References and further reading

Heaps' law was discovered by Heaps (1978). See also Baeza-Yates and Ribeiro-Neto (1999). A detailed study of vocabulary growth in large collections is (Williams and Zobel, 2005). Zipf's law is due to Zipf (1949). Witten and Bell (1990) investigate the quality of the fit obtained by the law. Other term distribution models, including K mixture and two-poisson model, are discussed by Manning and Schütze (1999, Chapter 15). Carmel et al. (2001), Büttcher and Clarke (2006), Blanco and Barreiro (2007), and Ntoulas and Cho (2007) show that lossy compression can achieve good compression with no or no significant decrease in retrieval effectiveness.

Dictionary compression is covered in detail by Witten et al. (1999, Chapter 4), which is recommended as additional reading.

Subsection 5.3.1 is based on (Scholer et al., 2002). The authors find that variable byte codes process queries two times faster than either bit-level compressed indexes or uncompressed indexes with a 30% penalty in compression ratio compared with the best bit-level compression method. They also show that compressed indexes can be superior to uncompressed indexes not only in disk usage, but also in query processing speed. Compared with VB codes, ``variable nibble'' codes showed 5% to 10% better compression and up to one third worse effectiveness in one experiment (Anh and Moffat, 2005). Trotman (2003) also recommends using VB codes unless disk space is at a premium. In recent work, Anh and Moffat (2006a;2005) and Zukowski et al. (2006) have constructed word-aligned binary codes that are both faster in decompression and at least as efficient as VB codes. Zhang et al. (2007) investigate the increased effectiveness of caching when a number of different compression techniques for postings lists are used on modern hardware.

codes (Exercise 5.3.2 ) and codes
were introduced by Elias (1975), who proved that
both codes are universal. In addition, codes are
asymptotically optimal for
.
codes perform better than codes if large
numbers (greater than 15) dominate. A good introduction to
information theory, including the concept of
*entropy* , is (Cover and Thomas, 1991).
While Elias codes are only asymptotically optimal,
arithmetic codes
(Witten et al., 1999, Section 2.4)
can be constructed to be
arbitrarily close to the optimum for any
.

Several additional index compression techniques are covered
by Witten et al. (1999; Sections 3.3 and 3.4 and Chapter 5).
They recommend using
*parameterized codes*
for
index compression, codes that explicitly model the
probability distribution of gaps for each term. For example,
they show that *Golomb codes* achieve better
compression ratios than codes for large
collections. Moffat and Zobel (1992) compare several
parameterized methods, including LLRUN
(Fraenkel and Klein, 1985).

The distribution of gaps in a postings list depends on the assignment of docIDs to documents. A number of researchers have looked into assigning docIDs in a way that is conducive to the efficient compression of gap sequences (Moffat and Stuiver, 1996; Blandford and Blelloch, 2002; Silvestri et al., 2004; Blanco and Barreiro, 2006; Silvestri, 2007). These techniques assign docIDs in a small range to documents in a cluster where a cluster can consist of all documents in a given time period, on a particular web site, or sharing another property. As a result, when a sequence of documents from a cluster occurs in a postings list, their gaps are small and can be more effectively compressed.

Different considerations apply to the compression of term frequencies and word positions than to the compression of docIDs in postings lists. See Scholer et al. (2002) and Zobel and Moffat (2006). Zobel and Moffat (2006) is recommended in general as an in-depth and up-to-date tutorial on inverted indexes, including index compression.

This chapter only looks at index compression for Boolean
retrieval. For *ranked retrieval*
(Chapter 6 ), it is advantageous to order postings
according to term frequency instead of docID. During query
processing, the scanning of many postings lists can then be
terminated early because smaller weights do not change the
ranking of the highest ranked documents found so
far.
It is not a good idea to precompute and
store weights in the index (as opposed to frequencies)
because they cannot be compressed as well as integers
(see impactordered).

*Document compression* can also be important in an
efficient information retrieval system. de Moura et al. (2000)
and Brisaboa et al. (2007) describe
compression schemes that allow direct searching of terms and
phrases in the compressed text, which is infeasible with
standard text compression utilities like gzip and
compress.

**Exercises.**

- We have defined unary codes as being ``10'': sequences of 1s
terminated by a 0. Interchanging the roles of 0s and 1s
yields an equivalent ``01'' unary code. When this 01 unary
code is used, the construction of a
code can be stated as follows: (1) Write
down in binary using
bits. (2) Prepend 0s.
(i) Encode the numbers in Table 5.5 in this alternative
code.
(ii)
Show that this method produces
a well-defined alternative
code in the sense that it has the same length and can
be uniquely decoded.
- Unary code is not a universal code in the sense
defined above. However, there exists a distribution over gaps
for which unary code is optimal. Which distribution is this?
- Give some examples of terms that violate the
assumption that gaps all have the same size (which we made
when estimating the space requirements of a -encoded
index). What are
general characteristics of these terms?
- Consider a term whose postings list has size ,
say, . Compare the size of the
-compressed gap-encoded postings list if the distribution of the
term is uniform (i.e., all gaps have the same size) versus
its size when the distribution is not uniform. Which
compressed postings list is smaller?
- Work out the sum
in Equation 12
and show it adds up to about
251 MB. Use the numbers in Table 4.2 , but do not round , , and the number of vocabulary blocks.

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2009-04-07