The codelengths are put all together into a sequence of numbers between 0 and 15 (the Huffman trees that are created must be kept to codelengths of no more than 15; this is the tricky part, not the part about constraining the order of the elements).

Not all the elements have to be given codelengths; if the last elements of an alphabet are of 0 codelengths, they can and probably should be left out. The number of elements in each of the two alphabets will be transmitted, so the trimmed alphabets go together into a single sequence.

Once this sequence of codelengths is assembled, it is compressed with a form of what is called run-length compression. When several elements in a row have the same codelength (often 0) special symbols may be used to indicate the number of elements with this codelength. Our sequence is now a sequence of numbers between 0 and 18 (possibly with extra bits forming integers to modify base values, as was the case with the length and distance codes).

A Huffman tree is created for this alphabet of 0-18. Sigh. The sequence of 0-18 codes and extra bits is prepared with the Huffman codes replacing the 0-18 elements.

This Huffman tree needs to be included with the data, of course. Like the other Huffman trees, it will be included by recording the codelengths of the elements. However... once again, *SIGH*. They are not recorded in the order 0, 1, 2, 3, 4 ... 16, 17, 18. They are recorded in a special order: 16, 17, 18, 0, 8, 7, 9, 6, 10, 5, 11, 4, 12, 3, 13, 2, 14, 1, 15. The logic behind this is that the elements near the end of sequence are most likely to have 0 codelengths (i.e. not be in the tree at all). If they are indeed 0, they can be trimmed from the end of the sequence; the number of elements that are being directly recorded is also included in the data.

Those are the trouble points, I think, of the DEFLATE specs. Along with the specification itself and a bit of time, it should hopefully clear things up. If you have any other questions, please ask.

- Antaeus Feldspar ( feldspar@netcom.com)