Introduction to Information Retrieval Introduction to Information Retrieval Hinrich Schtze and Christina Lioma Lecture 5: Index Compression 1 Introduction to Information Retrieval Overview

Recap Compression Term statistics Dictionary compression Postings compression

2 Introduction to Information Retrieval Outline Recap Compression Term statistics

Dictionary compression Postings compression 3 Introduction to Information Retrieval Blocked Sort-Based Indexing

4 Introduction to Information Retrieval Single-pass in-memory indexing Abbreviation: SPIMI Key idea 1: Generate separate dictionaries for each block no need to maintain term-termID mapping across blocks. Key idea 2: Dont sort. Accumulate postings in postings lists as they occur. With these two ideas we can generate a complete inverted index for each block. These separate indexes can then be merged into one big index.

5 Introduction to Information Retrieval SPIMI-Invert 6 Introduction to Information Retrieval MapReduce for index construction 7 Introduction to Information Retrieval

Dynamic indexing: Simplest approach Maintain big main index on disk New docs go into small auxiliary index in memory. Search across both, merge results Periodically, merge auxiliary index into big index 8

Introduction to Information Retrieval Roadmap Today: index compression Next 2 weeks: perspective of the user: how can we give the user relevant results, how can we measure relevance, what types of user interactions are effective? After Pentecost: statistical classification and clustering in information retrieval Last 3 weeks: web information retrieval 9 Introduction to Information Retrieval

Take-away today Motivation for compression in information retrieval systems How can we compress the dictionary component of the inverted index? How can we compress the postings component of the inverted index? Term statistics: how are terms distributed in document collections? 10 Introduction to Information Retrieval Outline

Recap Compression Term statistics Dictionary compression

Postings compression 11 Introduction to Information Retrieval Why compression? (in general) Use less disk space (saves money) Keep more stuff in memory (increases speed) Increase speed of transferring data from disk to memory (again, increases speed) [read compressed data and decompress in memory] is faster than [read uncompressed data]

Premise: Decompression algorithms are fast. This is true of the decompression algorithms we will use. 12 Introduction to Information Retrieval Why compression in information retrieval? First, we will consider space for dictionary Main motivation for dictionary compression: make it small enough to keep in main memory Then for the postings file Motivation: reduce disk space needed, decrease time needed to read from disk

Note: Large search engines keep significant part of postings in memory We will devise various compression schemes for dictionary and postings. 13 Introduction to Information Retrieval Lossy vs. lossless compression Lossy compression: Discard some information Several of the preprocessing steps we frequently use can be viewed as lossy compression: downcasing, stop words, porter, number elimination

Lossless compression: All information is preserved. What we mostly do in index compression 14 Introduction to Information Retrieval Outline Recap

Compression Term statistics Dictionary compression Postings compression 15 Introduction to Information Retrieval

Model collection: The Reuters collection symbol N L M T statistics documents avg. # tokens per document word types avg. # bytes per token (incl. spaces/punct.) avg. # bytes per token (without spaces/punct.)

avg. # bytes per term (= word type) non-positional postings value 800,000 200 400,000 6 4.5 7.5 100,000,000 16 Introduction to Information Retrieval

Effect of preprocessing for Reuters 17 Introduction to Information Retrieval How big is the term vocabulary?

That is, how many distinct words are there? Can we assume there is an upper bound? Not really: At least 7020 1037 different words of length 20. The vocabulary will keep growing with collection size. Heaps law: M = kTb M is the size of the vocabulary, T is the number of tokens in the collection. Typical values for the parameters k and b are: 30 k 100 and b 0.5. Heaps law is linear in log-log space. It is the simplest possible relationship between collection size and vocabulary size in log-log space. Empirical law 18

Introduction to Information Retrieval Heaps law for Reuters Vocabulary size M as a function of collection size T (number of tokens) for Reuters-RCV1. For these data, the dashed line log10M = 0.49 log log10 T + 1.64 is the best least squares fit. Thus, M = 101.64T0.49 and k = 101.64 44 and b = 0.49.

19 Introduction to Information Retrieval Empirical fit for Reuters Good, as we just saw in the graph. Example: for the first 1,000,020 tokens Heaps law predicts 38,323 terms: 44 1,000,0200.49 38,323 The actual number is 38,365 terms, very close to the prediction. Empirical observation: fit is good in general. 20

Introduction to Information Retrieval Exercise What is the effect of including spelling errors vs. automatically correcting spelling errors on Heaps law? Compute vocabulary size M Looking at a collection of web pages, you find that there are 3000 different terms in the first 10,000 tokens and 30,000 different terms in the first 1,000,000 tokens. Assume a search engine indexes a total of 20,000,000,000 (2 1010) pages, containing 200 tokens on average What is the size of the vocabulary of the indexed collection as predicted by Heaps law?

21 Introduction to Information Retrieval Zipfs law Now we have characterized the growth of the vocabulary in collections. We also want to know how many frequent vs. infrequent terms we should expect in a collection. In natural language, there are a few very frequent terms and very many very rare terms. Zipfs law: The ith most frequent term has frequency cfi proportional to 1/i . cfi is collection frequency: the number of occurrences of the

term ti in the collection. 22 Introduction to Information Retrieval Zipfs law Zipfs law: The ith most frequent term has frequency proportional to 1/i . cf is collection frequency: the number of occurrences of the term in the collection. So if the most frequent term (the) occurs cf1 times, then the second most frequent term (of) has half as many occurrences . . . and the third most frequent term (and) has a third as many occurrences

Equivalent: cfi = cik and log cfi = log c +k log i (for k = 1) Example of a power law 23 Introduction to Information Retrieval Zipfs law for Reuters Fit is not great. What is important is the key insight: Few frequent terms, many rare terms. 24

Introduction to Information Retrieval Outline Recap Compression Term statistics

Dictionary compression Postings compression 25 Introduction to Information Retrieval Dictionary compression The dictionary is small compared to the postings file. But we want to keep it in memory. Also: competition with other applications, cell phones,

onboard computers, fast startup time So compressing the dictionary is important. 26 Introduction to Information Retrieval Recall: Dictionary as array of fixed-width entries Space needed: 20 bytes 4 bytes 4 bytes for Reuters: (20+4+4)*400,000 = 11.2 MB 27

Introduction to Information Retrieval Fixed-width entries are bad. Most of the bytes in the term column are wasted. We allot 20 bytes for terms of length 1. We cant handle HYDROCHLOROFLUOROCARBONS and SUPERCALIFRAGILISTICEXPIALIDOCIOUS Average length of a term in English: 8 characters How can we use on average 8 characters per term? 28 Introduction to Information Retrieval

Dictionary as a string 29 Introduction to Information Retrieval Space for dictionary as a string 4 bytes per term for frequency 4 bytes per term for pointer to postings list 8 bytes (on average) for term in string

3 bytes per pointer into string (need log2 8 400000 < 24 bits to resolve 8 400,000 positions) Space: 400,000 (4 +4 +3 + 8) = 7.6MB (compared to 11.2 MB for fixed-width array) 30 Introduction to Information Retrieval Dictionary as a string with blocking 31 Introduction to Information Retrieval

Space for dictionary as a string with blocking Example block size k = 4 Where we used 4 3 bytes for term pointers without blocking . . . . . .we now use 3 bytes for one pointer plus 4 bytes for indicating the length of each term. We save 12 (3 + 4) = 5 bytes per block. Total savings: 400,000/4 log 5 = 0.5 MB This reduces the size of the dictionary from 7.6 MB to 7.1 MB. 32 Introduction to Information Retrieval Lookup of a term without blocking

33 Introduction to Information Retrieval Lookup of a term with blocking: (slightly) slower 34 Introduction to Information Retrieval Front coding One block in blocked compression (k = 4) . . . 8 a u t o m a t a 8 a u t o m a t e 9 a u t o m a t i c 10 a u t o m a t i o n

. . . further compressed with front coding. 8 a u t o m a t log a 1 e e 2 e i c 3 e i o n 35 Introduction to Information Retrieval Dictionary compression for Reuters: Summary data structure dictionary, fixed-width dictionary, term pointers into string , with blocking, k = 4 , with blocking & front coding

size in MB 11.2 7.6 7.1 5.9 36 Introduction to Information Retrieval Exercise Which prefixes should be used for front coding? What are the tradeoffs?

Input: list of terms (= the term vocabulary) Output: list of prefixes that will be used in front coding 37 Introduction to Information Retrieval Outline Recap Compression

Term statistics Dictionary compression Postings compression 38 Introduction to Information Retrieval

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 4-byte integers. Alternatively, we can use log2 800,000 19.6 < 20 bits per docID. Our goal: use a lot less than 20 bits per docID. 39 Introduction to Information Retrieval

Key idea: Store gaps instead of docIDs Each postings list is ordered in increasing order of docID. Example postings list: COMPUTER: 283154, 283159, 283202, . . . It suffices to store gaps: 283159-283154=5, 283202-283154=43 Example postings list using gaps : COMPUTER: 283154, 5, 43, . . . Gaps for frequent terms are small. Thus: We can encode small gaps with fewer than 20 bits.

40 Introduction to Information Retrieval Gap encoding 41 Introduction to Information Retrieval Variable length encoding Aim: For ARACHNOCENTRIC and other rare terms, we will use about 20 bits per gap (= posting). For THE and other very frequent terms, we will use only a

few bits per gap (= posting). In order to implement this, we need to devise some form of variable length encoding. Variable length encoding uses few bits for small gaps and many bits for large gaps. 42 Introduction to Information Retrieval Variable byte (VB) code Used by many commercial/research systems Good low-tech blend of variable-length coding and sensitivity to alignment matches (bit-level codes, see later).

Dedicate 1 bit (high bit) to be a continuation bit c. If the gap G fits within 7 bits, binary-encode it in the 7 available bits and set c = 1. Else: encode lower-order 7 bits and then use one or more 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 of the other bytes to 0 (c = 0). 43 Introduction to Information Retrieval VB code examples docIDs gaps

VB code 824 00000110 10111000 829 5 10000101 215406 214577 00001101 00001100 10110001 44

Introduction to Information Retrieval VB code encoding algorithm 45 Introduction to Information Retrieval VB code decoding algorithm 46 Introduction to Information Retrieval Other variable codes

Instead of bytes, we can also use a different unit of alignment: 32 bits (words), 16 bits, 4 bits (nibbles) etc Variable byte alignment wastes space if you have many small gaps nibbles do better on those. Recent work on word-aligned codes that efficiently pack a variable number of gaps into one word see resources at the end 47 Introduction to Information Retrieval Gamma codes for gap encoding You can get even more compression with another type of variable length encoding: bitlevel code.

Gamma code is the best known of these. First, we need unary code to be able to introduce gamma code. 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 70 is: 11111111111111111111111111111111111111111111111111111111111111111111110 48 Introduction to Information Retrieval Gamma code

Represent a gap G as a pair of length and offset. Offset is the gap in binary, with the leading bit chopped off. For example 13 1101 101 = offset Length is the length of offset. For 13 (offset 101), this is 3. Encode length in unary code: 1110. Gamma code of 13 is the concatenation of length and

offset: 1110101. 49 Introduction to Information Retrieval Gamma code examples 50 Introduction to Information Retrieval Exercise Compute the variable byte code of 130

Compute the gamma code of 130 51 Introduction to Information Retrieval Length of gamma code The length of offset is loglog2 G bits. bits. The length of length is loglog2 G bits. + 1 bits, So the length of the entire code is 2 x loglog2 G bits. + 1 bits. codes are always of odd length. Gamma codes are within a factor of 2 of the optimal encoding length log2 G. (assuming the frequency of a gap G is proportional to log2 G not really true)

52 Introduction to Information Retrieval Gamma code: Properties Gamma code is prefix-free: a valid code word is not a prefix of any other valid code. Encoding is optimal within a factor of 3 (and within a factor of 2 making additional assumptions). This result is independent of the distribution of gaps! We can use gamma codes for any distribution. Gamma code is universal. Gamma code is parameter-free.

53 Introduction to Information Retrieval Gamma codes: Alignment Machines have word boundaries 8, 16, 32 bits Compressing and manipulating at 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. 54

Introduction to Information Retrieval Compression of Reuters data structure dictionary, fixed-width dictionary, term pointers into string , with blocking, k = 4 , with blocking & front coding collection (text, xml markup etc) collection (text) T/D incidence matrix postings, uncompressed (32-bit words) postings, uncompressed (20 bits) postings, variable byte encoded postings, encoded

size in MB 11.2 7.6 7.1 5.9 3600.0 960.0 40,000.0 400.0 250.0 116.0 101.0 55

Introduction to Information Retrieval Term-document incidence matrix Entry is 1 if term occurs. Example: CALPURNIA occurs in Julius Caesar. Entry is 0 if term doesnt occur. Example: CALPURNIA doesnt occur in The tempest. 56 Introduction to Information Retrieval Compression of Reuters data structure dictionary, fixed-width

dictionary, term pointers into string , with blocking, k = 4 , with blocking & front coding collection (text, xml markup etc) collection (text) T/D incidence matrix postings, uncompressed (32-bit words) postings, uncompressed (20 bits) postings, variable byte encoded postings, encoded size in MB 11.2 7.6 7.1

5.9 3600.0 960.0 40,000.0 400.0 250.0 116.0 101.0 57 Introduction to Information Retrieval Summary We can now create an index for highly efficient Boolean

retrieval that is very space efficient. Only 10-15% of the total size of the text in the collection. However, weve ignored positional and frequency information. For this reason, space savings are less in reality. 58 Introduction to Information Retrieval Take-away today Motivation for compression in information retrieval systems How can we compress the dictionary component of the inverted index?

How can we compress the postings component of the inverted index? Term statistics: how are terms distributed in document collections? 59 Introduction to Information Retrieval Resources Chapter 5 of IIR Resources at http://ifnlp.org/ir Original publication on word-aligned binary codes by Anh and Moffat (2005); also: Anh and Moffat (2006a) Original publication on variable byte codes by Scholer, Williams, Yiannis and Zobel (2002)

More details on compression (including compression of positions and frequencies) in Zobel and Moffat (2006) 60