Compression
Now, we will consider compressing the space for the dictionary and postings
Basic Boolean index only
No study of positional indexes, etc.
We will consider compression schemes
Why compress the dictionary?
 Search begins with the dictionary
We want to keep it in memory
Memory footprint competition with other applications
Embedded/mobile devices may have very little memory
Even if the dictionary isn’t in memory, we want it to be small for a fast search startup time
So, compressing the dictionary is important
Dictionary storage  first cut
 Array of fixedwidth entries
 ~400,000 terms; 28 bytes/term = 11.2 MB.
↓ 20 bytes 4 bytes each
Dictionary search structure
Fixedwidth terms are wasteful
Most of the bytes in the Term column are wasted – we allot 20 bytes for 1 letter terms.
And we still can’t handle supercalifragilisticexpialidocious or hydrochlorofluorocarbons.
Written English averages ~4.5 characters/word.
Exercise: Why is/isn’t this the number to use for estimating the dictionary size?
Ave. dictionary word in English: ~8 characters
How do we use ~8 characters per dictionary term?
Short words dominate token counts but not type average.
Compressing the term list: DictionaryasaString
Store dictionary as a (long) string of characters:
Pointer to next word shows end of current word
 Hope to save up to 60% of dictionary space.
Space for dictionary as a string
 4 bytes per term for Freq.
4 bytes per term for pointer to Postings. → Now avg. 11 bytes/term,
3 bytes per term pointer ...Not 20
Avg. 8 bytes per term in term string
400K terms x 19 ⇒7.6 MB (against 11.2MB for fixed width)
Blocking
Store pointers to every kth term string.
Example below: k=4.
Need to store term lengths (1 extra byte)
Net
 Example for block size k = 4
Where we used 3 bytes/pointer without blocking
3 x 4 = 12 bytes,
now we use 3 + 4 = 7 bytes.
Shaved another ~0.5MB. This reduces the size of the dictionary from 7.6 MB to 7.1 MB.
We can save more with larger k.
Why not go with larger k?
Exercise
Estimate the space usage (and savings compared to 7.6 MB) with blocking, for block sizes of k = 4, 8 and 16.
Dictionary search without blocking

Dictionary search with blocking
Büinary search down to 4term block;
Then linear search through terms in block.
Blocks of 4 (binary tree), avg. = (1+2∙2+2∙3+2∙4+5)/8 = 3 compares
Exercise
Estimate the impact on search performance (and slowdown compared to k=1) with blocking, for block sizes of k = 4, 8 and 16.
Front coding
Frontcoding:
Sorted words commonly have long common prefix – store differences only
(for last k1 in a block of k)
8automata8automate9automatic10automation
→ 8automat*a1♦e2♦ic3♦ion
↑ ↑
Encodes automat Extra length beyond automat.
Begins to resemble general string compression.
RCV1 dictionary compression summary










Postings compression
The postings file is much larger than the dictionary, factor of at least 10.
Key desideratum: store each posting compactly.
A posting for our purposes is a docID.
For Reuters (800,000 documents), we would use 32 bits per docID when using 4byte integers.
Alternatively, we can use log2 800,000 ≈ 20 bits per docID.
Our goal: use far fewer than 20 bits per docID.
Postings: two conflicting forces
A term like arachnocentric occurs in maybe one doc out of a million – we would like to store this posting using log2 1M ~ 20 bits.
A term like the occurs in virtually every doc, so 20 bits/posting is too expensive.
Prefer 0/1 bitmap vector in this case
Postings file entry
We store the list of docs containing a term in increasing order of docID.
computer: 33,47,154,159,202 …
Consequence: it suffices to store gaps.
33,14,107,5,43 …
Hope: most gaps can be encoded/stored with far fewer than 20 bits.
Three postings entries
Variable length encoding
Aim:
For arachnocentric, we will use ~20 bits/gap entry.
For the, we will use ~1 bit/gap entry.
If the average gap for a term is G, we want to use ~log2G bits/gap entry.
Key challenge: encode every integer (gap) with about as few bits as needed for that integer.
This requires a variable length encoding
Variable length codes achieve this by using short codes for small numbers
Variable Byte (VB) codes
For a gap value G, we want to use close to the fewest bytes needed to hold log2 G bits
Begin with one byte to store G and dedicate 1 bit in it to be a continuation bit c
If G ≤127, binaryencode it in the 7 available bits and set c =1
Else encode G’s lowerorder 7 bits and then use additional bytes to encode the higher order bits using the same algorithm
At the end set the continuation bit of the last byte to 1 (c =1) – and for the other bytes c = 0.
Example
Other variable unit codes
Instead of bytes, we can also use a different “unit of alignment”: 32 bits (words), 16 bits, 4 bits (nibbles).
Variable byte alignment wastes space if you have many small gaps – nibbles do better in such cases.
Variable byte codes:
Used by many commercial/research systems
Good lowtech blend of variablelength coding and sensitivity to computer memory alignment matches (vs. bitlevel codes, which we look at next).
There is also recent work on wordaligned codes that pack a variable number of gaps into one word
Unary code
Represent n as n 1s with a final 0.
Unary code for 3 is 1110.
Unary code for 40 is
11111111111111111111111111111111111111110 .
Unary code for 80 is:
111111111111111111111111111111111111111111111111111111111111111111111111111111110
This doesn’t look promising, but….
Gamma codes
We can compress better with bitlevel codes
The Gamma code is the best known of these.
Represent a gap G as a pair length and offset
offset is G in binary, with the leading bit cut off
For example 13 → 1101 → 101
length is the length of offset
For 13 (offset 101), this is 3.
We encode length with unary code: 1110.
Gamma code of 13 is the concatenation of length and offset: 1110101
Gamma code examples
Gamma code properties
G is encoded using 2 ⌊ log G⌋ + 1 bits
Length of offset is ⌊ log G ⌋ bits
Length of length is ⌊log G ⌋ + 1 bits
All gamma codes have an odd number of bits
Almost within a factor of 2 of best possible, log_{2} G
Gamma code is uniquely prefixdecodable, like VB
Gamma code can be used for any distribution
Gamma code is parameterfree
Gamma seldom used in practice
Machines have word boundaries – 8, 16, 32, 64 bits
Operations that cross word boundaries are slower
Compressing and manipulating at the granularity of bits can be slow
Variable byte encoding is aligned and thus potentially more efficient
Regardless of efficiency, variable byte is conceptually simpler at little additional space cost
RCV1 compression
Index compression summary
 We can now create an index for highly efficient Boolean retrieval that is very space efficient
Only 4% of the total size of the collection
Only 1015% of the total size of the text in the collection
However, we’ve ignored positional information
Hence, space savings are less for indexes used in practice
But techniques substantially the same.