How to efficiently predict if data is compressible

Question

I want to write a storage backend to store larger chunks of data. The data can be anything, but it is mainly binary files (images, pdfs, jar files) or text files (xml, jsp, js, html, java...). I found most of the data is already compressed. If everything is compressed, about 15% disk space can be saved.

I am looking for the most efficient algorithm that can predict with high probability that a chunk of data (let's say 128 KB) can be compressed or not (lossless compression), without having to look at all the data if possible.

The compression algorithm will be either LZF, Deflate, or something similar (maybe Google Snappy). So predicting if data is compressible should be much faster than compressing the data itself, and use less memory.

Algorithms I already know about:

• Try to compress a subset of the data, let's say 128 bytes (this is a bit slow)

• Calculate the sum of 128 bytes, and if it's within a certain range then it's likely not compressible (within 10% of 128 * 127) (this is fast, and relatively good, but I'm looking for something more reliable, because the algorithm really only looks at the topmost bits for each byte)

• Look at the file headers (relatively reliable, but feels like cheating)

I guess the general idea is that I need an algorithm that can quickly calculate if the probability of each bit in a list of bytes is roughly 0.5.

Update

I have implemented 'ASCII checking', 'entropy calculation', and 'simplified compression', and all give good results. I want to refine the algorithms, and now my idea is to not only predict if data can be compressed, but also how much it can be compressed. Possibly using a combination of algorithms. Now if I could only accept multiple answers... I will accept the answer that gave the best results.

Additional answers (new ideas) are still welcome! If possible, with source code or links :-)

Update 2

A similar method is now implemented in Linux.

Answer 1

From my experience almost all of the formats that can effectively be compressed are non-binary. So checking if about 70-80% of the characters are within in the [0-127] rage should do the trick.

If you want to to it "properly" (even though I really can't see a reason to do that), you either have to run (parts of) your compression algorithm on the data or calculate the entropy, as tskuzzy already proposed.

Answer 2

I expect there's no way to check how compressible something is until you try to compress it. You could check for patterns (more patterns, perhaps more compressible), but then a particular compression algorithmn may not use the patterns you checked for - and may do better than you expect. Another trick may be to take the first 128000 bytes of data, push it through Deflate/Java compression, and see if it's less than the original size. If so - chances are it's worthwhile compressing the entire lot.

Answer 3

Also -- Why not try lzop? I can personally vouch for the fact that it's faster, much faster (compression and decompression) than bzip, gzip, zip, rar...

http://www.lzop.org

Using it for disk image compression makes the process disk-IO bound. Using any of the other compressors makes the process CPU-bound (i.e., the other compressors use all available CPU, lzop (on a reasonable CPU) can handle data at the same speed a 7200 RPM stock hard drive can dish it out...)

I'll bet if you tested it with the first X bytes of a 'test compression' string, it would be much faster than most other methods...

Answer 4

Fast compressor such as LZ4 already have built-in checks for data compressibility. They quickly skip the bad segments to concentrate on more interesting ones. To give a proper example, LZ4 on non-compressible data works at almost RAM speed limit (2GB/s on my laptop). So there is little room for a detector to be even faster. You can try it for yourself : http://code.google.com/p/lz4/

Answer 5

Calculate the entropy of the data. If it has high entropy (~1.0), it is not likely going to be further compressed. If it has low entropy (~0.0), then that means that there isn't a lot of "information" in it and can be further compressed.

It provides a theoretical measure of how compressed a piece of data can get.

Answer 6

I implemented a few methods to test if data is compressible.

Simplified Compression

This basically checks for duplicate byte pairs:

static boolean isCompressible(byte[] data, int len) {
    int result = 0;
    // check in blocks of 256 bytes, 
    // and sum up how compressible each block is
    for (int start = 0; start < len; start += 256) {
        result += matches(data, start, Math.min(start + 255, len));
    }
    // the result is proportional to the number of 
    // bytes that can be saved
    // if we can save many bytes, then it is compressible
    return ((len - result) * 777) < len * 100;
}

static int matches(byte[] data, int i, int end) {
    // bitArray is a bloom filter of seen byte pairs
    // match counts duplicate byte pairs
    // last is the last seen byte
    int bitArray = 0, match = 0, last = 0;
    if (i < 0 || end > data.length) {
        // this check may allow the JVM to avoid
        // array bound checks in the following loop
        throw new ArrayIndexOutOfBoundsException();
    }
    for (; i < end; i++) {
        int x = data[i];
        // the bloom filter bit to set
        int bit = 1 << ((last ^ x) & 31);
        // if it was already set, increment match
        // (without using a branch, as branches are slow)
        match -= (-(bitArray & bit)) >> 31;
        bitArray |= bit;
        last = x;
    }
    return match;
}

On my (limited) set of test data, this algorithm is quite accurate. It about 5 times faster than compressing itself if the data is not compressible. For trivial data (all zeroes), it is about half as fast however.

Partial Entropy

This algorithm estimates the entropy of the high nibbles. I wanted to avoid using too many buckets, because they have to be zeroed out each time (which is slow if the blocks to check are small). 63 - numberOfLeadingZeros is the logarithm (I wanted to avoid using floating point numbers). Depending on the data, it is faster or slower than the algorithm above (not sure why). The result isn't quite as accurate as the algorithm above, possibly because of using only 16 buckets, and only integer arithmetic.

static boolean isCompressible(byte[] data, int len) {
    // the number of bytes with 
    // high nibble 0, 1,.., 15
    int[] sum = new int[16];
    for (int i = 0; i < len; i++) {
        int x = (data[i] & 255) >> 4;
        sum[x]++;
    }
    // see wikipedia to understand this formula :-)
    int r = 0;
    for (int x : sum) {
        long v = ((long) x << 32) / len;
        r += 63 - Long.numberOfLeadingZeros(v + 1);
    }
    return len * r < 438 * len;
}