Introduction to Information Retrieval
Information Retrieval and Data Mining
(AT71.07)
Comp. Sc. and Inf. Mgmt.
Asian Institute of Technology
Instructor: Prof. Sumanta Guha
Slide Sources: Introduction to
Information Retrieval book slides
from Stanford University, adapted
and supplemented
Chapter 2: The term vocabulary and
postings lists
1
Introduction to Information Retrieval
Introduction to
Information Retrieval
CS276: Information Retrieval and Web Search
Christopher Manning and Prabhakar Raghavan
Lecture 2: The term vocabulary and postings
lists
Introduction to Information Retrieval
Recap of the previous lecture
 Basic inverted indexes:
 Structure: Dictionary and Postings
 Key step in construction: Sorting
 Boolean query processing
 Intersection by linear time “merging”
 Simple optimizations
 Overview of course topics
Ch. 1
Introduction to Information Retrieval
Plan for this lecture
Elaborate basic indexing
 Preprocessing to form the term vocabulary
 Documents
 Tokenization
 What terms do we put in the index?
 Postings
 Faster merges: skip lists
 Positional postings and phrase queries
Introduction to Information Retrieval
Recall the basic indexing pipeline
Documents to
be indexed.
Friends, Romans, countrymen.
Tokenizer
Token stream.
Friends Romans
Countrymen
Linguistic
modules
Modified tokens.
= terms
Inverted index.
friend
roman
countryman
Indexer friend
2
4
roman
1
2
countryman
13
16
Introduction to Information Retrieval
Sec. 2.1
Parsing a document
 What format is it in?
 pdf/word/excel/html?
 What language is it in?
 What character set is in use?
Each of these is a classification problem,
which we will study later in the course.
But these tasks are often done heuristically …
Introduction to Information Retrieval
Sec. 2.1
Complications: Format/language
 Documents being indexed can include docs from
many different languages
 A single index may have to contain terms of several
languages.
 Sometimes a document or its components can
contain multiple languages/formats
 French email with a German pdf attachment.
 What is a unit document?




A file?
An email? (Perhaps one of many in an mbox.)
An email with 5 attachments?
A group of files (PPT or LaTeX as HTML pages)
Introduction to Information Retrieval
TOKENS AND TERMS
Introduction to Information Retrieval
Definitions
 Word – A delimited string of characters as it appears
in the text.
 Term – A “normalized” word (case, morphology,
spelling etc); an equivalence class of words.
 Token – An instance of a word or term occurring in a
document.
 Type – The same as a term in most cases: an
equivalence class of tokens.
9
Introduction to Information Retrieval
Sec. 2.2.1
Tokenization
 Input: “Friends, Romans and Countrymen”
 Output: Tokens




Friends
Romans
and
Countrymen
 A token is an instance of a sequence of characters
 Each such token is now a candidate for an index
entry, after further processing
 Described below
 But what are valid tokens to emit?
Introduction to Information Retrieval
Sec. 2.2.1
Tokenization
 Issues in tokenization:
 Finland’s capital 
Finland? Finlands? Finland’s?
 Hewlett-Packard  Hewlett and Packard as two
tokens?
 state-of-the-art: break up hyphenated sequence.
 co-education
 lowercase, lower-case, lower case ?
 It can be effective to get the user to put in possible hyphens
 San Francisco: one token or two?
 How do you decide it is one token?
Introduction to Information Retrieval
Sec. 2.2.1
Numbers





3/20/91
Mar. 12, 1991
20/3/91
55 B.C.
B-52
My PGP key is 324a3df234cb23e
(800) 234-2333
 Often have embedded spaces
 Older IR systems may not index numbers
 But often very useful: think about things like looking up error
codes/stacktraces on the web
 (One answer is using n-grams: Lecture 3)
 Will often index “meta-data” separately
 Creation date, format, etc.
Introduction to Information Retrieval
Sec. 2.2.1
Tokenization: language issues
 French
 L'ensemble  one token or two?
 L ? L’ ? Le ?
 Want l’ensemble to match with un ensemble
 Until at least 2003, it didn’t on Google
 Internationalization!
 German noun compounds are not segmented
 Lebensversicherungsgesellschaftsangestellter
 ‘life insurance company employee’
 German retrieval systems benefit greatly from a compound splitter
module
 Can give a 15% performance boost for German
Sec. 2.2.1
Introduction to Information Retrieval
Tokenization: language issues
 Chinese, Japanese and Thai have no spaces
between words:
 莎拉波娃现在居住在美国东南部的佛罗里达。
 Not always guaranteed a unique tokenization
 Further complicated in Japanese, with multiple
alphabets intermingled
 Dates/amounts in multiple formats
フォーチュン500社は情報不足のため時間あた$500K(約6,000万円)
Katakana
Hiragana
Kanji
Romaji
End-user can express query entirely in hiragana!
Introduction to Information Retrieval
Sec. 2.2.1
Tokenization: language issues
 Arabic (or Hebrew) is basically written right to left,
but with certain items like numbers written left to
right
 Words are separated, but letter forms within a word
form complex ligatures

← → ←→
← start
 ‘Algeria achieved its independence in 1962 after 132
years of French occupation.’
 With Unicode, the surface presentation is complex, but the
stored form is straightforward
Introduction to Information Retrieval
Sec. 2.2.2
Stop words
 With a stop list, you exclude from the dictionary
entirely the commonest words. Intuition:
 They have little semantic content: the, a, and, to, be
 There are a lot of them: ~30% of postings for top 30 words
 But the trend is away from doing this:
 Good compression techniques (lecture 5) means the space for
including stopwords in a system is very small
 Good query optimization techniques (lecture 7) mean you pay little
at query time for including stop words.
 You need them for:
 Phrase queries: “King of Denmark”
 Various song titles, etc.: “Let it be”, “To be or not to be”
 “Relational” queries: “flights to London”
Introduction to Information Retrieval
Sec. 2.2.3
Normalization to terms
 We need to “normalize” words in indexed text as
well as query words into the same form
 We want to match U.S.A. and USA
 Result is terms: a term is a (normalized) word type,
which is an entry in our IR system dictionary
 We most commonly implicitly define equivalence
classes of terms by, e.g.,
 deleting periods to form a term
 U.S.A., USA  USA
 deleting hyphens to form a term
 anti-discriminatory, antidiscriminatory  antidiscriminatory
Introduction to Information Retrieval
Sec. 2.2.3
Normalization: other languages
 Accents: e.g., French résumé vs. resume.
 Umlauts: e.g., German: Tuebingen vs. Tübingen
 Should be equivalent
 Most important criterion:
 How are your users like to write their queries for these
words?
 Even in languages that standardly have accents, users
often may not type them
 Often best to normalize to a de-accented term
 Tuebingen, Tübingen, Tubingen  Tubingen
Sec. 2.2.3
Introduction to Information Retrieval
Normalization: other languages
 Normalization of things like date forms
 7月30日 vs. 7/30
 Japanese use of kana vs. Chinese characters
 Tokenization and normalization may depend on the
language and so is intertwined with language
detection
Morgen will ich in MIT …
Is this
German “mit”?
 Crucial: Need to “normalize” indexed text as well as
query terms into the same form
Introduction to Information Retrieval
Case folding
 Reduce all letters to lower case
 exception: upper case in mid-sentence?
 e.g., General Motors
 Fed vs. fed
 SAIL vs. sail
 Often best to lower case everything, since
users will use lowercase regardless of
‘correct’ capitalization…
 Google example:
 Query C.A.T.
 #1 result is for “cat” (well, Lolcats) not
Caterpillar Inc.
Sec. 2.2.3
Sec. 2.2.3
Introduction to Information Retrieval
Normalization to terms
 An alternative to equivalence classing is to do
asymmetric query expansion
 An example of where this may be useful
 Enter: window
 Enter: windows
 Enter: Windows
Search: window, windows
Search: Windows, windows, window
Search: Windows
 Potentially more powerful, but less efficient
Introduction to Information Retrieval
Thesauri and soundex
 Do we handle synonyms and homonyms?
 E.g., by hand-constructed equivalence classes
 car = automobile
color = colour
 We can rewrite to form equivalence-class terms
 When the document contains automobile, index it under carautomobile (and vice-versa)
 Or we can expand a query
 When the query contains automobile, look under car as well
 What about spelling mistakes?
 One approach is soundex, which forms equivalence classes
of words based on phonetic heuristics (Muller = Mueller,
color = colour)
 More in lectures 3 and 9
Introduction to Information Retrieval
Sec. 2.2.4
Lemmatization
 Reduce inflectional/variant forms to base form
 E.g.,
 am, are, is  be
 car, cars, car's, cars'  car
 the boy's cars are different colors  the boy car be
different color
 Lemmatization implies doing “proper” reduction to
dictionary headword form
Sec. 2.2.4
Introduction to Information Retrieval
Stemming
 Reduce terms to their “roots” before indexing
 “Stemming” suggest crude affix chopping
 language dependent
 e.g., automate(s), automatic, automation all reduced to
automat.
for example compressed
and compression are both
accepted as equivalent to
compress.
for exampl compress and
compress ar both accept
as equival to compress
Introduction to Information Retrieval
Sec. 2.2.4
Porter’s algorithm
 Commonest algorithm for stemming English
 Results suggest it’s at least as good as other stemming
options
 Conventions + 5 phases of reductions
 phases applied sequentially
 each phase consists of a set of commands
 sample convention: Of the rules in a compound command,
select the one that applies to the longest suffix.
Introduction to Information Retrieval
Typical rules in Porter




sses  ss
ies  i
ational  ate
tional  tion
 Weight of word sensitive rules
 (m>1) EMENT →
 replacement → replac
 cement → cement
Sec. 2.2.4
Introduction to Information Retrieval
Sec. 2.2.4
Other stemmers
 Other stemmers exist, e.g., Lovins stemmer
 http://www.comp.lancs.ac.uk/computing/research/stemming/general/lovins.htm
 Single-pass, longest suffix removal (about 250 rules)
 Full morphological analysis – at most modest
benefits for retrieval
 Do stemming and other normalizations help?
 English: very mixed results. Helps recall for some queries but
harms precision on others
 E.g., operative (dentistry) ⇒ oper
 Definitely useful for Spanish, German, Finnish, …
 30% performance gains for Finnish!
Introduction to Information Retrieval
Sec. 2.2.4
Language-specificity
 Many of the above features embody transformations
that are
 Language-specific and
 Often, application-specific
 These are “plug-in” addenda to the indexing process
 Both open source and commercial plug-ins are
available for handling these
Sec. 2.2
Introduction to Information Retrieval
Dictionary entries – first cut
ensemble.french
時間.japanese
MIT.english
mit.german
guaranteed.english
entries.english
sometimes.english
tokenization.english
These may be
grouped by
language (or
not…).
More on this in
ranking/query
processing.
Introduction to Information Retrieval
Sec. 1.3
Stemming and Lemmatization

a.
b.
c.
d.

a.
b.
c.
d.
e.
Exercise 2.1: Are the following statements true or false:
In a Boolean retrieval system, stemming never lowers precision.
In a Boolean retrieval system, stemming never lowers recall.
Stemming increases the size of the vocabulary.
Stemming should be invoked at index time, but not while
processing a query.
Exercise 2.2: Suggest what normalized form should be used for
these words (including the word itself as a possibility):
‘Cos
Shi’ite
cont’d
Hawai’i
O’Rourke
30
Introduction to Information Retrieval
Sec. 1.3
Stemming and Lemmatization
 Exercise 2.3: The following pairs of words are stemmed to
the same form by the Porter stemmer. Which pairs,
would you argue, should not be conflated? Give your
reasoning.
a. abandon/abandonment
b. absorbency/absorbent
c. marketing/markets
d. university/universe
e. volume/volumes
31
Introduction to Information Retrieval
Sec. 1.3
Stemming and Lemmatization
Porter’s stemming algorithm, phase 1 rules (select rule which applies to
longest suffix)
Rule
Example
SSES → SS
caresses → caress
IES → I
ponies → poni
SS → SS
caress → caress
S
→
cats
→ cat
 Exercise 2.4: For the Porter stemmer rule group shown above:
a. What is the purpose of including an identity rule SS → SS
b. Applying just this rule group, what will the following words be stemmed
to?
circus
canaries
boss
c. What rule should be added to correctly stem pony?
d. The stemming for ponies and pony might seem strange. Does it have a
32
deleterious effect on retrieval? Why or why not?
Introduction to Information Retrieval
FASTER POSTINGS MERGES:
SKIP POINTERS/SKIP LISTS
Sec. 2.3
Introduction to Information Retrieval
Recall basic merge
 Walk through the two postings simultaneously, in
time linear in the total number of postings entries
2
8
2
4
8
41
1
2
3
8
48
11
64
17
128
21
Brutus
31 Caesar
If the list lengths are m and n, the merge takes O(m+n)
operations.
Can we do better?
Yes (if index isn’t changing too fast).
Sec. 2.3
Introduction to Information Retrieval
Augment postings with skip pointers
(at indexing time)
128
41
2
4
8
41
64
128
31
11
1
48
2
3
8
11
17
21
31
 Why?
 To skip postings that will not figure in the search
results.
 How?
 Where do we place skip pointers?
Sec. 2.3
Introduction to Information Retrieval
Query processing with skip pointers
128
41
2
4
8
41
64
128
31
11
1
48
2
3
8
11
17
21
31
Suppose we’ve stepped through the lists until we
process 8 on each list. We match it and advance.
We then have 41 and 11 on the lower. 11 is smaller.
But the skip successor of 11 on the lower list is 31, so
we can skip ahead past the intervening postings.
Introduction to Information Retrieval
IntersectWithSkips(p1, p2)
1 answer
← <>
2 while p1 ≠ NIL and p2 ≠ NIL
3 do if docID(p1) = docID(p2)
4
then ADD(answer, docID(p1))
5
p1 ← next(p1)
6
7
p2 ← next(p2)
else if docID(p1) < docID(p2)
8
9
10
11
12
13
14
15
16 return answer
then if hasSkip(p1) and (docID(skip(p1)) ≤ docID(p2))
then while hasSkip(p1) and (docID(skip(p1)) ≤ docID(p2))
do p1 ← skip(p1)
else p1 ← next(p1)
else if hasSkip(p2) and (docID(skip(p2)) ≤ docID(p1))
then while hasSkip(p2) and (docID(skip(p2)) ≤ docID(p1))
do p2 ← skip(p2)
else p2 ← next(p2)
37
Introduction to Information Retrieval
Sec. 2.3
Where do we place skips?
 Tradeoff:
 More skips  shorter skip spans  more likely to skip.
But lots of comparisons to skip pointers.
 Fewer skips  few pointer comparison, but then long skip
spans  few successful skips.
Introduction to Information Retrieval
Sec. 2.3
Placing skips
 Simple heuristic: for postings of length L, use L
evenly-spaced skip pointers.
 This ignores the distribution of query terms.
 Easy if the index is relatively static; harder if L keeps
changing because of updates.
 This definitely used to help; with modern hardware it
may not (Bahle et al. 2002) unless you’re memorybased
 The I/O cost of loading a bigger postings list can outweigh
the gains from quicker in memory merging!
Introduction to Information Retrieval
Skips
 Exercise 2.5: Why are skip pointers not useful for queries of
the form x OR y?
 Exercise 2.6: We have a two-word query. For one term the
postings list consists of the 16 entries
[4, 6, 10, 12, 14, 16, 18, 20, 22, 32, 47, 81, 120, 122, 157, 180]
and for the other it consists of only 1 entry
[47]
Work out how many comparisons would be done to intersect
the two postings lists with the following strategies.
a. Using standard postings lists.
b. Using postings lists with skip pointers, with the suggested
skip length of √P.
40
Introduction to Information Retrieval
Skips
 Exercise 2.7: Consider a postings intersection between this
postings list with skip pointers:
3 5 9 15 24 39 60 68 75 81 84 89 92 96 97 100 115
and the following intermediate result posting list (which hence
has no skip pointers):
3 5 89 95 97 99 100 101
Trace through the postings intersection algorithm.
a. How often is a skip pointer followed (i.e., p1 advanced to
skip(p1))?
b. How many postings comparisons will be made by this
algorithm while intersecting the two lists?
c. How many comparisons would be made if the postings lists
are intersected without the use of skip pointers?
41
Introduction to Information Retrieval
PHRASE QUERIES AND POSITIONAL
INDEXES
Introduction to Information Retrieval
Sec. 2.4
Phrase queries
 Want to be able to answer queries such as “stanford
university” – as a phrase
 Thus the sentence “I went to university at Stanford”
is not a match.
 The sentence “The inventor Stanford Ovshinsky never
went to university” is not a match.
 The concept of phrase queries has proven easily
understood by users; one of the few “advanced search”
ideas that works
 Many more queries are implicit phrase queries
 For this, it no longer suffices to store only
<term : docs> entries
Introduction to Information Retrieval
Sec. 2.4.1
A first attempt: Biword indexes
 Index every consecutive pair of terms in the text as a
phrase
 For example the text “Friends, Romans,
Countrymen” would generate the biwords
 friends romans
 romans countrymen
 Each of these biwords is now a dictionary term
 Two-word phrase query-processing is now
immediate.
Sec. 2.4.1
Introduction to Information Retrieval
Longer phrase queries
 Longer phrases are processed as we did with wildcards:
 stanford university palo alto can be broken into the
Boolean query on biwords:
stanford university AND university palo AND palo alto
Without the docs, we cannot verify that the docs
matching the above Boolean query do contain the
phrase.
Can have false positives!
Introduction to Information Retrieval
Sec. 2.4.1
Extended biwords
 Parse the indexed text and perform part-of-speech-tagging
(POST).
 Bucket the terms into (say) Nouns (N) and
articles/prepositions (X).
 Call any string of terms of the form NX*N an extended biword.
 Each such extended biword is now made a term in the
dictionary.
 Example: catcher in the rye
N
X X N
 Query processing: parse it into N’s and X’s
 Segment query into enhanced biwords
 Look up in index: catcher rye
Introduction to Information Retrieval
Sec. 2.4.1
Issues for biword indexes
 False positives, as noted before
 Index blowup due to bigger dictionary
 Infeasible for more than biwords, big even for them
 Biword indexes are not the standard solution (for all
biwords) but can be part of a compound strategy
Introduction to Information Retrieval
Sec. 2.4.2
Solution 2: Positional indexes
 In the postings, store, for each term the position(s) in
which tokens of it appear:
<term, number of docs containing term;
doc1: frequency of the term; position1, position2 … ;
doc2: frequency of the term; position1, position2 … ;
etc.>
Sec. 2.4.2
Introduction to Information Retrieval
Positional index example
<be: 993427;
1: 6; 7, 18, 33, 72, 86, 231;
2: 2; 3, 149;
4: 5; 17, 191, 291, 430, 434;
5: 11; 363, 367, …>
Which of docs 1,2,4,5
could contain “to be
or not to be”?
 For phrase queries, we use a merge algorithm
recursively at the document level
 But we now need to deal with more than just
equality
Introduction to Information Retrieval
Sec. 2.4.2
Processing a phrase query
 Extract inverted index entries for each distinct term: to, be,
or, not.
 Merge their doc:position lists to enumerate all positions
with “to be or not to be”.
 to:
 2: 5; 1,17,74,222,551; 4: 5; 8,16,190,429,433; 7: 3; 13,23,191;
...
 be:
 1: 2;17,19; 4: 5;17,191,291,430,434; 5: 3; 14,19,101; ...
 Same general method for proximity searches
Introduction to Information Retrieval
Sec. 2.4.2
Proximity queries
 LIMIT! /3 STATUTE /3 FEDERAL /2 TORT
 Again, here, /k means “within k words of”.
 Clearly, positional indexes can be used for such
queries; biword indexes cannot.
 Exercise: Adapt the linear merge of postings to
handle proximity queries. Can you make it work for
any value of k?
 This is a little tricky to do correctly and efficiently
 See Figure 2.12 of IIR
 There’s likely to be a problem on it!
Introduction to Information Retrieval
The algorithm finds places where the two terms appear within k words of each other
and returns a list of triples giving docID and the term position in p1 and p2.
PositionalIntersect(p1, p2, k)
1 answer ← <>
l is a moving window of positions
2 while p1 ≠ nil and p2 ≠ nil
of the second word in the current
3 do if docID(p1) = docID(p2)
doc which are within k of the
4
then l ← <>
current position of the first word.
5
pp1 ← positions(p1)
6
pp2 ← positions(p2)
7
while pp1 ≠ nil
8
do while pp2 ≠ nil
9
do if |pos(pp1) – pos(pp2)| ≤ k
10
then ADD(l , pos(pp2))
11
else if pos(pp2) > pos(pp1)
12
then break
13
pp2 ←next(pp2)
14
while l ≠ <> and |l[0] – pos(pp1)| > k
15
do DELETE(l[0])
16
for each ps ∈ l
17
do ADD(answer, <docID(p1), pos(pp1), ps>)
18
pp1 ← next(pp1)
For each successive position of the
19
p1 ← next(p1)
first word: moves the “head” of the
20
p2 ← next(p2)
window l up till at most k away from
21
else if docID(p1) < docID(p2)
the new position of the first word;
22
then p1 ← next(p1)
deletes “tail” of the window till within
23
else p2 ← next(p2)
52
k of the first word.
24 return answer
Introduction to Information Retrieval
Sec. 2.4.2
Positional index size
 You can compress position values/offsets: we’ll talk
about that in lecture 5
 Nevertheless, a positional index expands postings
storage substantially
 Nevertheless, a positional index is now standardly
used because of the power and usefulness of phrase
and proximity queries … whether used explicitly or
implicitly in a ranking retrieval system.
Sec. 2.4.2
Introduction to Information Retrieval
Positional index size
 Need an entry for each occurrence, not just once per
document
 Index size depends on average document size
Why?
 Average web page has <1000 terms
 SEC filings, books, even some epic poems … easily 100,000
terms
 Consider a term with frequency 0.1%
Document size
Postings
Positional postings
1000
1
1
100,000
1
100
Introduction to Information Retrieval
Sec. 2.4.2
Rules of thumb
 A positional index is 2–4 as large as a non-positional
index
 Positional index size 35–50% of volume of original
text
 Caveat: all of this holds for “English-like” languages
Introduction to Information Retrieval
Sec. 2.4.3
Combination schemes
 These two approaches can be profitably
combined
 For particular phrases (“Michael Jackson”, “Britney
Spears”) it is inefficient to keep on merging positional
postings lists
 Even more so for phrases like “The Who”
 Williams et al. (2004) evaluate a more
sophisticated mixed indexing scheme
 A typical web query mixture was executed in ¼ of the
time of using just a positional index
 It required 26% more space than having a positional
index alone
Introduction to Information Retrieval
Positional Postings and Phrase Queries
 Exercise 2.8: Assume a biword index. Give an example of a
document that will be returned for a query of
New York University but is actually a false positive.
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Introduction to Information Retrieval
Positional Postings and Phrase Queries
 Exercise 2.9: Shown below is a portion of a positional index:
angels: 2: <36, 174, 252, 651>; 4: <12, 22, 102, 432>; 7: <17>;
fools: 2: <1, 17, 74, 222>; 4: <8, 78, 108, 458>; 7: <3, 13, 23, 193>;
fear: 2: <87, 704, 722, 901>; 4: <13, 43, 113, 433>; 7: <18, 328, 528>;
in: 2: <3, 37, 76, 444, 851>; 4: <10, 20, 110, 470, 500>; 7: <5, 15, 25, 195>;
rush: 2: <2, 66, 194, 321, 702>; 4: <9, 69, 149, 429, 569>; 7: <4, 14, 404>;
to: 2: <47, 86, 234, 999>; 4: <14, 24, 774, 944>; 7: <199, 319, 599, 709>;
tread: 2: <57, 94, 333>; 4: <15, 35, 155>; 7: <20, 320>;
where: 2: <67, 124, 393, 1001>; 4: <11, 41, 101, 421, 431>; 7: <16, 36, 736>;
Which docs if any match each of the following queries where expressions
within quotes are phrase queries?
a. “fools rush in”
b. “fools rush in” AND “angels fear to tread”
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Introduction to Information Retrieval
Positional Postings and Phrase Queries
 Exercise 2.10: Consider the following fragment of a positional
index:
Gates : 1: <3>; 2: <6>; 3: <2, 17>; 4: <1>;
IBM : 4: <3>; 7: <14>;
Microsoft : 1: <1>; 2: <1, 21>; 3: <3>; 5: <16, 22, 51>;
a. Describe the set of documents that satisfy the query
Gates /2 Microsoft.
b. Describe each set of values for k for which the query
Gates /k Microsoft returns a different set of documents
as the answer.
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Introduction to Information Retrieval
Resources for today’s lecture
 IIR 2
 MG 3.6, 4.3; MIR 7.2
 Porter’s stemmer:
http://www.tartarus.org/~martin/PorterStemmer/

Skip Lists theory: Pugh (1990)

 Multilevel skip lists give same O(log n) efficiency as trees
H.E. Williams, J. Zobel, and D. Bahle. 2004. “Fast Phrase
Querying with Combined Indexes”, ACM Transactions on
Information Systems.
http://www.seg.rmit.edu.au/research/research.php?author=4

D. Bahle, H. Williams, and J. Zobel. Efficient phrase querying with an
auxiliary index. SIGIR 2002, pp. 215-221.