number | unary code | length | offset | code | ||
0 | 0 | |||||
1 | 10 | 0 | 0 | |||
2 | 110 | 10 | 0 | 10,0 | ||
3 | 1110 | 10 | 1 | 10,1 | ||
4 | 11110 | 110 | 00 | 110,00 | ||
9 | 1111111110 | 1110 | 001 | 1110,001 | ||
13 | 1110 | 101 | 1110,101 | |||
24 | 11110 | 1000 | 11110,1000 | |||
511 | 111111110 | 11111111 | 111111110,11111111 | |||
1025 | 11111111110 | 0000000001 | 11111111110,0000000001 |
VB codes use an adaptive number of bytes depending on the size of the gap. Bit-level codes adapt the length of the code on the finer grained bit level. The simplest bit-level code is unary code . The unary code of is a string of 1s followed by a 0 (see the first two columns of Table 5.5 ). Obviously, this is not a very efficient code, but it will come in handy in a moment.
How efficient can a code be in principle? Assuming the gaps with are all equally likely, the optimal encoding uses bits for each . So some gaps ( in this case) cannot be encoded with fewer than bits. Our goal is to get as close to this lower bound as possible.
A method that is within a factor of optimal is encoding . codes implement variable-length encoding by splitting the representation of a gap into a pair of length and offset. Offset is in binary, but with the leading 1 removed.^{} For example, for 13 (binary 1101) offset is 101. Length encodes the length of offset in unary code. For 13, the length of offset is 3 bits, which is 1110 in unary. The code of 13 is therefore 1110101, the concatenation of length 1110 and offset 101. The right hand column of Table 5.5 gives additional examples of codes.
A code is decoded by first reading the unary code up to the 0 that terminates it, for example, the four bits 1110 when decoding 1110101. Now we know how long the offset is: 3 bits. The offset 101 can then be read correctly and the 1 that was chopped off in encoding is prepended: 101 1101 = 13.
The length of offset is bits and the length of length is bits, so the length of the entire code is bits. codes are always of odd length and they are within a factor of 2 of what we claimed to be the optimal encoding length . We derived this optimum from the assumption that the gaps between and are equiprobable. But this need not be the case. In general, we do not know the probability distribution over gaps a priori.
The characteristic of a discrete probability distribution^{} that determines its coding properties (including whether a code is optimal) is its entropy , which is defined as follows:
(4) |
It can be shown
that the lower bound for the expected length of a
code is if certain conditions hold (see the references). It can
further be shown that for
,
encoding is within a factor of 3 of this optimal encoding,
approaching 2 for large :
(5) |
In addition to universality, codes have two other properties that are useful for index compression. First, they are prefix free , namely, no code is the prefix of another. This means that there is always a unique decoding of a sequence of codes - and we do not need delimiters between them, which would decrease the efficiency of the code. The second property is that codes are parameter free . For many other efficient codes, we have to fit the parameters of a model (e.g., the binomial distribution) to the distribution of gaps in the index. This complicates the implementation of compression and decompression. For instance, the parameters need to be stored and retrieved. And in dynamic indexing, the distribution of gaps can change, so that the original parameters are no longer appropriate. These problems are avoided with a parameter-free code.
How much compression of the inverted index do codes
achieve? To answer this question we use Zipf's law, the term
distribution model introduced in Section 5.1.2 .
According to Zipf's law, the collection frequency is proportional to the
inverse of the rank , that is, there is a constant such that:
(6) |
(7) | |||
(8) |
(9) |
(10) |
Now we have derived term statistics that characterize the distribution of terms in the collection and, by extension, the distribution of gaps in the postings lists. From these statistics, we can calculate the space requirements for an inverted index compressed with encoding. We first stratify the vocabulary into blocks of size . On average, term occurs times per document. So the average number of occurrences per document is for terms in the first block, corresponding to a total number of gaps per term. The average is for terms in the second block, corresponding to gaps per term, and for terms in the third block, corresponding to gaps per term, and so on. (We take the lower bound because it simplifies subsequent calculations. As we will see, the final estimate is too pessimistic, even with this assumption.) We will make the somewhat unrealistic assumption that all gaps for a given term have the same size as shown in Figure 5.10. Assuming such a uniform distribution of gaps, we then have gaps of size 1 in block 1, gaps of size 2 in block 2, and so on.
Encoding the gaps of size with codes, the number of bits needed for the postings list of a term in the th block (corresponding to one row in the figure) is:
For Reuters-RCV1,
400,000
and
When we run compression on Reuters-RCV1, the actual size of the compressed index is even lower: 101 MB, a bit more than one tenth of the size of the collection. The reason for the discrepancy between predicted and actual value is that (i) Zipf's law is not a very good approximation of the actual distribution of term frequencies for Reuters-RCV1 and (ii) gaps are not uniform. The Zipf model predicts an index size of 251 MB for the unrounded numbers from Table 4.2 . If term frequencies are generated from the Zipf model and a compressed index is created for these artificial terms, then the compressed size is 254 MB. So to the extent that the assumptions about the distribution of term frequencies are accurate, the predictions of the model are correct.
data structure | size in MB | |
dictionary, fixed-width | 19.211.2 | |
dictionary, term pointers into string | 10.8 7.6 | |
, with blocking, | 10.3 7.1 | |
, with blocking & front coding | 7.9 5.9 | |
collection (text, xml markup etc) | 3600.0 | |
collection (text) | 960.0 | |
term incidence matrix | 40,000.0 | |
postings, uncompressed (32-bit words) | 400.0 | |
postings, uncompressed (20 bits) | 250.0 | |
postings, variable byte encoded | 116.0 | |
postings, encoded | 101.0 |
Table 5.6 summarizes the compression techniques covered in this chapter. The term incidence matrix (Figure 1.1 , page 1.1 ) for Reuters-RCV1 has size bits or 40 GB. The numbers were the collection (3600 MB and 960 MB) are for the encoding of RCV1 of CD, which uses one byte per character, not Unicode.
codes achieve great compression ratios - about 15% better than variable byte codes for Reuters-RCV1. But they are expensive to decode. This is because many bit-level operations - shifts and masks - are necessary to decode a sequence of codes as the boundaries between codes will usually be somewhere in the middle of a machine word. As a result, query processing is more expensive for codes than for variable byte codes. Whether we choose variable byte or encoding depends on the characteristics of an application, for example, on the relative weights we give to conserving disk space versus maximizing query response time.
The compression ratio for the index in Table 5.6 is about 25%: 400 MB (uncompressed, each posting stored as a 32-bit word) versus 101 MB () and 116 MB (VB). This shows that both and VB codes meet the objectives we stated in the beginning of the chapter. Index compression substantially improves time and space efficiency of indexes by reducing the amount of disk space needed, increasing the amount of information that can be kept in the cache, and speeding up data transfers from disk to memory.
Exercises.
encoded gap sequence of run 1 | 1110110111111001011111111110100011111001 | ||
encoded gap sequence of run 2 | 11111010000111111000100011111110010000011111010101 |