Question

What would be the encoded version of the message xxy yyx xxy xxy yyx if LZW compression, starting with the dictionary containing \(x, y\), and a space (as described in Section 1.8), were used?

Short answer

Encoded version: 0, 1, 2, 3, 1, 0, 4.

Step-by-step solution

Initialize Dictionary

Start with an initial dictionary containing the individual characters and a space: Dictionary: {0: 'x', 1: 'y', 2: ' '}. Each pattern ("x", "y", or " ") starts with a code already assigned.

Read First Symbol

Read the first symbol of the message, which is 'x'. Write out its existing code from the dictionary, which is 0.

Build Patterns

Begin with 'x' (from Step 2), try to extend the sequence with the next symbol 'x'. The pattern 'xx' is not in the dictionary, so add 'xx' to the dictionary with code 3. Read up to 'xx', encode with code 0.

Encode and Expand

Move forward. Extend 'x' by adding 'y' to make 'xy'. This pattern 'xy' is not in the dictionary, so add 'xy' with code 4. To represent 'xx' followed by 'y', encode with code 1 for 'y'.

Continue Process

Read the next ' ', yielding the pattern 'y '. This pattern is new, add 'y ' to the dictionary with code 5. Encode 'y' followed by ' ', with 2.

Process Subsequent Patterns

Proceed to the next 'x'. Extend it with the following 'x' to form 'xx', which is already in the dictionary and thus encoded with code 3. For the next 'y', extend to make 'yy', add to dictionary with code 6, and 'x' again makes 'yyx', add to dictionary with code 7.

Encode Remaining Symbols

The remaining sequence 'yyx' is recognized as a continuous pattern already added. Encode 'yyx' with code 1 for 'y' and 2 for ' '. Then 'x' and 'xy' are 0 and 4 respectively.

Key concepts

Dictionary Encoding

Dictionary encoding is a technique used in data compression to replace redundant data patterns with shorter binary codes. It capitalizes on the principle of substitution, where repeated strings of symbols in data are replaced with shorter identifiers, known as codes.

In the context of LZW (Lempel-Ziv-Welch) compression, dictionary encoding initializes with a predefined set of individual characters. For example, in our scenario with the message "xxy yyx xxy xxy yyx", this dictionary began with single characters: 'x', 'y', and the space character. Each of these is assigned a unique code which is stored in a dictionary.

As LZW processes the data, it incrementally builds upon this initial dictionary by adding new sequences (or patterns) of symbols that are not yet present. This dynamic enhancement of the dictionary allows new patterns to be represented by new, unique codes, optimizing the space needed for representation. As the dictionary expands, well-encoded sequences create a more efficient encoded output by substituting longer strings with shorter codes.

Pattern Recognition

Pattern recognition is a core functionality of dictionary encoding, especially in LZW compression, enabling the system to identify and memorize new patterns during the process of encoding a message. This involves continuously searching through the sequence of symbols to find repeated combinations and adding these new patterns to the dictionary for future reference.

For the message "xxy yyx xxy xxy yyx", LZW begins by recognizing the repetition of 'x' and 'y'. For instance, 'xx' is quickly identified as a new pattern not initially in the dictionary. As such, 'xx' is added with its own code for efficient future encoding. The necessity of recognizing new patterns efficiently ensures that the dictionary holds all possible recurring sequences to minimize file size.

This dynamic pattern recognition allows LZW to swiftly adapt to the nature of the data being compressed, making it suitable for a wide range of data types where pattern repetition is expected. This adaptability is what makes LZW both robust and flexible.

Symbol Encoding

Symbol encoding is essentially the assignment of binary codes to symbols in the data, which in LZW, encompasses both individual characters and patterns. This encoding not only involves initial assignment but also the dynamic assignment of codes to newly identified patterns during the compression process.

Initially, in the LZW algorithm used in our example, each known symbol 'x', 'y', and ' ' is given a unique code, such as 0, 1, and 2. As the sequence is processed, new symbols (or patterns) are recognized, such as 'xx', 'xy', and 'yyx'. These, in turn, receive new unique identifiers as they are encountered.

Symbol encoding goes beyond just assigning numbers; it forms the fundamental structure for efficient data storage and transmission. By encoding sequences in as few bits as possible, LZW reduces the overall size of the data, making it very efficient for storage and quicker for data transfer, hence maintaining the integrity and speed of information handling.