Index Compression

Ferrol Aderholdt

Motivation

Uncompressed indexes are large It might be useful for some modern devices to

support information retrieval techniques that would not be able to do with uncompressed indexes

Motivation (cont.)

Disk I/O is slow

Types of Compression

Lossy Compression that involves the removal of data.

Loseless Compression that involves no removal of data.

Overview

A lossy compression scheme Static Index Pruning

Loseless compression Elias Codes n-s encoding Golomb encoding Variable Byte Encoding (vByte) Fixed Binary Codewords CPSS-Tree

Static Index Pruning

Goal is to reduce the size of the index without reducing the precision such that a human can’t tell the difference between a pruned index and non-pruned index

Focuses on the top k or top δ results Assumes there is a scoring function

Assumes the function is based off of some table A such that A(t,d) > 0 if t is within d and A(t,d) = 0 otherwise

Static Index Pruning (cont.)

Two approaches1. Defined as Uniform pruning.

The removal of “all posting entries whose corresponding table values are bounded above by some fixed cutoff threshold”

Could have a term’s entire posting list pruned

2. Defined as Term based pruning An approach that attempts to guarantee that every

term will have at least some entries remaining in the index

Static Index Pruning (cont.)

Scoring functions are fuzzy Only need to find some scoring function S’

such that S’ is within a factor of epsilon of S Carmel et al proved this mathematically for

both uniform and term-based methods

Static Index Pruning (cont.)

Static Index Pruning results

Found that the idealized top k pruning algorithm did not work very well The smallest value in the posting list was almost

always above their threshold so little pruning was done

Modified the algorithm to apply a shift Subtracted the smallest value from all positive

scores with the list Greatly increased the pruning

Static Index Pruning results (cont.)

Static Index Pruning results (cont.)

Static Index Pruning results (cont.)

Overview

Loseless Compression

Elias Codes

Non-parameterized bitwise method of coding integers

Gamma Codes Represent a positive integer k with

stored as a unary code. This is followed by

the binary representation of the number

without the most significant bit Not efficient for numbers larger than 15

k2log1

Elias Codes (cont.)

Delta Codes Represent a positive integer k with

stored as a gamma code. This is followed by

the binary representation of the number

without the most significant bit Not efficient for small values

k2log1

n-s coding

Parameterized, bitwise encoding Uses a block of n bits followed by s stop bits. Also contains a parameter b which refers to

the base of the number. Meaning, the numbers represented in the blocks of n size cannot be greater than or equal to b.

n-s coding example

Let n=3, s=2, and the base be 6. Valid data blocks are 000, 001, 010, 011,

100, and 101. 101 100 001 11 would have the value of 5416

n-s coding (cont.)

[2] used n-s codes with prefix omission and run-length encoding

Ex.

n-s coding (cont.)

Run-length encoding is the process of replacing non-initial elements of a sequence with differences between adjacent elements. E.g.

n-s coding results

Golomb coding

Better compression and faster retrieval than Elias codes

Is parameterized This is usually stored separate using some other

compression scheme

vByte coding

A very simple bytewise compression scheme Uses 7 bits to code the data portion and the

most significant bit is reserved as a flag bit.

Scholer et. al.

Defined an inverted list to be the following:

Where the list is <freq,doc,[offsets]> Example inverted list for term “Matthew”:

<3,7,[6,51,117]><1,44,[12]><2,117,[14,1077]> Uses different coding schemes per part

E.g. Golomb for freq, Gamma for doc, and vByte for offset

tdotdodftdftd ,,,...,,,,,,0,

Scholer et al. (cont.)

One optimization is to require encoding to be byte aligned so that decompression can be faster

Another optimization when referring to Boolean or ranked queries is to ignore the offsets and only take into account flag bits within the offset. Referred to as scanning

Scholer et al. (cont.)

Third optimization is called signature blocks. An eight bit block that stores the flag bits of up to

eight blocks that follow. For example: 11100101

Represents 5 integers that are stored in the eight blocks Requires more space but allows the data blocks

to use all 8 bits instead of 7.

Scholer et al. results

Scholer et al. results (cont.)

Scholer et al. results (cont.)

Fixed Binary Codes

Often times the inverted list will be stored as a series of difference gaps between documents like so,

This reduces the amount of bits required to represent a document IDs on average

1,...,,,, 23121 tft ddddddf

Fixed Binary Codes (cont.)

Take for example the following list of d-gaps:

<12; 38, 17, 13, 34, 6 ,4 ,1, 3, 1, 2, 3, 1> If a binary code was used to encode this list,

6 bits would be used on each codeword when that would be unnecessary

Fixed Binary Codes (cont.)

Instead encode as spans:

<12; (6,4 : 38, 17, 13, 34),(3,1: 6),

(2,7 : 4, 1, 3, 1, 2, 3, 1)>

where the notation

would indicate that w-bit binary codes are to be used to code each of the next s values.

Similar to the approach of Anh and Moffat

sdddsw ,...,,:, 21

Anh and Moffat

Uses a selector then data representation for encoding A selector can be thought of as the unary portion

of gamma codes Data representation would be the binary portion of

gamma codes The selector uses a table of values where

each case is determined on the w-value and is relative to the previous case.

Anh and Moffat (cont.)

Anh and Moffat (cont.)

Using this list and assuming s1= 1, s2= 2, and s3= 4

From the table on the previous slide we get the following

With each selector as 4 bits (2 bits for w ± 3, 2 bits to choose s1-s3) it takes 16 bits plus the summation of all of the w x s pairs. So, 57 bits are used to encode this list. It would take 60 bits for gamma code.

Anh and Moffat (cont.)

Anh and Moffat (cont.)

The use of parsing is involved to discover segments. A graph is used in combination with shortest path

labeling Each node is a d-gap and the width to code it Each outgoing edge is a different way in which selector

might be used to cover some subsequent gaps.

Anh and Moffat (cont.)

A multiplier is used since every list can be different but the values for s1, s2, and s3 are fixed.

For example, if m=2 and s1= 1, s2= 2, and s3= 4, or 1-2-4, then they would be equal to 2-4-8.

An escape sequence can also be used on lists that have gaps that span larger than s3 would allow. This is the addition of an extra 4 bits stating that up to 15m

gaps can be placed under one selector

Anh and Moffat results (cont.)

Anh and Moffat results (cont.)

Anh and Moffat

Speeding up decoding

Need to exploit the cache and reduce both cache misses and TLB misses

Use CSS-trees or CPSS-trees CSS-trees are cache-sensitive search trees that

are a variation on m-ary trees. By making each node contiguous this reduces the need

for child pointers This allows for each node to fit into a cache line (32/64 bit)

CSS-Tree vs m-ary Tree

CPSS-trees

Cache/Page sensitive search trees main purpose is to reduce number cache/TLB misses during random searches Accomplished by making each node, except the

root, 4 KB in size and contains several CSS-Trees The CSS-Trees are the same size as a cache line and

contain the postings Either 32 or 64 bit

CPSS-trees results

CPSS-trees results (cont.)

Compressed CPSS-trees results

Compressed CPSS-tree results

Questions??

Questions??

References

[1] David Carmel, Doron Cohen, Ronald Fagin, Eitan Farchi, Michael Herscovici, Yoelle S. Maarek, Aya Soffer. Static Index Pruning for Information Retrieval Systems. SIGIR ’01: Proceedings of the 24th annual international ACM SIGIR conference on Research and development in information retrieval, pgs 43-50, 2001.

[2] Gordon Linoff, Craig Stanfill. Compression of Indexes with Full Positional Information in Very Large Text Databases. SIGIR ’93: Proceedings of the 16th annual international ACM SIGIR conference on Research and development in information retrieval, pgs 88-95, 1993.

References

[3] Falk Scholer, Hugh Williams, John Yiannis, and Justin Zobel. Compression of Inverted Indexes for Fast Query Evaluation. SIGIR ’02: Proceedings of the 25th annual international ACM SIGIR conference on Research and development in information retrieval, pgs 222-229, 2002.

[4] V. N. Anh and A. Moffat. Index Compression using Fixed Binary Codewords. ADC ’04: Proceedings of the 15th Australasian database conference, pg 61-67, 2004

References

[5] Stefan Buttcher and Charles L. A. Clarke. Index Compression is Good, Especially for Random Access. CIKM ’07: Proceedings of the sixteenth ACM conference on Conference on information and knowledge management, pgs 761-770, 2007.