Lempel–Ziv–Welch

A high-level view of the encoding algorithm is shown here:

1. Initialize the dictionary to contain all strings of length one.
2. Find the longest string W in the dictionary that matches the current input.
3. Emit the dictionary index for W to output and remove W from the input.
4. Add W followed by the next symbol in the input to the dictionary.
5. Go to Step 2.

A dictionary is initialized to contain the single-character strings corresponding to all the possible input characters (and nothing else except the clear and stop codes if they're being used). The algorithm works by scanning through the input string for successively longer substrings until it finds one that is not in the dictionary. When such a string is found, the index for the string without the last character (i.e., the longest substring that is in the dictionary) is retrieved from the dictionary and sent to output, and the new string (including the last character) is added to the dictionary with the next available code. The last input character is then used as the next starting point to scan for substrings.

In this way, successively longer strings are registered in the dictionary and available for subsequent encoding as single output values. The algorithm works best on data with repeated patterns, so the initial parts of a message see little compression. As the message grows, however, the compression ratio tends asymptotically to the maximum (i.e., the compression factor or ratio improves on an increasing curve, and not linearly, approaching a theoretical maximum inside a limited time period rather than over infinite time).[2]

A high-level view of the decoding algorithm is shown here:

1. Initialize the dictionary to contain all strings of length one.
2. Read the next encoded symbol: Is it encoded in the dictionary?
   1. Yes:
      1. Emit the corresponding string W to output.
      2. Concatenate the previous string emitted to output with the first symbol of W. Add this to the dictionary.
   2. No:
      1. Concatenate the previous string emitted to output with its first symbol. Call this string V.
      2. Add V to the dictionary and emit V to output.
3. Repeat Step 2 until end of input string

The decoding algorithm works by reading a value from the encoded input and outputting the corresponding string from the dictionary. However, the full dictionary is not needed, only the initial dictionary that contains single-character strings (and that is usually hard coded in the program, instead of sent with the encoded data). Instead, the full dictionary is rebuilt during the decoding process the following way: after decoding a value and outputting a string, the decoder concatenates it with the first character of the next decoded string (or the first character of current string, if the next one can't be decoded; since if the next value is unknown, then it must be the value added to the dictionary in this iteration, and so its first character is the same as the first character of the current string), and updates the dictionary with the new string. The decoder then proceeds to the next input (which was already read in the previous iteration) and processes it as before, and so on until it has exhausted the input stream.

If variable-width codes are being used, the encoder and decoder must be careful to change the width at the same points in the encoded data so they don't disagree on boundaries between individual codes in the stream. In the standard version, the encoder increases the width from p to p + 1 when a sequence ω + s is encountered that is not in the table (so that a code must be added for it) but the next available code in the table is 2^p (the first code requiring p + 1 bits). The encoder emits the code for ω at width p (since that code does not require p + 1 bits), and then increases the code width so that the next code emitted is p + 1 bits wide.

The decoder is always one code behind the encoder in building the table, so when it sees the code for ω, it generates an entry for code 2^p − 1. Since this is the point where the encoder increases the code width, the decoder must increase the width here as well—at the point where it generates the largest code that fits in p bits.

Unfortunately, some early implementations of the encoding algorithm increase the code width and then emit ω at the new width instead of the old width, so that to the decoder it looks like the width changes one code too early. This is called "early change"; it caused so much confusion that Adobe now allows both versions in PDF files, but includes an explicit flag in the header of each LZW-compressed stream to indicate whether early change is being used. Of the graphics file formats that support LZW compression, TIFF uses early change, while GIF and most others don't.

When the table is cleared in response to a clear code, both encoder and decoder change the code width after the clear code back to the initial code width, starting with the code immediately following the clear code.

Since the codes emitted typically do not fall on byte boundaries, the encoder and decoder must agree on how codes are packed into bytes. The two common methods are LSB-first ("least significant bit first") and MSB-first ("most significant bit first"). In LSB-first packing, the first code is aligned so that the least significant bit of the code falls in the least significant bit of the first stream byte, and if the code has more than 8 bits, the high-order bits left over are aligned with the least significant bits of the next byte; further codes are packed with LSB going into the least significant bit not yet used in the current stream byte, proceeding into further bytes as necessary. MSB-first packing aligns the first code so that its most significant bit falls in the MSB of the first stream byte, with overflow aligned with the MSB of the next byte; further codes are written with MSB going into the most significant bit not yet used in the current stream byte.

GIF files use LSB-first packing order. TIFF files and PDF files use MSB-first packing order.

Buffer input characters in a sequence ω until ω + next character is not in the dictionary. Emit the code for ω, and add ω + next character to the dictionary. Start buffering again with the next character. (The string to be encoded is "TOBEORNOTTOBEORTOBEORNOT#".)

Unencoded length = 25 symbols × 5 bits/symbol = 125 bits

Encoded length = (6 codes × 5 bits/code) + (11 codes × 6 bits/code) = 96 bits.

Using LZW has saved 29 bits out of 125, reducing the message by more than 23%. If the message were longer, then the dictionary words would begin to represent longer and longer sections of text, sending repeated words very compactly.

To decode an LZW-compressed archive, one needs to know in advance the initial dictionary used, but additional entries can be reconstructed as they are always simply concatenations of previous entries.

At each stage, the decoder receives a code X; it looks X up in the table and outputs the sequence χ it codes, and it conjectures χ + ? as the entry the encoder just added – because the encoder emitted X for χ precisely because χ + ? was not in the table, and the encoder goes ahead and adds it. But what is the missing letter? It is the first letter in the sequence coded by the next code Z that the decoder receives. So the decoder looks up Z, decodes it into the sequence ω and takes the first letter z and tacks it onto the end of χ as the next dictionary entry.

This works as long as the codes received are in the decoder's dictionary, so that they can be decoded into sequences. What happens if the decoder receives a code Z that is not yet in its dictionary? Since the decoder is always just one code behind the encoder, Z can be in the encoder's dictionary only if the encoder just generated it, when emitting the previous code X for χ. Thus Z codes some ω that is χ + ?, and the decoder can determine the unknown character as follows:

1. The decoder sees X and then Z, where X codes the sequence χ and Z codes some unknown sequence ω.
2. The decoder knows that the encoder just added Z as a code for χ + some unknown character c, so ω = χ + c.
3. Since c is the first character in the input stream after χ, and since ω is the string appearing immediately after χ, c must be the first character of the sequence ω.
4. Since χ is an initial substring of ω, c must also be the first character of χ.
5. So even though the Z code is not in the table, the decoder is able to infer the unknown sequence and adds χ + (the first character of χ) to the table as the value of Z.

This situation occurs whenever the encoder encounters input of the form cScSc, where c is a single character, S is a string and cS is already in the dictionary, but cSc is not. The encoder emits the code for cS, putting a new code for cSc into the dictionary. Next it sees cSc in the input (starting at the second c of cScSc) and emits the new code it just inserted. The argument above shows that whenever the decoder receives a code not in its dictionary, the situation must look like this.

Although input of form cScSc might seem unlikely, this pattern is fairly common when the input stream is characterized by significant repetition. In particular, long strings of a single character (which are common in the kinds of images LZW is often used to encode) repeatedly generate patterns of this sort.