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 4-byte integers.
Alternatively, we can use log2 800,000 ≈ 20 bits per docID.
Our goal: use far fewer than 20 bits per docID.
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
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.
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
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, binary-encode it in the 7 available bits and set c =1
Else encode G’s lower-order 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.
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 low-tech blend of variable-length coding and sensitivity to computer memory alignment matches (vs. bit-level codes, which we look at next).
There is also recent work on word-aligned codes that pack a variable number of gaps into one word
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….
We can compress better with bit-level 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
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 prefix-decodable, like VB
Gamma code can be used for any distribution
Gamma code is parameter-free
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