Does It Make Sense to Compress a File Multiple Times?

1. Overview

In this tutorial, we’ll explore whether compressing a file multiple times is practical and beneficial.

We’ll discuss the concepts of file compression, examine the effects of applying compression algorithms more than once, and discuss alternative strategies for achieving better compression.

2. Understanding File Compression

Before we assess the merits of multiple compressions, it’s essential to understand how file compression works. It reduces data size by encoding information more efficiently, often eliminating redundancies. Let’s explore this further.

2.1. How Do Compression Algorithms Work?

Compression algorithms analyze data to find patterns and repetitions. They replace these patterns with shorter representations, effectively reducing the file size. There are two main types of compression:

• Lossless compression: This method reduces file size without losing any original data. When we decompress a losslessly compressed file, we retrieve a replica of the original. Common formats include ZIP, GZIP, and PNG.
• Lossy Compression: This method achieves higher compression ratios by discarding some data, which may result in a loss of quality. It’s commonly used for images, audio, and video files where a perfect reproduction isn’t necessary. Examples include JPEG and MP3.

3. Compressing a File Multiple Times

Let’s investigate what happens when we compress a file more than once using the same or different compression algorithms.

3.1. Reapplying the Same Compression Algorithm

When we compress a file using an algorithm like GZIP and then compress the resulting file again with the same algorithm, we typically observe little to no further reduction in size. In some cases, the file size may increase due to additional compression headers and metadata.

Compression algorithms are designed to find and eliminate redundancies in data. After the first compression, most of these redundancies are removed. Reapplying the same algorithm doesn’t find new patterns to compress, leading to negligible size reduction.

3.2. Using Different Compression Algorithms

Using a different compression algorithm on an already compressed file might seem like a way to achieve further size reduction. However, this approach usually provides minimal benefits. Although each algorithm has its strengths and may handle specific data types better, most redundancies are eliminated once the data is compressed.

For example, if we compress a file first with GZIP and then with BZIP2, we might achieve a slightly smaller file, but the difference is often insignificant compared to the original size reduction from the first compression.

4. Practical Experiments

Let’s conduct practical experiments by compressing files multiple times and observing the outcomes to solidify our understanding.

4.1. Experiment with a Text File

Let’s create a large text file with repetitive content to maximize the potential for compression.

First, let’s generate a text file containing 100,000 identical lines:

yes "This is a sample line of text to test compression." | head -n 100000 > sample.txt

We compress the file using GZIP:

gzip sample.txt

This produces sample.txt.gz.

Next, we compress the already compressed file:

gzip sample.txt.gz

This results in sample.txt.gz.gz. Some operating systems might warn us that the compression has already happened and that the file has a .gz extension, and they won’t perform any more operations.

Let’s now compare sizes:

File

Size

sample.txt

Approximately 4.5 MB

sample.txt.gz

Approximately 10 KB

sample.txt.gz.gz

Approximately 11 KB

The initial compression reduced the file size due to the high redundancy in the text file. The second compression showed no significant size reduction because GZIP couldn’t find additional patterns to compress. In fact, we even observe a slight increase in size due to added compression headers!

4.2. Experiment with a Binary File

Now, let’s use a JPEG image, which is already compressed with a lossy algorithm.

Let’s start with compressing the following JPEG file:

gzip image.jpg

We then compress it again:

gzip image.jpg.gz

Let’s compare the output sizes:

File

Size

image.jpg

2 MB

image.jpg.gz

Approximately 2 MB (no significant change)

image.jpg.gz.gz

2.01 MB (slightly larger than the original)

Since JPEG images are compressed efficiently, applying GZIP doesn’t reduce the size. The compressed file might be slightly larger due to added headers. The data within the JPEG doesn’t contain redundant patterns that GZIP can exploit.

5. Theoretical Perspectives

The key concept is entropy from information theory, which provides a mathematical framework for the limits of compression.

5.1. Entropy and Data Compression

Entropy, in the context of information theory, measures the randomness or unpredictability of data. Introduced by Claude Shannon in 1948, entropy quantifies the minimum number of bits needed to encode a string of symbols based on the frequency of those symbols. The more unpredictable the data, the higher its entropy, and the more bits are required to represent it without loss.

Shannon’s Source Coding Theorem states that it’s impossible to compress data losslessly beyond its entropy limit. This theorem defines the entropy H of a source as:

   

In this equation, the summation is over all possible symbols. This calculation determines the average minimum number of bits required to represent each symbol in the data set.

5.2. Implications of Compression Limits

After the first compression, most of the data’s redundancy is eliminated, and the file appears more random to compression algorithms. Subsequent compression attempts cannot find additional patterns to exploit, as the data now resembles random noise. Compressed data often looks indistinguishable from random data because the compression process removes predictable structures.

Moreover, each compression algorithm introduces its overhead in the form of headers, footers, and metadata required for decompression. When we compress an already compressed file, we add more of this overhead, which can increase the total file size rather than reduce it.

When applying a compression algorithm to data, we aim to reduce the data entropy by eliminating redundancy and representing common patterns more efficiently. After compression, the data entropy approaches the theoretical lower limit as many redundancies are removed in the process. Once we reach this limit, further compression without loss of information becomes impossible because there are no additional patterns or redundancies to exploit.

For instance, highly redundant data, like text files with repeated phrases, has low entropy and is highly compressible. In contrast, data that is already random or lacks patterns, such as encrypted files, has high entropy and cannot be compressed effectively.

5.3. The Pigeonhole Principle

The Pigeonhole Principle in mathematics states that if we have more items than containers, at least one container must hold more than one item.

Applied to compression, the principle explains that if we try to map a larger set of data representations to a smaller set (attempting to compress data beyond its entropy limit), we’ll inevitably encounter collisions where different data inputs produce the same compressed output. The collisions lead to a loss of information upon decompression, which is unacceptable in lossless compression.

Understanding these theoretical limits reinforces the practical observations from our experiments. The initial compression brings the data closer to its entropy limit, leaving little to no room for further size reduction through additional compression. Therefore, compressing a file multiple times does not yield significant benefits and can sometimes be counterproductive due to added overhead.

6. Alternative Approaches for Better Compression

We should consider other strategies to achieve maximum compression rather than compressing a file multiple times.

6.1. Choosing the Right Compression Algorithm

Different algorithms are optimized for different data types. By selecting the most appropriate one, we can achieve better results.

Algorithms like BZIP2 and LZMA (used by 7-Zip) often outperform GZIP for text files due to their advanced compression techniques:

• BZIP2 offers better compression ratios but is slower
• LZMA provides high compression with a balance between speed and efficiency

For executables or archives, algorithms that handle binary data efficiently are preferable:

• PAQ delivers excellent compression but is significantly slower
• ZPAQ is an incremental journaling archiver offering good compression

6.2. Preprocessing Data Before Compression

Transforming data to increase redundancy can enhance compression effectiveness.

For example, when dealing with text data, we can remove unnecessary formatting and whitespace or convert data to a more compressible format to improve compression ratios. On the other hand, when working with time series, it’s more efficient to store the differences between data segments rather than the entire data.

6.3. Archiving Multiple Files

Combining multiple files into a single archive before compression can improve the overall compression ratio.

By archiving files with TAR and then compressing, we allow the algorithm to find redundancy across files:

tar -cf archive.tar file1.txt file2.txt
gzip archive.tar

7. Practical Considerations

While compressing files multiple times isn’t beneficial, there are practical considerations to be aware of.

7.1. Increased Processing Time

Repeated compression consumes additional CPU resources and time without proportional gains. This extra processing can be significant for large files or datasets, leading to inefficiencies in system performance.

It’s important that we balance the time spent compressing against the actual space saved.

7.2. Complexity and Errors

Multiple layers of compression can complicate file handling and increase the risk of errors.

Extraction complexity is increased as we must decompress files multiple times.

There’s a risk of error propagation as corruption in one layer affects all subsequent layers.

7.3. Compatibility Issues

Not all systems or applications can handle files compressed multiple times or with uncommon algorithms, leading to accessibility problems. Users may encounter errors or be unable to decompress files if they lack the necessary software or knowledge.

This can hinder collaboration and data sharing across different platforms.

8. Conclusion

In this article, we explored whether compressing a file multiple times makes sense. We determined through theoretical analysis and practical experiments that repeatedly applying the same compression algorithm doesn’t yield significant size reductions. In some cases, it can even lead to a slight increase in file size due to additional metadata.

Using different compression algorithms on already compressed data offers minimal advantages, as the redundancy necessary for compression has been eliminated. For better results, it’s more effective to choose the most suitable algorithm for the data type, preprocess the data to enhance redundancy or combine multiple files into a single archive before compressing.