Question

Show how run-length encoding can be used to compress the following text stream: xxxyyyyy zzzzzAAxxxx What is the compression ratio? (Assume each digit and letter requires 8 bits.)

Short answer

The compression uses 80 bits instead of 144, giving a compression ratio of 0.56.

Step-by-step solution

Identify Consecutive Characters

Identify groups of consecutive repeating characters in the text stream. We have four such groups: 'xxx', 'yyyyy', 'zzzzz', 'AA', and 'xxxx'.

Encode Each Group

For each group of consecutive characters, write down the count followed by the character. This is the run-length encoded version. For the text 'xxxyyyyy zzzzzAAxxxx', this will become '3x5y1 5z2A4x'.

Calculate Original Bit Length

Calculate the bit length of the original string. Each character (letter or space) uses 8 bits. The original text contains 5 x's, 5 y's, 1 space, 5 z's, and 2 A's, totaling 18 characters: \(18 \times 8 = 144\) bits.

Calculate Encoded Bit Length

Calculate the bit length of the encoded string. Each digit and each character use 8 bits. For '3x5y1 5z2A4x', we have five digits and five letters, totaling 10 characters: \(10 \times 8 = 80\) bits.

Calculate Compression Ratio

The compression ratio is calculated by dividing the encoded bit length by the original bit length: \(\frac{80}{144}\approx 0.56\).

Key concepts

Data Compression

Data compression is a fundamental concept in computing and data science. It involves reducing the size of data by encoding it more efficiently. This process is essential for saving storage space and reducing transmission times over networks.
In everyday computing, data compression makes file storage more efficient and allows faster processing. For example, photos, videos, and documents are often compressed to take up less space on your devices or when being sent over networks.
There are several techniques for compressing data, each with its strengths and trade-offs. One widely used method is run-length encoding (RLE), which is especially useful for compressing data with lots of repetitive elements. It works by replacing sequences of repeating data elements with a single data value and a count of its repetitions.

Compression Ratio

The compression ratio is a measure of how much data has been compressed, and it's a critical indicator of the efficiency of a compression algorithm.
Simply put, it is calculated by dividing the compressed (or encoded) size by the original size. It tells us how much smaller the compressed data is compared to the original.
Using run-length encoding, you might compress a string and get a compression ratio. For instance, in our problem, we found a compression ratio of approximately 0.56. Generally, values closer to 0 indicate highly effective compression, whereas values closer to 1 suggest minimal compression.
It's important to recognize that not all data compresses well. Some data, if already efficiently stored, may not see as significant a reduction. However, higher ratios are typically desired as they imply more substantial space savings.

Bit Length

Bit length refers to the total number of bits used to store or represent data. It's a crucial factor to consider when assessing both size and efficiency in data storage and transmission.

• In our example, each character, whether a letter or space, is represented using 8 bits. This is typical in many encoding systems, like ASCII, where each character has a fixed bit length.
• The original text had 18 characters, resulting in a total of 144 bits (calculated as \(18 \times 8\) bits).
• For the encoded string, because each digit and each letter also use 8 bits, the total bit length was 80 bits (calculated as \(10 \times 8\) bits).

Choosing efficient encoding practices, like run-length encoding for compressible data, helps in reducing bit length significantly, thus lowering storage requirements and transmission times.

Encoded String

An encoded string is the result of applying a specific encoding algorithm to a dataset to store it more efficiently. In compression, encoding transforms longer sequences into a more compact form.
In our exercise using run-length encoding, the original string 'xxxyyyyy zzzzzAAxxxx' was translated into the encoded string '3x5y1 5z2A4x'.
This encoded form contains two parts for each sequence: a number representing the count of consecutive characters and the character itself. Such encoding effectively reduces the size of the data by minimizing redundancy.
This transformation makes use of numbers to denote the frequency of characters instead of repeating the characters themselves, creating a compact representation. Hence, it enables the efficient transmission and storage of data, thereby optimizing resources. Understanding the format and structure of encoded strings is helpful for decoding them back to their original form.