Question

If the message xxy yyx xxy xxyy were compressed using LZW with a starting dictionary whose first, second, and third entries were \(x, y\), and space, respectively, what would be the entries in the final dictionary?

Short answer

The final dictionary entries are 'x', 'y', ' ', 'xx', 'xy', 'y ', ' y', 'yx', 'x ', ' x', 'xxy', and 'yy'.

Step-by-step solution

Initialize the Dictionary

We start with a dictionary that includes the initial characters and a space as follows: 1. 'x' 2. 'y' 3. ' ' These form our initial dictionary entries with codes 1, 2, and 3 respectively.

Begin Compressing and Update Dictionary

Start compressing the message: `xxy yyx xxy xxyy`. Initialize the string `w` as empty. - Read the first character 'x': - Since 'x' is in the dictionary, set `w = x`. - Read the second character 'x': - 'xx' is not in the dictionary, so add 'xx' as a new entry (code 4). - Output the code for 'x' (1), and set `w = x`. - Process 'y': - 'xy' is not in the dictionary, add 'xy' as code 5. - Output the code for 'x' (1), set `w = y`. - Read the space character: - Since 'y ' is not in the dictionary, add 'y ' as code 6. - Output the code for 'y' (2) and set `w = ' '`.

Continue Compression

Continue compressing the rest of the message: - Read 'y': - ' y' is not in the dictionary, add ' y' as code 7. - Output the code for ' ' (3) and set `w = y`. - Read 'x': - Since 'yx' is not in the dictionary, add 'yx' as code 8. - Output code for 'y' (2) and set `w = x`. - Process space: - 'x ' is not in the dictionary, add 'x ' as code 9. - Output code for 'x' (1), set `w = ' '`.

Finalize Compression

Finish compressing: - Read 'x': - ' x' is not in the dictionary, add ' x' as code 10. - Output code for ' ' (3), set `w = x`. - Read the next 'x': - 'xx' is in the dictionary, so continue. - Read 'y': - 'xxy' is not in the dictionary, add 'xxy' as code 11. - Output code for 'xx' (4), set `w = y`. - Read 'y': - 'yy' is not in the dictionary, add 'yy' as code 12. - Output code for 'y' (2), set `w = y`, and then restart for the next segment, which is done now.

Final Dictionary

The final dictionary after completing the LZW compression is: 1. 'x' 2. 'y' 3. ' ' 4. 'xx' 5. 'xy' 6. 'y ' 7. ' y' 8. 'yx' 9. 'x ' 10. ' x' 11. 'xxy' 12. 'yy' Each new string detected during processing was added to the dictionary.

Key concepts

Dictionary Initialization

When using LZW compression, the first step is to initialize the dictionary. This dictionary holds the unique patterns or substrings found in the text. At the start, it usually contains only the individual characters that appear in the input and sometimes additional special symbols like spaces or punctuation marks.
For example, in the problem provided, our dictionary starts with:

• 'x'
• 'y'
• ' '

These characters are assigned initial codes:

• 1 for 'x'
• 2 for 'y'
• 3 for ' '

Starting with a small dictionary ensures that as new patterns are found, they'll be added, expanding our collection of codes. This sets up the foundation for the LZW compression process.

Code Assignment

After initializing the dictionary, each character or pattern in the string is assigned a code. Code assignment is a critical part of LZW compression, as it translates the input string into a sequence of codes.
During processing, whenever a new pattern is recognized, it is added to the dictionary, and assigned the next available number. For instance:

• 'xx' is added as code 4 when 'x' is followed by another 'x'.
• Then 'xy' becomes code 5 when a 'y' follows 'x'.

This process continues throughout the input string. The numeric codes replace the original characters, which allows for more efficient storage and transmission. By reusing dictionary entries for common sequences, the LZW method makes repetitive data consume less space.

String Processing

String processing is the core task in LZW where we read and interpret the input string to update our dictionary and produce compressed output. We begin the process by setting an empty string 'w', to record subsequences.
We examine the input character by character, forming longer strings when possible. Consider the following steps from the solution:

• Start with the first 'x', since it exists in the dictionary, 'w' becomes 'x'.
• Upon encountering the next 'x', and noticing 'xx' is new, we add 'xx' to the dictionary.
• Each time 'w' is updated, until it pairs with a character that forms a new string not yet in our dictionary.

Reading through characters and continually updating 'w', allows the addition of new strings into the dictionary, crucially expanding the compression benefits as we progress through more of the text.

Compression Algorithm

The LZW compression algorithm elegantly combines dictionary initialization, code assignment, and string processing to compress data.
With our initial dictionary set, the algorithm uses a systematic approach:

• An input string is scanned from start to finish.
• Patterns not currently in the dictionary are added with sequential codes.
• Already known patterns are replaced by their respective codes, reducing size.

By following these steps, we achieve a compressed output where repeated patterns are minimized, recording just their code representations. The efficiency of LZW comes from its ability to rapidly adapt and expand the dictionary, allowing for substantial compression gains without needing to store lengthy strings, only their numeric code translations. This algorithm is widely used where space efficiency is as important as preserving data integrity.