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How Much Can a Webpage Inflate?

August 23rd, 2013
tech  [html]
When your browser loads a page it's probably using gzip compression. When it asks for http://en.wikipedia.org/wiki/Cabbage it adds a note telling the server that it understands gzip:
  GET /wiki/Cabbage HTTP/1.1
  Host: en.wikipedia.org
  Accept-Encoding: gzip
Wikipedia's server responds:
  HTTP/1.0 200 OK
  Content-Encoding: gzip
  Content-Length: 37924
This tells us that the server did decide to use gzip to compress the page, and that after compression it was 37924 bytes. We can verify this manually:
$ curl -H 'Accept-Encoding: gzip' \
       -s http://en.wikipedia.org/wiki/Cabbage \
       | wc -c
Your browser can't work with gzipped data directly, so it has to expand it first:
$ curl -H 'Accept-Encoding: gzip' \
       -s http://en.wikipedia.org/wiki/Cabbage \
       | gunzip | wc -c
Unzipping it we have 157194 bytes, which means gzip managed to make the page 4x smaller. Great! Compression saves you gobs of time, bandwidth, and mobile data charges every year.

You might wonder, though: how much bigger can a file when you inflate it? That wikipedia page got 4x bigger, but could we have a small page that gets 100x bigger? A million times bigger? If someone with a devious attitude crafted a maximally pathological page, how bad could it get?

To answer this we need to look what gzip is doing. Like zip, gzip uses the deflate compression algorithm. (rfc1951) It has two pieces: first it condenses duplicate sections and then encodes symbols in length inversely proportional to their frequency. What does that mean?

Imagine I have "abc a abc abc ab ab a a cab abq". There's a lot of duplication: "abc" is repeated over an over. When "abc " comes up for the second time, however, we can replace it with a note saying "go back six and take four from there". That's what LZ77 does. Another thing I can do is notice that in those 31 characters there are 10 'a's, 9 spaces, 7 'b's, 4 'c's and a 'q'. While normal writing uses just as much space for 'q' as 'a', we can notice that 'a' is being used more often and use a small number of bytes to represent it while letting 'q' take up many more bytes. That's the main idea of Huffman coding.

The most likely candidate for the worst possible file is one that is all the same character: "aaaaaaaaaaa...". This can be encoded as "go back 1 and copy the maximum number of characters allowed". That requires two symbols: Code 0 for "go back one" and Code 285 for "copy the maximum allowed" which happens to be 258 bytes. [1] Huffman coding two symbols is easy: '0' for one symbol and '1' for the other. A single byte of "01010101" can then expand to 258*4 output bytes, or 1032x expansion. [2]

With real usage there's overhead, but it should be small compared to the total size. What do we see if we feed this to gzip?

dd if=/dev/zero bs=1000 count=1000 | gzip | wc -c
input bytes output bytes expansion potential
1000 (1K) 29 34x
10000 (10K) 45 222x
100000 (100K) 132 758x
1000000 (1M) 1003 997x
10000000 (10M) 9737 1027x
100000000 (100M) 97071 1030x
1000000000 (1G) 970501 1030x
10000000000 (10G) 9704731 1030x
We very quickly get close to 1032x, but even with 10G to compress we don't quite get there, so either I'm missing something about the format or the compressor I'm using could be slightly more efficient.

It's surprising that the limitation on the potential size of a gzip bomb turns out to be that the longest supported backreference is 258 bytes. I wonder how much thought went into that limit?

(If we used a format that supported run length encoding, for example bzip2, then we could have far larger potential for expansion.)

[1] You would think we would need a third symbol to represent the initial character to copy, but Code 0 and Code 285 are going to be repeated so many more times that we really want to keep their encodings small. Luckily deflate lets us divide the stream into blocks, each with their own encoding system. We can start with a tiny block that just encodes a single character, and then follow that with one very large block that is optimized for repeating that character over and over in sequences of 258.

[2] Technically I've only shown a lower bound on the amount of expansion, but after reading the gzip format several times I don't see how you can do better.

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