Using Suffix Arrays to Compute Term 
Frequency and Document Frequency for 
All Substrings in a Corpus 
Mikio Yamamoto* 
University of Tsukuba 
Kenneth W. Churcht 
AT&T Labs--Research 
Bigrams and trigrams are commonly used in statistical natural language processing; this paper 
will describe techniques for working with much longer n-grams. Suffix arrays (Manber and My- 
ers 1990) were /irst introduced to compute the frequency and location of a substring (n-gram) 
in a sequence (corpus) of length N. To compute frequencies over all N(N + 1)/2 substrings in a 
corpus, the substrings are grouped into a manageable number of equivalence classes. In this way, 
a prohibitive computation over substrings is reduced to a manageable computation over classes. 
This paper presents both the algorithms and the code that were used to compute term frequency 
(tf) and document frequency (dr)for all n-grams in two large corpora, an English corpus of 
50 million words of Wall Street Journal and a Japanese corpus of 216 million characters of 
Mainichi Shimbun. 
The second half of the paper uses these frequencies to find "interesting" substrings. Lexi- 
cographers have been interested in n-grams with high mutual information (MI) where the joint 
term frequency is higher than what would be expected by chance, assuming that the parts of the 
n-gram combine independently. Residual inverse document frequency (RIDF) compares docu- 
ment frequency to another model of chance where terms with a particular term frequency are 
distributed randomly throughout the collection. MI tends to pick out phrases with noncompo- 
sitional semantics (which often violate the independence assumption) whereas RIDF tends to 
highlight technical terminology, names, and good keywords for information retrieval (which tend 
to exhibit nonrandom distributions over documents). The combination of both MI and RIDF is 
better than either by itself in a Japanese word extraction task. 
1. Introduction 
We will use suffix arrays (Manber and Myers 1990) to compute a number of type/token 
statistics of interest, including term frequency and document frequency, for all n-grams 
in large corpora. Type/token statistics model the corpus as a sequence of N tokens 
(characters, words, terms, n-grams, etc.) drawn from a vocabulary of V types. Different 
tokenizing rules will be used for different corpora and for different applications. In 
this work, the English text is tokenized into a sequence of English words delimited by 
white space and the Japanese text is tokenized into a sequence of Japanese characters 
(typically one or two bytes each). 
Term frequency (tf) is the standard notion of frequency in corpus-based natural 
language processing (NLP); it counts the number of times that a type (term/word/n- 
gram) appears in a corpus. Document frequency (df) is borrowed for the information 
* Institute of Information Sciences and Electronics, 1-1-1 Tennodai, Tsukuba 305-8573, Japan 
t 180 Park Avenue, Florham Park, NJ 07932 
Computational Linguistics Volume 27, Number 1 
retrieval literature (Sparck Jones 1972); it counts the number of documents that con- 
tain a type at least once. Term frequency is an integer between 0 and N; document 
frequency is an integer between 0 and D, the number of documents in the corpus. 
The statistics, tf and df, and functions of these statistics such as mutual information 
(MI) and inverse document frequency (IDF), are usually computed over short n-grams 
such as unigrams, bigrams, and trigrams (substrings of 1-3 tokens) (Charniak 1993; 
Jelinek 1997). This paper will show how to work with much longer n-grams, including 
million-grams and even billion-grams. 
In corpus-based NLP, term frequencies are often converted into probabilities, using 
the maximum likelihood estimator (MLE), the Good-Turing method (Katz 1987), or 
Deleted Interpolation (Jelinek 1997, Chapter 15). These probabilities are used in noisy 
channel applications such as speech recognition to distinguish more likely sequences 
from less likely sequences, reducing the search space (perplexity) for the acoustic 
recognizer. In information retrieval, document frequencies are converted into inverse 
document frequency (IDF), which plays an important role in term weighting (Sparck 
Jones 1972). 
dr(t) IDF(t) = -log 2 D 
IDF(t) can be interpreted as the number of bits of information the system is given if it 
is told that the document in question contains the term t. Rare terms contribute more 
bits than common terms. 
Mutual information (MI) and residual IDF (RIDF) (Church and Gale 1995) both 
compare tf and df to what would be expected by chance, using two different notions of 
chance. MI compares the term frequency of an n-gram to what would be expected if the 
parts combined independently, whereas RIDF combines the document frequency of a 
term to what would be expected if a term with a given term frequency were randomly 
distributed throughout the collection. MI tends to pick out phrases with noncompo- 
sitional semantics (which often violate the independence assumption) whereas RIDF 
tends to highlight technical terminology, names, and good keywords for information 
retrieval (which tend to exhibit nonrandom distributions over documents). 
Assuming a random distribution of a term (Poisson model), the probability p~(k) 
that a document will have exactly k instances of the term is: 
e-O Ok 
pa(k) = rc(O,k) - k! ' 
where 0 = np, n is the average length of a document, and p is the occurrence probability 
of the term. That is, 
Ntf tf 8-- 
DN D" 
Residual IDF is defined as the following formula. 
Residual IDF = observed IDF - predicted IDF 
df log(1 = - log + - 
df ---- -log~ +log(1 - e-~) 
The rest of the paper is divided into two sections. Section 2 describes the algorithms 
and the code that were used to compute term frequencies and document frequencies 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
for all substrings in two large corpora, an English corpus of 50 million words of the 
Wall Street Journal, and a Japanese corpus of 216 million characters of the Mainichi 
Shimbun. 
Section 3 uses these frequencies to find "interesting" substrings, where what counts 
as "interesting" depends on the application. MI finds phrases of interest to lexicog- 
raphy, general vocabulary whose distribution is far from random combination of the 
parts, whereas RIDF picks out technical terminology, names, and keywords that are 
useful for information retrieval, whose distribution over documents is far from uni- 
form or Poisson. These observations may be particularly useful for a Japanese word 
extraction task. Sequences of characters that are high in both MI and RIDF are more 
likely to be words than sequences that are high in just one, which are more likely than 
sequences that are high in neither. 
2. Computing tf and df for All Substrings 
2.1 Suffix Arrays 
This section will introduce an algorithm based on suffix arrays for computing tf and 
df and many functions of these quantities for all substrings in a corpus in O(NlogN) 
time, even though there are N(N + 1)/2 such substrings in a corpus of size N. The 
algorithm groups the N(N + 1)/2 substrings into at most 2N - 1 equivalence classes. 
By grouping substrings in this way, many of the statistics of interest can be computed 
over the relatively small number of classes, which is manageable, rather than over the 
quadratic number of substrings, which would be prohibitive. 
The suffix array data structure (Manber and Myers 1990) was introduced as a 
database indexing technique. Suffix arrays can be viewed as a compact representa- 
tion of suffix trees (McCreight 1976; Ukkonen 1995), a data structure that has been 
extensively studied over the last thirty years. See Gusfield (1997) for a comprehensive 
introduction to suffix trees. Hui (1992) shows how to compute df for all substrings 
using generalized suffix trees. The major advantage of suffix arrays over suffix trees 
is space. The space requirements for suffix trees (but not for suffix arrays) grow with 
alphabet size: O(N\]~\]) space, where \]~\] is the alphabet size. The dependency on al- 
phabet size is a serious issue for Japanese. Manber and Myers (1990) reported that 
suffix arrays are an order of magnitude more efficient in space than suffix trees, even 
in the case of relatively small alphabet size (IGI = 96). The advantages of suffix arrays 
over suffix trees become much more significant for larger alphabets such as Japanese 
characters (and English words). 
The suffix array data structure makes it convenient to compute the frequency 
and location of a substring (n-gram) in a long sequence (corpus). The early work was 
motivated by biological applications such as matching of DNA sequences. Suffix arrays 
are closely related to PAT arrays (Gonnet, Baeza-Yates, and Snider 1992), which were 
motivated in part by a project at the University of Waterloo to distribute the Oxford 
English Dictionary with indexes on CD-ROM. PAT arrays have also been motivated 
by applications in information retrieval. A similar data structure to suffix arrays was 
proposed by Nagao and Mori (1994) for processing Japanese text. 
The alphabet sizes vary considerably in each of these cases. DNA has a relatively 
small alphabet of just 4 characters, whereas Japanese has a relatively large alphabet of 
more than 5,000 characters. The methods such as suffix arrays and PAT arrays scale 
naturally over alphabet size. In the experimental section (Section 3) using the Wall Street 
Journal corpus, the suffix array is applied to a large corpus of English text, where the al- 
phabet is assumed to be the set of all English words, an unbounded set. It is sometimes 
assumed that larger alphabets are more challenging than smaller ones, but ironically, 
Computational Linguistics Volume 27, Number 1 
it can be just the reverse because there is often an inverse relationship between the size 
of the alphabet and the length of meaningful or interesting substrings. For expository 
convenience, this section will use the letters of the alphabet, a-z, to denote tokens. 
This section starts by reviewing the construction of suffix arrays and how they 
have been used to compute the frequency and locations of a substring in a sequence. 
We will then show how these methods can be applied to find not only the frequency 
of a particular substring but also the frequency of all substrings. Finally, the methods 
are generalized to compute document frequencies as well as term frequencies. 
A suffix array, s, is an array of all N suffixes, sorted alphabetically. A suffix, s\[i\], 
also known as a semi-infinite string, is a string that starts at position i in the corpus 
and continues to the end of the corpus. In practical implementations, it is typically 
denoted by a four-byte integer, i. In this way, a small (constant) amount of space is 
used to represent a very long substring, which one might have thought would require 
N space. 
A substring, sub(i,j), is a prefix of a suffix. That is, sub(i,j), is the first j characters 
of the suffix s\[i\]. The corpus contains N(N + 1)/2 substrings. 
The algorithm, suffix_array, presented below takes a corpus and its length N as 
input, and outputs the suffix array, s. 
suffix_array *-- function(corpus, N){ 
Initialize s to be a vector of integers from 0 to N - 1. 
Let each integer denote a suffix starting at s\[i\] in the corpus. 
Sort s so that the suffixes are in alphabetical order. 
Return s. } 
The C program below implements this algorithm. 
char *corpus ; 
int suffix_compare(int *a, int *b) 
{ return strcmp(corpus+*a, corpus+*b);} 
int *suffix_array(int n){ 
int i, *s = (int *)malloc(n*sizeof(int)); 
for(i=O; i < n; i++) s\[i\] = i; /* initialize */ 
qsort(s, n, sizeof(int), suffix_compare); /* sort */ 
return s;} 
Figures i and 2 illustrate a simple example where the corpus ("to_be_ormot_to_be") 
consists of N = 18 (19 bytes): 13 alphabetic characters plus 5 spaces (and 1 null 
termination). The C program (above) starts by allocating memory for the suffix array 
(18 integers of 4 bytes each). The suffix array is initialized to the integers from 0 
to 17. Finally, the suffix array is sorted by alphabetical order. The suffix array after 
initialization is shown in Figure 1. The suffix array after sorting is shown in Figure 2. 
As mentioned above, suffix arrays were designed to make it easy to compute the 
frequency (tf) and locations of a substring (n-gram or term) in a sequence (corpus). 
Given a substring or term, t, a binary search is used to find the first and last suffix 
that start with t. Let s\[i\] be the first such suffix and s~\] be the last such suffix. Then 
tf(t) = j - i + 1 and the term is located at positions: {s\[i\], s\[i + 1\] .... ,s~\]}, and only 
these positions. 
Figure 2 shows how this procedure can be used to compute the frequency and 
locations of the term "to_be" in the corpus "to_be_or_not_to_be". As illustrated in the 
figure, s\[i = 16\] is the first suffix to start with the term "to_be" and s~ = 17\] is the last 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
Input corpus: "to_be_or_not_to_be" 
Position: 
Characters: 
Initialized 
Suffix Array 
s\[O\] 0 
s\[l\] 1 
s\[2\] 2 
s\[3\] 3 
s\[13\] 13 
s\[18\] 14 
s \[15\] 15 
s \[16\] 15 
s\[17\] 17 
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Itlol_lblel_lolrl_lnloltl_ltlol_lblel 
.~" 
: : : : ""{ i i i ! 
' ' ' ! Suffixes denoted by s \[ i\] i i i i i 
-" / ." / to b~ o~ not to b~ I/ / i i i 
.... .. j...: o or to j :" 'ii'' I ........ " ..,'" I be or not to be I ." / / / / I"~ be orNot_to_be ...... ,'"'" ...'/" .y ./" / 
Figure 1 
Illustration of a suffix array, s, that has just been initialized and not yet sorted. Each element in 
the suffix array, s\[i\], is an integer denoting a suffix or a semi-infinite string, starting at 
position i in the corpus and extending to the end of the corpus. 
Suffix Array 
s\[0\] 1151 
s \[1\] 2., 
s\[2\] 8 
s\[3\] 5 
s \[4\] 12 
s \[5\] 16 
s\[6\] 3 
s \[7\] 17 
s \[8\] 4 
s\[9\] 9 
s \[10\] 14 
sill\] 1 
s\[12\] 6 
s\[13\] 10 
s \[14\] 7 
s \[15\] 11 
s \[16\] 13 
s\[17\] 0 
Figure 2 
Suffixes denoted by s \[ i \] 
_De 
be_or_not_to_be 
_not to be 
_or_not_to_be 
_to_be 
be 
be_or_no t_t o_be 
e 
e_or_no t_t o_be 
not_to_be 
obe 
o be or_not_to_be 
or_not to be 
ot_to_be 
r_not_to_be 
t to be 
to_be 
to be or_not_to_be 
Illustration of the suffix array in Figure 1 after sorting. The integers in s are sorted so that the 
semi-infinite strings are now in alphabetical order. 
Computational Linguistics Volume 27, Number 1 
suffix to start with this term. Consequently, tf("to_be") = 17-16+ 1 = 2. Moreover, the 
term appears at positions("to_be") = {s\[16\], s\[17\] } = {13, 0}, and only these positions. 
Similarly, the substring "to" has the same tf and positions, as do the substrings, "to_" 
and "to_b". Although there may be N(N + 1)/2 ways to pick i and j, it will turn out 
that we need only consider 2N - 1 of them when computing tf for all substrings. 
Nagao and Mori (1994) ran this procedure quite successfully on a large corpus 
of Japanese text. They report that it takes O(NlogN) time, assuming that the sort 
step performs O(N log N) comparisons, and that each comparison takes constant time. 
While these are often reasonable assumptions, we have found that if the corpus con- 
tains long repeated substrings (e.g., duplicated articles), as our English corpus does 
(Paul and Baker 1992), then the sort can consume quadratic time, since each compar- 
ison can take order N time. Like Nagao and Mori (1994), we were also able to apply 
this procedure quite successfully to our Japanese corpus, but for the English corpus, 
after 50 hours of CPU time, we gave up and turned to Manber and Myers's (1990) al- 
gorithm, which took only two hours. 1 Manber and Myers' algorithm uses some clever, 
but difficult to describe, techniques to achieve O(N log N) time, even for a corpus with 
long repeated substrings. For a corpus that would otherwise consume quadratic time, 
the Manber and Myers algorithm is well worth the effort, but otherwise, the procedure 
described above is simpler, and can even be a bit faster. 
The "to_be_or_not_to_be" example used the standard English alphabet (one byte per 
character). As mentioned above, suffix arrays can be generalized in a straightforward 
way to work with larger alphabets such as Japanese (typically two bytes per character). 
In the experimental section (Section 3), we use an open-ended set of English words as 
the alphabet. Each token (English word) is represented as a four-byte pointer into a 
symbol table (dictionary). The corpus "to_be_or_not_to_be", for example, is tokenized 
into six tokens: "to", "be", "or", "not", "to", and "be", where each token is represented 
as a four-byte pointer into a dictionary. 
2.2 Longest Common Prefixes (LCPs) 
Algorithms from the suffix array literature make use of an auxiliary array for storing 
LCPs (longest common prefixes). The lcp array contains N + 1 integers. Each element, 
lcp\[i\], indicates the length of the common prefix between s\[i - 1\] and s\[i\]. We pad the 
lcp vector with zeros (lcp\[O\] = lcp\[N\] = 0) to simplify the discussion. The padding 
avoids the need to test for certain end conditions. 
Figure 3 shows the lcp vector for the suffix array of "to_be_or_not_to_be". For 
example, since s\[10\] and s\[11\] both start with the substring "o_be", lcp\[11\] is set to 4, the 
length of the longest common prefix. Manber and Myers (1990) use the Icp vector in 
their O(P+log N) algorithm for computing the frequency and location of a substring of 
length P in a sequence of length N. They showed that the Icp vector can be computed 
in O(N log N) time. These algorithms are much faster than the obvious straightforward 
implementation when the corpus contains long repeated substrings, though for many 
corpora, the complications required to avoid quadratic behavior are unnecessary. 
2.3 Classes of Substrings 
Thus far we have seen how to compute tf for a single n-gram, but how do we compute 
tf and df for all n-grams? As mentioned above, the N(N + 1)/2 substrings will be 
clustered into a relatively small number of classes, and then the statistics will be 
1 We used Doug McIlroy's implementation, available on the Web at: http://cm.belMabs.com/cm/cs/ who/doug/ssort.c. 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
computed over the classes rather than over the substrings, which would be prohibitive. 
The reduction of the computation over substrings to a computation over classes is 
made possible by four properties. 
Properties 1-2: all substrings in a class have the same statistics (at least 
for the statistics of interest, namely tf and dJ), 
Property 3: the set of all substrings is partitioned by the classes, and 
Property 4: there are many fewer classes (order N) than substrings 
(order N2). 
Classes are defined in terms of intervals. Let (i,j) be an interval on the suffix array, 
s\[i\], s\[i + 1\] ..... s~\]. Class((i,j)) is the set of substrings that start every suffix within the 
interval and no suffix outside the interval. It follows from this construction that all 
substrings in a class have tf = j - i + 1. 
The set of substrings in a class can be constructed from the lcp vector: 
class((i,j)) = {s\[i\]ml max(Icp\[i\], lcp~ + 1\]) < m ~ min(lcp\[i + 1\], lcp\[i + 2\] ..... lcp~'\])}, 
where s\[i\]m denotes the first m characters of the suffix s\[i\]. We will refer to lcp\[i\] and 
Icp~+l\] as bounding lcps and Icp\[i+l\], lcp\[i+2\] ..... lcp~\] as interior lcps. The equation 
Suffix Array i Suffix denoted by s \[ i \] 
s\[O\] 15 _be; 
s\[l\] 2 _b~_or_not to be 
s\[2\] 8 _}not to be 
s\[3\] 5 br not to be --\] 
-- 
s\[4\] 12 :--to_be 
s\[5\] IB ibei 
s\[6\] 3 :he'or_not to be 
s\[7\] 
s \[8\] i e~_or_not to be 
s \[ 9 \] not_to_be 
s \[I0\] o_b4 len~=4 
sill\] ~ or not to be 
s\[12\] qr_not to be 
s\[13\] q't to be 
s\[14\] r_not to be 
s\[15\] tJ to be 
s \[ 16 \] rio_be, 
s \[ 17 \] to_be\[~or_not to be 
The doted lines denote lcp's. 
Figure 3 
Lcp vector 
icp\[O O" 
icp \[i 3 
icp \[2 I 
icp \[3 I 
Icp \[4 I 
icp\[5 0 
icp\[6\] 2 
icp\[7\] 0 
icp\[8\] I 
Icp\[9\] 0 
icp\[lO\] 0 
lcp \[12\] 1 
Icp\[13\] 1 
icp \[14\] 0 
icp\[15\] 0 
icp\[16\] I 
icp\[17\] 5 
icp\[18\] O- 
-- always 0 
always 0 
The longest common prefix is a vector of N + 1 integers, lcp\[i\] denotes the length of the 
common prefix between the suffix s\[i - 1\] and the suffix s\[i\]. Thus, for example, s\[10\] and s\[11\] 
share a common prefix of four characters, and therefore lcp\[11\] = 4. The common prefix is 
highlighted by a dotted line in the suffix array. The suffix array is the same as in the previous figure. 
7 
Computational Linguistics Volume 27, Number 1 
Vertical lines denote lcps. Gray area denotes Boundaries 
endpoints of substrings in class(<10,11>). 
S \[9\]lli' n o t to be ~'~f7 °f<10'll> 
I,Ill 0 __ b e - ~ X<";flO,11> .......... / s \[ 10 \] 
s\[ll\]\]~o\]_ b el_or_not_to_be ~1 4_ I 
s\[12\]I lo:;Ir n~t to be ........ <10,13> 
s\[14\]s\[13\]\]i!I~to ~-t~_be e ............ ~ 
Bounding lcps, LBL, SIL, Interior lcp of <10,11> 
LCP-delimited Class 
interval 
<10,11> 
<10,13> 
<9,9> 
<10,10> 
<11,11> 
<12,12> 
<13,13> 
<14,14> 
LBL SIL 
Figure 4 
{"O_", "o_b", "o_be"} 1 4 
{"o"} 0 1 
{"n", "no", "not", ...} 0 infinity 
{} 4 infinity 
{"o_be_","o_beo",...} 4 infinity 
{"or", "or_", "or_n", ...} 1 infinity 
{"ot", "ot_", "oCt", ...} 1 infinity 
"r", "r " " " _, r_n,...} 0 infinity 
tf 
Six suffixes are copied from Figure 3, s\[9\]-s\[14\], along with eight of their lcp-delimited 
intervals. Two of the lcp-delimited intervals are nontrivial (tf > 1), and six are trivial (tf = 1). 
Intervals are associated with classes, sets of substrings. These substrings start every suffix 
within the interval and no suffix outside the interval. All of the substrings within a class have 
the same term frequency (and document frequency). 
above can be rewritten as 
class((i, jl ) = {s\[i\]mlLBL((i, jl) < m <_ SIL((i, jl)}, 
where LBL (longest bounding lcp) is 
LBL((i,j)) = max(lcp\[i\], lcp~ + 1\]), 
and SIL (shortest interior lcp) is 
sIn((i,j)) = min(lcp\[i + 1\], Icp\[i + 2\] ..... lcp~\]). 
By construction, the class will be empty unless there is some room for m between 
the LBL and SIL. We say that an interval is lcp-delimited when this room exists 
(that is, LBL < SIL). Except for trivial intervals where tf = 1 (see below), classes are 
nonempty iff the interval is lcp-delimited. Moreover, the number of substrings in a 
nontrivial class depends on the amount of room between the LBL and the SIL. That 
is, \[class((i,j))l = SIL((i,j)) - LBL((i,j)). 
Figure 4 shows eight examples of lcp-delirnited intervals. The top part of the figure 
highlights the interval (10,11) with dashed horizontal lines. Solid vertical lines denote 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
bounding lcps, and thin vertical lines denote interior lcps (there is only one interior lcp 
in this case). The interval {10, 11 / is lcp-delimited because the bounding lcps, Icp\[lO\] --= 0 
and lcp\[12\] = 1, are smaller than the interior lcp, lcp\[11\] = 4. That is, the LBL (= 1) is 
less than the SIL (= 4). Thus there is room for m between the LBL of s\[10\]m and the SIL 
of s\[10\]m. The endpoints m between LBL and SIL are highlighted in gray. The class is 
nonempty. Its size depends on the width of the gray area: class((lO, 11)) = {s\[10\]mll < 
m _< 4} = {"o_", "o_b", "o_be"}. These substrings have the same tf: tf = j - i + 1 = 
11 - 10 + 1 = 2. Each of these substrings occurs exactly twice in the corpus. 
Every substring in the class starts every suffix in the interval (10,11), and no suffix 
outside (10,11). In particular, the substring "o" is excluded from the class, because it 
is shared by suffixes outside the interval, namely s\[12\] and s\[13\]. The longer substring, 
"o_be_", is excluded from the class because it is not shared by s\[10\], a suffix within 
the interval. 
We call an interval trivial if the interval starts and ends at the same place: (i, i). 
The remaining six intervals mentioned in Figure 4 are trivial intervals. We call the 
class of a trivial interval a trivial class. As in the nontrivial case, the class contains 
all (and only) the substrings that start every suffix within the interval and no suf- 
fix outside the interval. We can express the class of a trivial interval, class((i, i)), as 
{s\[i\]m\]LBL < m <_ SIL}. The trivial case is the same as the nontrivial case, except that 
the SIL of a trivial interval is defined to be infinite. As a result, trivial classes are 
usually quite large, because they contain all prefixes of s\[i\] that are longer than the 
LBL. They cover all (and only) the substrings with tf = 1, typically the bulk of the 
N(N+ 1)/2 substrings in a corpus. The trivial class of the interval (11,11), for example, 
contains 13 substrings: "o_be_", "o_be_o", "o_be_or", and so on. Of course, there are 
some exceptions: the trivial class, class((lO, 10)), in Figure 4, for example, is very small 
(= empty set). 
Not every interval is lcp-delimited. The interval, Ill, 12), for example, is not lcp- 
delimited because there is no room for m of s\[ll\]m between the LBL (= 4) and the 
SIL (= 1). When the interval is not lcp-delimited, the class is empty. There are no 
substrings starting all the suffixes within the interval (11,12), and not starting any 
suffix outside the interval. 
It is possible for lcp-delimited intervals to be nested, as in the case of (10,11) and 
(10, 13). We say that one interval (i,j) is nested within another (u,v) if i < u < v < j 
(and (i,j) ~ (u, v)). Nested intervals have distinct SILs and disjoint classes. (Two classes 
are disjoint if the corresponding sets of substrings are disjoint.) 2 The substrings in the 
class of the nested interval, (u, v), are longer than the substrings in the class of the 
outer interval, (i,j). 
Although it is possible for lcp-delimited intervals to be nested, it is not possible for 
lcp-delimited intervals to overlap. We say that one nontrivial interval Ca, b) overlaps 
another nontrivial interval (c, d) if a < c < b < d. If two intervals overlap, then at 
least one of the intervals is not lcp-delimited and has an empty class. If an interval 
(a, b) is lcp-delimited, an overlapped interval (c, d) is not lcp-delimited. Because a 
bounding lcp of /a, b) must be within (c, d) and an interior lcp of (a, b) must be a 
bounding lcp of (c,d), SIL((c,d)) <_ LBL((a,b)) <_ SIL((a,b)) <_ LBL((c,d)). That is, the 
overlapped interval (c, d) is not lcp-delimited. The fact that lcp-delimited intervals are 
nested and do not overlap will turn out to be convenient for enumerating lcp-delimited 
intervals. 
2 Because (u, v I is lcp-delimited, there must be a bounding lcp of lu, v I that is smaller than any lcp 
within (u, v). This bounding lcp must be within (i,j), and as a result, class((i,j)) and class((u, v)) must be disjoint. 
Computational Linguistics Volume 27, Number 1 
2.4 Four Properties 
As mentioned above, classes are constructed so that it is practical to reduce the com- 
putation of various statistics over substrings to a computation over classes. This sub- 
section will discuss four properties of classes that help make this reduction feasible. 
The first two properties are convenient because they allow us to associate tf and 
df with classes rather than with substrings. The substrings in a class all have the same 
tf value (property 1) and the same df value (property 2). That is, if Sl and s2 are two 
substrings in class((i,j)) then 
Property 1: tf(sl) = tf(s2) = j - i + 1 
Property 2: dr(s1) = rid(s2 ). 
Both of these properties follow straightforwardly from the construction of intervals. 
The value of tf is a simple function of the endpoints; the calculation of df is more 
complicated and will be discussed in Section 2.6. While tf and df treat each member 
of a class as equivalent, not all statistics do. Mutual information (MI) is an important 
counter example; in most cases, MI(sl) ~ MI(s2). 
The third property is convenient because it allows us to iterate over classes rather 
than substrings, without worrying about missing any of the substrings. 
Property 3: The classes partition the set of all substrings. 
There are two parts to this argument: every substring belongs to at most one class 
(property 3a), and every substring belongs to at least one class (property 3b). 
Demonstration of property 3a (proof by contradiction): Suppose there is a sub- 
string, s, that is a member of two distinct classes: class((i,j}) and class((u,v)). There 
are three possibilities: one interval precedes the other, they are properly nested, or 
they overlap. In all three cases, s cannot be a member of both classes. If one interval 
precedes the other, then there must be a bounding lcp between the two intervals which 
is shorter than s. And therefore, s cannot be in both classes. The nesting case was men- 
tioned previously where it was noted that nested intervals have disjoint classes. The 
overlapping case was also discussed previously where it was noted that two overlap- 
ping intervals cannot both be lcp-delimited, and therefore at least one of the classes 
would have to be empty. 
Demonstration of property 3b (constructive argument): Let s be an arbitrary sub- 
string in the corpus. There will be at least one suffix in the suffix array that starts with 
s. Let i be the first such suffix and let j be the last such suffix. By construction, the 
interval (i,j} is lcp-delimited (LBL((i,j}) < Is I and SIL((i,j}) >__ Is\[), and therefore, s is 
an element of class((i,j}). 
Finally, as mentioned above, computing over classes is much more efficient than 
computing over the substrings themselves because there are many fewer classes (at 
most 2N - 1) than substrings (N(N + 1)/2). 
Property 4: There are at most N nonempty classes with tf = 1 and at 
most N - 1 nonempty classes with tf > 1. 
The first clause is relatively straightforward. There are N trivial intervals (i, i/. 
These are all and only the intervals with tf = 1. By construction, these intervals are 
lcp-delimited, though it is possible that a few of the classes could be empty. 
To argue the second clause, we make use of a uniqueness property: an lcp- 
delimited interval (i, jl can be uniquely determined by an SIL and a representative ele- 
10 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
ment k, where i < k < j. For convenience, we will choose k such that SIL(<i,j>) = Icp\[k\], 
but we could have uniquely determined the lcp-delimited interval by choosing any k 
such that i < k G j. 
The uniqueness property can be demonstrated using a proof by contradiction. 
Suppose there were two distinct lcp-delimited intervals, <i,j> and <u, v>, with the same 
representative k, where i < k < j and u < k G v. Since they share a common repre- 
sentative, k, one interval must be nested inside the other. But nested intervals have 
disjoint classes and different SILs. 
Given this uniqueness property, we can determine the N- 1 upper bound on 
the number of lcp-delimited intervals by considering the N - 1 elements in the Icp 
vector. Each of these elements, lcp\[k\], has the opportunity to become the SIL of an 
lcp-delimited interval <i,j> with a representative k. Thus there could be as many as 
N- 1 lcp-delimited intervals (though there could be fewer if some of the opportunities 
don't work out). Moreover, there cannot be any more intervals with tf > 1, because if 
there were one, its SIL should have been in the Icp vector. (Note that this lcp counting 
argument does not count trivial intervals because their S1Ls \[= infinity\] are not in the 
lcp vector; the Icp vector contains integers less than N.) 
From property 4, it follows that there are at most N distinct values of RIDE The 
N trivial intervals <i, i> have just one RIDF value since tf = df = 1 for these intervals. 
The other N - 1 intervals could have as many as another N - 1 RIDF values. Similar 
arguments hold for many other statistics that make use of tf and dr, and treat all 
members of a class as equivalent. 
In summary, the four properties taken collectively make it practical to compute 
tf, dr, and RIDF over a relatively small number of classes; it would be prohibitively 
expensive to compute these quantities directly over the N(N + 1)/2 substrings. 
2.5 Computing All Classes Using Suffix Arrays 
This subsection describes a single-pass procedure, print_LDIs, for computing t/for 
all LDIs (lcp-delimited intervals). Since lcp-delimited intervals are properly nested, the 
procedure is based on a push-down stack. The procedure outputs four quantities for 
each lcp-delimited interval, <i,j>. The four quantities are the two endpoints (i and j), the 
term frequency (tf) and a representative (k), such that i < k _< j and lcp\[k\] : SIL(<i,j>). 
This procedure will be described twice. The first implementation is expressed in a 
recursive form; the second implementation avoids recursion by implementing the stack 
explicitly. 
The recursive implementation is presented first, because it is simpler. The function 
print_LDIs is initially called with print LDIs (0,0), which will cause the function to 
be called once for each value of k between 0 and N - 1. k is a representative in the 
range: i < k < j, where i and j are the endpoints of an interval. For each of the 
N values of k, a trivial LDI is reported at <k, k>. In addition, there could be up to 
N- 1 nontrivial intervals, where k is the representative and lcp\[k\] is the SIL. Recall 
that lcp-delimited intervals are uniquely determined by a representative k such that 
i < k < j where SIL(<i,j)) = Icp\[k\]. Not all of these candidates will produce LDIs. 
The recursion searches for j's such that LBL((i,j>) < SIL(<i,j>), but reports intervals at 
(i,j> only when the inequality is a strict inequality, that is, LBL(<i,j>) < SIL(<i,j>). The 
program stack keeps track of the left and right edges of these intervals. While Icp\[k\] 
is monotonically increasing, the left edge is remembered on the stack, as pr±nt LDIs 
is called recursively. The recursion unwinds as lcp\[j\] < lcp\[k\]. Figure 5 illustrates the 
function calls for computing the nontrivial lcp-delimited intervals in Figure 4. C code 
is provided in Appendix A. 
11 
Computational Linguistics Volume 27, Number 1 
s\[9 
s\[lO 
s\[ll 
s\[12 
s\[13\] o"t ..... t ... 
s\[14\] r _ n o . . . 
Vertical lines denote lcps. 
~.print LDIs(i=9,k=10) 
......... ~ \[print CDIs(i=10,k=ll) 
\] n o.-~-"j_ .... ,-'" | print <10,11>, rep=11, tf=2" 
\] "5~"" b e .,'"'" \[return(j=11) 
.... , "print_LDIs(i=10,k=12) 
\] o _ b e. ........ -, ,\[print_LDIs(i=12,k=13) 
\] o "±'"'---_.n ...... -:- | print nothing, 
| (<12,13> isn t lcp-delimited) 
Lreturn(j=13) 
print "<10,13>, rep=13, tf=4" 
return(j-13) 
Figure 5 
Trace of function calls for computing the nontrivial lcp-delimited intervals in Figure 4. In this 
trace, trivial intervals are omitted. Print_LDIs(i = x,k = y) represents a function call with 
arguments, i and k. Indentation represents the nest of recursive calls. Print_LDIs(i,k) searches 
the right edge, j, of the non-trivial lcp-delimited interval, <i,j>, whose SIL is lcp\[k\]. Each 
representative, k, value is given to the function print_LDIs just once (dotted arcs). 
print_LDIs +- function(i, k) { 
j~k. 
Output a trivial lcp-delimited interval <k, k> with tf = 1. 
While lcp\[k\] <_ lcp~ + 1\] and j + 1 < N, do j *-- print_LDIs(k, j + 1). 
Output an interval <i,j> with tf = j - i + 1 and rep = k, if it is lcp-delimited. 
Return j. } 
The second implementation (below) introduces its own explicit stack, a complica- 
tion that turns out to be important in practice, especially for large corpora. C code is 
provided in Appendix B. 
print_LDIs_stack *-- function(N){ 
stack_i ~-- an integer array for the stack of the left edges, i. 
stack_k ~ an integer array for the stack of the representatives, k. 
stack_i\[O\] *-- O. 
stack_k\[O\] *-- O. 
sp *--- 1 (a stack pointer). 
Forj *-- 0,1,2 ..... N- 1 do 
Output an lcp-delimited interval (j,j> with tf = 1. 
While Icp~ + 1\] < lcp\[stackd~\[sp - 1\]\] do 
Output an interval <i,j> with tf = j - i + 1, if it is lcp-delimited. 
sp*---sp-1. 
stack_i\[sp\] ~-- stack_k\[sp - 1\]. 
stack_k \[sp\] *--- j + 1. 
sp*---sp+l.} 
12 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
Suffix Array 
s\[O\] 2 I 
s\[l\] 15l 
s\[2\] !2_\] 
s\[3\] b I 
s \[4\] 8 1 
s\[5\] 181 
------4 
s\[6\] 3 I 
s\[7\] 161 
s\[8\] 4 i 
s\[9\] 17\[ 
s \[i0\] 9 I 
s\[ll\] 
s\[12\] 141 
s\[13\] 6 1 
s\[14\] 1--51 
s\[15\] 
s\[16\] 111 
s\[iv\] o--I 
s\[18\] 1~1 
Suffixes denoted 
by s \[i\] 
_be$ 
_be$ 
_to_be$ 
$ 
$ 
$ 
be$ 
be$ 
e$ 
e$ 
not_to_be$ 
o_be$ 
o_be$ 
orS 
ot to be$ 
r$ 
t_to_be$ 
to_be$ 
to_be$ 
Document id's 
icp\[i\] ofs\[i\] 
0 0 
4 2 
I 2 
0 0 
I I 
I 2 
0 0 
3 2 
0 0 
2 2 
0 2 
0 0 
5 2 
1 I 
I 2 
0 I 
0 2 
I 0 
6 2 
0 
Input documents: dO = "to_be$" -- 
dl = "orS" 
d2 = "not_to_be$" 
Corpus = dO + dl + d2 = "to_be$or$not_to_be$" 
Resulting non-trivial lcp-delimited intervals: 
('rep' means a representative, k.) 
(0,1>, rep= 1, tf=2, df=2 
(0, 2>, rep= 2, tf=3, df=2 
(3, 5), rep= 4, tf=3, df=3 
(6, 7>, rep= 7, tf=2, df=2 
<8, 9/, rep= 9, tf=2, df=2 
(11,12>, rep=12, tf=2, df=2 
<11,14}, rep=13, tf=4, df=3 
(17,18>, rep=18, tf=2, df=2 
<16,18), rep=17, tf=3, dr=2 
Figure 6 
A suffix array for a corpus consisting of three documents. The special character $ denotes the 
end of a document. The procedure outputs a sequence of intervals with their term frequencies 
and document frequencies. These results are also presented for the nontrivial intervals. 
2.6 Computing df for All Classes 
Thus far we have seen how to compute term frequency, tf, for all substrings (n-grams) 
in a sequence (corpus). This section will extend the solution to compute document 
frequency, dr, as well as term frequency. The solution runs in O(NlogN) time and 
O(N) space. C code is provided in Appendix C. 
This section will use the running example shown in Figure 6, where the corpus is: 
"to_be$or$not_to_be$'. The corpus consists of three documents, "to_be$', "orS", and 
"not_to_be$'. The special character $ is used to denote the end of a document. The 
procedure outputs a sequence of intervals with their term frequencies and document 
frequencies. These results are also presented for the nontrivial intervals. 
The suffix array is computed using the same procedures discussed above. In ad- 
dition to the suffix array and the lcp vector, Figure 6 introduces a new third table 
that is used to map from suffixes to document ids. This table of document ids will be 
13 
Computational Linguistics Volume 27, Number 1 
used by the function get_docnum to map from suffixes to document ids. Suffixes are 
terminated in Figure 6 after the first end of document symbol, unlike before, where 
suffixes were terminated with the end of corpus symbol. 
A straightforward method for computing df for an interval is to enumerate the 
suffixes within the interval and then compute their document ids, remove duplicates, 
and return the number of distinct documents. Thus, for example, df("o') in Figure 6, 
can be computed by finding corresponding interval, (11,14}, where every suffix within 
the interval starts with "o" and no suffix outside the interval starts with "o'. Then 
we enumerate the suffixes within the interval {s\[11\],s\[12\],s\[13\],s\[14\]}, compute their 
document ids, {0, 2,1, 2}, and remove duplicates. In the end we discover that df("o") -- 
3. That is, "o" appears in all three documents. 
Unfortunately, this straightforward approach is almost certainly too slow. Some 
document ids will be computed multiple times, especially when suffixes appear in 
nested intervals. We take advantage of the nesting property of lcp-delimited intervals 
to compute all df's efficiently. The df of an lcp-delimited interval can be computed 
recursively in terms of its constituents (nested subintervals), thus avoiding unnecessary 
recomputation. 
The procedure print_LDIs_w±th_df presented below is similar to print_LDIs_- 
stack but modified to compute df as well as tf. The stack keeps track of i and k, as 
before, but now the stack also keeps track of df. 
i, the left edge of an interval, 
k, the representative (SIL = lcp\[k\]), 
df, partial results for dr, counting documents seen thus far, minus duplicates. 
print_LDIs with_dr +--- function(N){ 
stack_i ~- an integer array for the stack of the left edges, i. 
stack_k ~ an integer array for the stack of the representatives, k. 
stackdf ~-- an integer array for the stack of the df counter. 
doclink\[O..D\] : an integer array for the document link initialized with -1. 
D = the number of documents. 
stack_i\[O\] ~ O. 
stack_k\[O\] *--- O. 
stack_dr\[O\] *--- 1. 
sp ~ 1 (a stack pointer). 
(1) Forj ~ 0,1,2,...,N- 1 do 
(2) (Output a trivial lcp-delimited interval q,j> with tf = 1 and d/= 1.) 
(3) doc *-- get_docnum(s~\]) 
(4) if doclink\[doc\] ~ -1, do 
(5) let x be the largest x such that doclink\[doc\] >_ stack_i N. 
(6) stack_dr\[x\] *--- stack_df\[x\] - 1. 
(7) doclink\[ doc\] ~-- j. 
(8) df ,-- 1. 
(9) While lcp~" + 1\] < lcp\[stack_k\[sp - 1\]\] do 
(10) df *--- stack~tf\[sp - 1\] + df . 
(11) Output a nontrivial interval (i,j) with tf = j - i + 1 and dr, 
if it is lcp-delimited. 
(12) sp ~-- sp - 1. 
(13) stack_i\[sp\] ~-- stack_k\[sp - 1\]. 
(14) stack_k \[sp\] ~-- j + 1. 
14 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
(15) staek_e/\[sp\] ~ ad. 
(16) sp *-- sp + 1. } 
Lines 5 and 6 take care of duplicate documents. The duplication processing makes 
use of doclink (an array of length D, the number of documents in the collection), which 
keeps track of which suffixes have been seen in which document, doclink is initialized 
with -1 indicating that no suffixes have been seen yet. As suffixes are processed, doclink 
is updated (on line 7) so that doclink \[d\] contains the most recently processed suffix in 
document d. As illustrated in Figure 7, when j = 16 (snapshot A), the most recently 
processed suffix in document 0 is s\[11\] ("o_be$"), the most recently processed suffix 
in document 1 is s\[15\] ("r$"), and the most recently processed suffix in document 2 
is s\[16\] ("t_to_be$"). Thus, doclink\[0\]= 11, doclink\[1\]= 15, and doclink\[2\]= 16. After 
processing s\[17\] ("to_be$"), which is in document 0, doclink\[0\] is updated from 11 to 
17, as shown in snapshot B of Figure 7. 
Stackdf keeps track of document frequencies as suffixes are processed. The invari- 
ant is: stack_dr\[x\] contains the document frequency for suffixes seen thus far starting 
at i = stack_i\[x\]. (x is a stack offset.) When a new suffix is processed, line 5 checks for 
double counting by searching for intervals on the stack (still being processed) that have 
suffixes in the same document as the current suffix. If there is any double counting, 
stackdf is decremented appropriately on line 6. 
There is an example of this decrementing in snapshot C of Figure 7, highlighted 
by the circle around the binding of df to 0 on the stack element: \[i = 0, k = 17, 
df = 0\]. Note that df was previously bound to 1 in snapshot B. The binding of df was 
decremented when processing s\[18\] because s\[18\] is in the same document as s\[16\]. 
This duplication was identified by line 5. The decrementing was performed by line 6. 
Intervals are processed in depth-first order, so that more deeply nested intervals 
are processed before less deeply nested intervals. In this way, double counting is only 
an issue for intervals higher on the stack. The most deeply nested intervals are trivial 
intervals. They are processed first. They have a df of 1 (line 8). For the remaining 
nontrivial intervals, staekdf contains the partial results for intervals in process. As the 
stack is popped, the df values are aggregated up to compute the df value for the outer 
intervals. The aggregation occurs on line 10 and the popping of the stack occurs on line 
12. The aggregation step is illustrated in snapshots C and D of Figure 7 by the two ar- 
rows with the "+" combination symbol pointing at a value of df in an output statement. 
2.7 Class Arrays 
The classes identified by the previous calculation are stored in a data structure we call 
a class array, to make it relatively easy to look up the term frequency and document 
frequency for an arbitrary substring. The class array is a stored list of five-tuples: (SIL, 
LBL, tf, df, longest suffix I. The fifth element of the five-tuple is a canonical member of 
the class (the longest suffix). The five-tuples are sorted by the alphabetical order of 
the canonical members. In our C code implementation, classes are represented by five 
integers, one for each element in the five-tuple. Since there are N trivial classes and 
at most N - 1 nontrivial classes, the class array will require at most 10N - 5 integers. 
However, for many practical applications, the trivial classes can be omitted. 
Figure 8 shows an example of the nontrivial class array for the corpus: "to_be$or$ 
not_to_be$". The class array makes it relatively easy to determine that the substring 
"o" appears in all three documents. That is, df("o") = 3. We use a binary search to 
find that tuple c\[5\] is the relevant five-tuple for "o". Having found the relevant tuple, 
it requires a simple record access to return the document frequency field. 
15 
Computational Linguistics Volume 27, Number 1 
Doclink 
doclink\[0\]=11 
doclink\[1\]=15 
doclink\[2\]=16 
Local variables Stack 
Ij=16 df:Ll'  I I 
lines 13-16: push(i=16, k=l 7, df=l ) 
line l:j~j+l 
doclink\[0\]=114 J =17 df=l 
doclink\[1\]=15 I~ i=16, k=17, df=l 
doclink\[_2\]--16 I ~ i=15, k=16, df=l \ 
lop\[k=17\] ~. Icp\[j+1=18\] lines 3-6: x 
1 6 ~line 7: doclink\[doc(j=17)=0\] .~j=17~ 
~lines 13-16:push(i=17, k=l 8, df=l ) 
J:'7 df:\Li=17, k:18, df=1, l) 
doclink\[1\]=15/ "'--.~=16, k=17, df=l~ / 
doclink\[2\]=16.J.............._~ I df=l l Y 
linesl-6:x__l ~ I J\ + - 
Icp\[k=17\] > Icp\[j+1=18\] _/line 7: ~"'"~ 5 0 ~ lines 8-11: output LDI--<17,18~=2~=2 
/" line12:pap0 ~~~~ 
~7---~//\]vj: 18 ~~-- l .... : ........ ------ ) 
doclink\[1\]= 15 dt / -'----~-.=16, k= 17,(.df=O.), " 
~doclink\[2\]=18)l l i--~, df=l \ 
Icp\[k=16\] > Icp\[j+1=18\] ~ lines 9-11" output LD1=<16,18>, tf=3, df=2 - 
1 0 ' line 12: pop() 
Ij=18 df=2 
..................... 
doclink\[0\]=17 
doclink\[1 \]=15 
doclink\[2\]=l 8 i=15, k=16, df=l 
(A) 
(B) 
(c) 
(D) 
Figure 7 
Snapshots of the doclink array and the stack during the processing of print_LDIs_with_df on 
the corpus: "to_be$or$not_to_be$'. The four snapshots A-D illustrate the state as j progresses 
from 16 to 18. Two nontrivial intervals are emitted while j is in this range: (17,18) and (16,18). 
The more deeply nested interval is emitted before the less deeply nested interval. 
16 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
Class array The longest suffix 
(Pointer to cor ~us) denoted by c\[i\] SIL 
c \[0\] 0 _ 1 
c \[i\] 0 _be$ 4 
c\[2\] 5 $ 1 
c \[3\] 3 be$ 3 
c \[4\] 4 e$ 2 
c\[5\] 1 o 1 
c \[6\] 1 o_be$ 5 
c\[7\] 11 t 1 
c\[8\] 0 to_be$ 6 
Figure 8 
An example of the class array for the cor 
LBL tf df 
0 3 2 
1 2 2 
0 3 3 
0 2 2 
0 2 2 
0 4 3 
1 2 2 
0 3 2 
1 2 2 
pus: "to _be$or$not_to_be$". 
3. Experimental Results 
3.1 RIDF and MI for English and Japanese 
We used the methods described above to compute dr, tf, and RIDF for all substrings in 
two corpora of newspapers summarized in Table 1. MI was computed for the longest 
substring in each class. The entire computation took a few hours on a MIPS10000 with 
16 Gbytes of main memory. The processing time was dominated by the calculation of 
the suffix array. 
The English collection consists of 50 million words (113 thousand articles) of the 
Wall Street Journal (distributed by the ACL/DCI) and the Japanese collection consists of 
216 million characters (436 thousand articles) of the CD-Mainichi Shimbun from 1991- 
1995 (which are distributed in CD-ROM format). The English corpus was tokenized 
into words delimited by white space, whereas the Japanese corpus was tokenized into 
characters (typically two bytes each). 
Table I indicates that there are a large number of nontrivial classes in both corpora. 
The English corpus has more substrings per nontrivial class than the Japanese corpus. 
It has been noted elsewhere that the English corpus contains quite a few duplicated 
articles (Paul and Baker 1992). The duplicated articles could explain why there are so 
many substrings per nontrivial class in the English corpus when compared with the 
Japanese corpus. 
Table 1 
Statistics of the English and Japanese corpora. 
Statistic Wall Street Journal Mainichi Shimbun 
N (corpus size in tokens) 
V (vocabulary in types) 
# articles 
# nontrivial classes 
# substrings in nontrivial classes 
substrings per class (in nontrivial 
classes) 
49,810,697 words 215,789,699 characters 
410,957 5,509 
112,915 435,859 
16,519,064 82,442,441 
2,548,140,677 1,388,049,631 
154.3 16.8 
17 
Computational Linguistics Volume 27, Number 1 
25 
20 
15 
\[~10 
5 
0 
-5 
-10 
5f, I 
ll.. ............ __, , .-. ,_- ........ i1:  
.... ,~--15 --20 25 3'0 35 40 f---5 .... ,(f 15 20 25 3() 35 40 
• . Length -, Length 
Figure 9 
The left panel plots MI as a function of the length of the n-gram; the right panel plots RIDF as 
a function of the length of the n-gram. Both panels were computed from the Japanese corpus. 
Note that while there is more dynamic range for shorter n-grams than for longer n-grams, 
there is plenty of dynamic range for n-grams well beyond bigrams and trigrams. 
For subsequent processing, we excluded substrings with tf < 10 to avoid noise, 
resulting in about 1.4 million classes (1.6 million substrings) for English and 10 million 
classes (15 million substrings) for Japanese. We computed RIDF and MI values for the 
longest substring in each of these 1.4 million English classes and 10 million Japanese 
classes. These values can be applied to the other substrings in these classes for RIDF, 
but not for MI. (As mentioned above, two substrings in the same class need not have 
the same MI value.) 
MI of each longest substring, t, is computed by the following formula. 
MI(t = xYz) p(xvz) = log p(zy)p(zly ) 
q(~Yz) 
N l°r:, q(xY) q(Yz) 
N q(Y) 
ff(~g~)ff(Y) 
= log tf(xY)tf(Yz)" 
where x and z are tokens, and Y and zYz are n-grams (sequences of tokens). When Y 
is the empty string, tf(Y) -- N. 
Figure 9 plots RIDF and MI values of 5,000 substrings randomly selected as a func- 
tion of string length. In both cases, shorter substrings have more dynamic range. That 
is, RIDF and MI vary more for bigrams than million-grams. But there is considerable 
dynamic range for n-grams well beyond bigrams and trigrams. 
3.2 Little Correlation between RIDF and MI 
We are interested in comparing and contrasting RIDF and MI. Figure 10 shows that 
RIDF and MI are largely independent. There is little if any correlation between the 
RIDF of a string and the MI of the same string. Panel (a) compares RIDF and MI 
for a sample of English word sequences from the WSJ corpus (excluding unigrams); 
panel (b) makes the same comparison but for Japanese phrases identified as keywords 
on the CD-ROM. In both cases, there are many substrings with a large RIDF value 
and a small MI, and vice versa. 
18 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
2 
1 
2 'i° ~ 
: , • ~ .~ : a ..," 
~, • ~ ¥" * . 
i • • 
'• # •• • • 1 • • • • • • 
;'.'. - • • l L--M. • 
l ,%, ; • 
.... ':" ..... 0 -~:: 
i i i 0 MI fo 2o -10 0 MI 10 20 
(a) English strings 
(word sequences) 
(b) Japanese strings 
(character sequences) 
Figure 10 
Both panels plot RIDF versus MI. Panel (a) plots RIDF and MI for a sample of English 
n-grams; panel (b) plots RIDF and MI for Japanese phrases identified as keywords on the 
CD-ROM. The right panel highlights the 10% highest RIDF and 10% lowest MI with a box, as 
well as the 10% lowest RIDF and the 10% highest MI. Arrows point to the boxes for clarity. 
We believe the two statistics are both useful but in different ways. Both pick out 
interesting n-grams, but n-grams with large MI are interesting in different ways from 
n-grams with large RIDF. Consider the English word sequences in Table 2, which all 
contain the word having. These sequences have large MI values and small RIDF values. 
In our collaboration with lexicographers, especially those working on dictionaries for 
learners, we have found considerable interest in statistics such as MI that pick out 
these kinds of phrases. Collocations can be quite challenging for nonnative speakers 
of the language. On the other hand, these kinds of phrases are not very good keywords 
for information retrieval. 
Table 2 
English word sequences containing the word having. Note that these phrases have large MI 
and low RIDF. They tend to be more interesting for lexicography than information retrieval. 
The table is sorted by MI. 
tf df RIDF MI Phrase 
18 18 -0.0 10.5 admits to having 
14 14 -0.0 9.7 admit to having 
25 23 0.1 8.9 diagnosed as having 
20 20 -0.0 7.4 suspected of having 
301 293 0.0 7.3 without having 
15 13 0.2 7.0 denies having 
59 59 -0.0 6.8 avoid having 
18 18 -0.0 6.0 without ever having 
12 12 -0.0 5.9 Besides having 
26 26 -0.0 5.8 denied having 
19 
Computational Linguistics Volume 27, Number 1 
Table 3 
English word sequences containing the word Mr. (sorted by RIDF). The word sequences near 
the top of the list are better keywords than the sequences near the bottom of the list. None of 
them are of much interest to lexicography. 
tf df RIDF MI Phrase 
11 3 1.9 0.6 . Mr. Hinz 
18 5 1.8 6.6 Mr. Bradbury 
51 16 1.7 6.7 Mr. Roemer 
67 25 1.4 6.8 Mr. Melamed 
54 27 1.0 5.8 Mr. Burnett 
11 8 0.5 1.1 Mr. Eiszner said 
53 40 0.4 0.3 Mr. Johnson. 
21 16 0.4 0.2 Mr. Nichols said. 
13 10 0.4 0.4 . Mr. Shulman 
176 138 0.3 0.5 Mr. Bush has 
13 11 0.2 1.5 to Mr. Trump's 
13 11 0.2 -0.9 Mr. Bowman, 
35 32 0.1 1.2 wrote Mr. 
12 11 0.1 1.7 Mr. Lee to 
22 21 0.1 1.4 facing Mr. 
11 11 -0.0 0.7 Mr. Poehl also 
13 13 -0.0 1.4 inadequate. " Mr. 
16 16 -0.0 1.6 The 41-year-old Mr. 
19 19 -0.0 0.5 14. Mr. 
26 26 -0.0 0.0 in November. Mr. 
27 27 -0.0 -0.0 " For his part, Mr. 
38 38 -0.0 1.4 . AMR, 
39 39 -0.0 -0.3 for instance, Mr. 
Table 3 shows MI and RIDF values for a sample of word sequences containing 
the word Mr. The table is sorted by RIDE The sequences near the top of the list are 
better keywords than the sequences further down. None of these sequences would be 
of much interest to a lexicographer (unless he or she were studying names). Many of 
the sequences have rather small MI values. 
Table 4 shows a few word sequences starting with the word the with large MI 
values. All of these sequences have high MI (by construction), but some are high 
in RIDF as well (labeled B), and some are not (labeled A). Most of the sequences 
are interesting in one way or another, but the A sequences are different from the 
B sequences. The A sequences would be of more interest to someone studying the 
grammar in the WSJ subdomain, whereas the B sequences would be of more interest 
to someone studying the terminology in this subdomain. The B sequences in Table 4 
tend to pick out specific events in the news, if not specific stories. The phrase, the Basic 
Law, for example, picks out a pair of stories that discuss the event of the handover of 
Hong Kong to China, as illustrated in the concordance shown in Table 5. 
Table 6 shows a number of word sequences with high MI containing common 
prepositions. The high MI indicates an interesting association, but again most have 
low RIDF and are not particularly good keywords, though there are a few exceptions 
(Just for Men, a well-known brand name, has a high RIDF and is a good keyword). 
The Japanese substrings are similar to the English substrings. Substrings with high 
RIDF pick out specific documents (and/or events) and therefore tend to be relatively 
good keywords. Substrings with high MI have nonindependent distributions (if not 
noncompositional semantics), and are therefore likely to be interesting to a lexicogra- 
pher or linguist. Substrings that are high in both are more likely to be meaningful units 
20 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
Table 4 
English word sequence containing the. All of these phrases have high MI. Some have high 
RIDF, and some do not. 
(A): Low RIDF (poor keywords) 
tf df RIDF MI Phase 
11 11 -0.0 11.1 the up side 
73 66 0.1 9.3 the will of 
16 16 -0.0 8.6 the sell side 
17 16 0.1 8.5 the Stock Exchange of 
16 15 0.1 8.5 the buy side 
20 20 -0.0 8.4 the down side 
55 54 0.0 8.3 the will to 
14 14 -0.0 8.1 the saying goes 
15 15 -0.0 7.6 the going gets 
(B): High RIDF (better keywords) 
tf df RIDF MI Phase 
37 3 3.6 2.3 the joint commission 
66 8 3.0 3.6 the SSC 
55 7 3.0 2.0 The Delaware & 
37 5 2.9 3.6 the NHS 
22 3 2.9 3.4 the kibbutz 
22 3 2.9 4.1 the NSA's 
29 4 2.9 4.2 the DeBartolos 
36 5 2.8 2.3 the Basic Law 
21 3 2.8 2.3 the national output 
(words or phrases) than substrings that are high in just one or the other. Meaningless 
fragments tend to be low in both MI and RIDE 
We grouped the Japanese classes into nine cells depending on whether the RIDF 
was in the top 10%, the bottom 10%, or in between, and whether the MI was in the 
top 10%, the bottom 10%, or in between. Substrings in the top 10% in both RIDF and 
MI tend to be meaningful words such as (in English translation) merger, stock certi~'cate, 
dictionary, wireless, and so on. Substrings in the bottom 10% in both RIDF and MI tend 
to be meaningless fragments, or straightforward compositional combinations of words 
such as current regular-season game. Table 7 shows examples where MI and RIDF point 
in opposite directions (see highlighted rectangles in panel (b) of Figure 10). 
We have observed previously that MI is high for general vocabulary (words found 
in dictionary) and RIDF is high for names, technical terminology, and good keywords 
for information retrieval. Table 7 suggests an intriguing pattern. Japanese uses different 
character sets for general vocabulary and loan words. Words that are high in MI tend 
to use the general vocabulary character sets (hiragana and kanji) whereas words that 
are high in RIDF tend to use the loan word character sets (katakana and English). 
(There is an important exception, though, for names, which will be discussed in the 
next subsection.) 
The character sets largely reflect the history of the language. Japanese uses four 
character sets (Shibatani 1990). Typically, functional words of Japanese origin are writ- 
ten in hiragana. Words that were borrowed from Chinese many hundreds of years ago 
are written in kanji. Loan words borrowed more recently from Western languages are 
written in katakana. Truly foreign words are written in the English character set (also 
known as romaji). We were pleasantly surprised to discover that MI and RIDF were 
distinguishing substrings on the basis of these character set distinctions. 
21 
Computational Linguistics Volume 27, Number 1 
Table 5 
Concordance of the phrase the Basic Law. Note that most of the instances of the Basic Law appear 
in just two stories, as indicated by the doc-id (the token-id of the first word in the document). 
token-id left context right context doc-id 
2229521: line in the drafting of the Basic Law that will determine how Hon 2228648 
2229902: s policy as expressed in the Basic Law - as Gov. Wilson's debut s 2228648 
9746758: he U.S. Constitution and the Basic Law of the Federal Republic of 9746014 
11824764: any changes must follow the Basic Law, Hong Kong's miniconstitut 11824269 
33007637: sts a tentative draft of the Basic Law, and although this may be 33007425 
33007720: the relationship between the Basic Law and the Chinese Constitutio 33007425 
33007729: onstitution. Originally the Basic Law was to deal with this topic 33007425 
33007945: wer of interpretation of the Basic Law shall be vested in the NPC 33007425 
33007975: tation of a provision of the Basic Law, the courts of the HKSAR { 33007425 
33008031: interpret provisions of the Basic Law. If a case involves the in 33007425 
33008045: tation of a provision of the Basic Law concerning defense, foreig 33007425 
33008115: etation of an article of the Basic Law regarding " defense, forei 33007425 
33008205: nland representatives of the Basic Law Drafting Committee fear tha 33007425 
33008398: e : Mainland drafters of the Basic Law simply do not appreciate th 33007425 
33008488: pret all the articles of the Basic Law. While recognizing that th 33007425 
33008506: y and power to interpret the Basic Law, it should irrevocably del 33007425 
33008521: pret those provisions of the Basic Law within the scope of Hong Ko 33007425 
33008545: r the tentative draft of the Basic Law, I cannot help but conclud 33007425 
33008690: d of being guaranteed by the Basic Law, are being redefined out o 33007425 
33008712: uncilor, is a member of the Basic Law Drafting Committee. 33007425 
39020313: sts a tentative draft of the Basic Law, and although this may be 39020101 
39020396: the relationship between the Basic Law and the Chinese Constitutio 39020101 
39020405: onstitution. Originally the Basic Law was to deal with this topic 39020101 
39020621: wer of interpretation of the Basic Law shall be vested in the NPC 39020101 
39020651: tation of a provision of the Basic Law, the courts of the HKSAR { 39020101 
39020707: interpret provisions of the Basic Law . If a case involves the in 39020101 
39020721: tation of a provision of the Basic Law concerning defense, foreig 39020101 
39020791: etation of an article of the Basic Law regarding " defense, forei 39020101 
39020881: nland representatives of the Basic Law Drafting Committee fear tha 39020101 
39021074: e : Mainland drafters of the Basic Law simply do not appreciate th 39020101 
39021164: pret all the articles of the Basic Law. While recognizing that th 39020101 
39021182: y and power to interpret the Basic Law, it should irrevocably del 39020101 
39021197: pret those provisions of the Basic Law within the scope of Hong Ko 39020101 
39021221: r the tentative draft of the Basic Law, I cannot help but conclud 39020101 
39021366: d of being guaranteed by the Basic Law, are being redefined out o 39020101 
39021388: uncilor, is a member of the Basic Law Drafting Committee. 39020101 
3.3 Names 
As mentioned above, names are an important exception to the rule that kanji (Chinese 
characters) are used for general vocabulary (words found in the dictionary) that were 
borrowed hundreds of years ago and katakana characters are used for more recent 
loan words (such as technical terminology). As illustrated in Table 7, kanji are also 
used for the names of Japanese people and katakana are used for the names of people 
from other countries. 
Names are quite different in English and Japanese. Figure 11 shows a striking con- 
trast in the distributions of MI and RIDF values. MI has a more compact distribution 
in English than Japanese. Japanese names cluster into two groups, but English names 
do not. 
The names shown in Figure 11 were collected using a simple set of heuristics. For 
English, we selected substrings starting with the titles Mr., Ms., or Dr. For Japanese, we 
selected keywords (as identified by the CD-ROM) ending with the special character 
22 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
Table 6 
English word sequences containing common prepositions. All have high MI (by construction); 
most do not have high RIDF (though there are a few exceptions such as Just for Men, a 
well-known brand name). 
tf df RIDF MI Preposition = "for" 
14 14 -0.0 14.6 feedlots for fattening 
15 15 -0.0 14.4 error for subgroups 
12 12 -0.0 14.1 Voice for Food 
10 5 1.0 13.8 Quest for Value 
12 4 1.6 13.8 Friends for Education 
13 13 -0.0 13.7 Commissioner for Refugees 
23 21 0.1 13.7 meteorologist for Weather 
10 2 2.3 13.4 Just for Men 
10 9 0.2 13.4 Witness for Peace 
19 16 0.2 12.9 priced for reoffering 
tf df RIDF MI Preposition = "on" 
1l 5 1.1 14.3 Terrorist on Trial 
11 10 0.1 13.1 War on Poverty 
13 12 0.1 12.7 Institute on Drug 
16 16 -0.0 12.6 dead on arrival 
12 12 -0.0 11.6 from on high 
12 12 -0.0 11.6 knocking on doors 
22 18 0.3 11.3 warnings on cigarette 
11 11 -0.0 11.2 Subcommittee on Oversight 
17 12 0.5 11.2 Group on Health 
22 20 0.1 11.1 free on bail 
tf df RIDF MI Preposition = "by" 
11 11 -0.0 12.9 piece by piece 
13 13 -0.0 12.6 guilt by association 
13 13 -0.0 12.5 step by step 
15 15 -0.0 12.4 bit by bit 
16 16 -0.0 ll.8 engineer by training 
61 59 0.0 11.5 side by side 
17 17 -0.0 11.5 each by Korea's 
12 12 -0.0 11.3 hemmed in by 
11 11 -0.0 10.8 dictated by formula 
20 20 -0.0 10.7 70%-owned by Exxon 
tf df RIDF MI Preposition = "of" 
11 10 0.1 16.8 Joan of Arc 
12 5 1.3 16.2 Ports of Call 
16 16 -0.0 16.1 Articles of Confederation 
14 13 0.1 16.1 writ of mandamus 
10 9 0.2 15.9 Oil of Olay 
11 11 -0.0 15.8 shortness of breath 
10 9 0.2 15.6 Archbishop of Canterbury 
10 8 0.3 15.3 Secret of My 
12 12 -0.0 15.2 Lukman of Nigeria 
16 4 2.0 15.2 Days of Rage 
(-shi), which is roughly the equivalent of the English titles Mr. and Ms. In both cases, 
phrases were required to have tf > 10. 3 
3 This procedure produced the interesting substring, Mr. From, where both words would normally 
appear on a stop list. This name has a large RIDF. (The MI, though, is small because the parts are so 
high in frequency.) 
23 
Computational Linguistics Volume 27, Number 1 
Table 7 
Examples of keywords with extreme values of RIDF and MI that point in opposite directions. 
The top half (high RIDF and low MI) tends to have more loan words, largely written in 
katakana and English. The bottom half (low RIDF and high MI) tends to have more general 
vocabulary, largely written in Chinese kanji. 
RIDF MI Substrings Features 
High Low :~k~E(native last name) 
10% 10% SUN (company name) 
U -- if(foreign name) 
~'c~ U(brush) 
',1 7 7' -- (sofa) 
Low High ~;~< f'~.~,~ (huge) 
10% 10% ~ ~J 1~ (passive) 
~t,~ O (determination) 
J~j(native full name) 
~j~(native full name) 
kanji character 
English character 
katakana character 
hiragana character 
loan word, katakana 
general vocabulary 
general vocabulary, kanji 
general vocabulary 
kanji character 
kanji character 
5 
4 
1 
qZ 
0 
° +-+ 
.~ ° 
" ++I +~C ° 
+.+ . . 
'o * 0++ ,  <~ 
'."° ,+.,,<~%,~ " 
: .+';+ . 
i i 
0 4 MI 8 12 
(a) English names 
Figure 11 
MI and RIDF of people's names. 
2 
I 
+ o + ¢¢++ ¢¢ 
o+~.~.o~,...~$°,,,'~ °°o° ,.. +,#.,.~_.,,.~ le++,++.+.®~ °° 
• -'.IL',+I~+,,.~,.~ + ,I~<,Q" ++CP++ .,, 
° ° <+~lh;,IllmllIl"\]IR~.& °*<,° ° ,,~'<,Ir...,~b-~. <, o,. o 
.+ ~+~mlL',ll,,l-l-l-l-l~ ";ll ~+° ° ,,¢ t + <, <+ 
+ • . 
+ "~."r'I,l~lll3P~+V ." 
. • . • ° . ° . i 
-8 -4 0 MI 4 8 12 
(b) Japanese names 
The English names have a sharp cutoff around MI = 7 due in large part to 
the title Mr. MI('Mr.', x) = log 2 ~N -- l°g2 tf('Mr.',x)tf(x) ---- 7.4 - log 2 ~'tf(x)' Since 
log 2 ~tf(x) is a small positive number, typically 0-3, MI('Mr',x) < 7.4. 
Names generally have RIDF values ranging from practically nothing (for com- 
mon names like Jones) to extremely large values for excellent keywords. The Japanese 
names, however, cluster into two groups, those with RIDF above 0.5, and those with 
RIDF below 0.5. The separation above and below RIDF = 0.5, we believe, is a reflec- 
24 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
tion of the well-known distinction between new information and given information in 
discourse structure. It is common in both English and Japanese, for the first mention 
of a name in a news article to describe the name in more detail than subsequent uses. 
In English, for example, terms like spokesman or spokeswoman and appositives are quite 
common for the first use of a name, and less so, for subsequent uses. In Japanese, the 
pattern appears to be even more rigid than in English. The first use will very often 
list the full name (first name plus last name), unlike subsequent uses, which almost 
always omit the first name. As a consequence, the last name exhibits a large range 
of RIDF values, as in English, but the full name will usually (90%) fall below the 
RIDF = 0.5 threshold. The MI values have a broader range as well, depending on the 
compositionality of the name. 
To summarize, RIDF and MI can be used to identify a number of interesting 
similarities and differences in the use of names. Names are interestingly different 
from general vocabulary. Many names are very good keywords and have large RIDE 
General vocabulary tends to have large MI. Although we observe this basic pattern 
over both English and Japanese, names bring up some interesting differences between 
the two languages such as the tendency for Japanese names to fall into two groups 
separated by the RIDF = 0.5 threshold. 
3.4 Word Extraction 
RIDF and MI may be useful for word extraction. In many languages such as Chinese, 
Japanese, and Thai, word extraction is not an easy task, because, unlike English, many 
of these languages do not use delimiters between words. Automatic word extraction 
can be applied to the task of dictionary maintenance. Since most NLP applications 
(including word segmentation for these languages) are dictionary based, word extrac- 
tion is very important for these languages. Nagao and Mori (1994) and Nagata (1996) 
proposed n-gram methods for Japanese. Sproat and Shih (1990) found MI to be useful 
for word extraction in Chinese. 
We performed the following simple experiment to see if both MI and RIDF could 
be useful for word extraction in Japanese. We extracted four random samples of 100 
substrings each. The four samples cover all four combinations of high and low RIDF 
and high and low MI, where high is defined to be in the top 10% and low is defined 
to be in the bottom 10%. Then we manually scored each sample substring using our 
own subjective judgment. Substrings were labeled "good" (the substring is a word), 
"bad" (the substring is not a word), or "gray" (the judge is not sure). The results are 
presented in Table 8, which shows that substrings with high scores in both dimensions 
are more likely to be words than substrings that score high in just one dimension. 
Substrings with low scores in both dimensions are very unlikely to be words. These 
results demonstrate plausibility for the use of multiple statistics. The approach could 
be combined with other methods in the literature such as Kita et al. (1994) to produce 
a more practical system. In any case, automatic word extraction is not an easy task for 
Japanese (Nagata 1996). 
4. Conclusion 
Bigrams and trigrams are commonly used in statistical natural language processing; 
this paper described techniques for working with much longer n-grams, including 
million-grams and even billion-grams. We presented algorithms (and C code) for com- 
puting term frequency (tf) and document frequency (dr) for all n-grams (substrings) 
in a corpus (sequence). The method took only a few hours to compute tf and df for 
all the n-grams in two large corpora, an English corpus of 50 million words of Wall 
25 
Computational Linguistics Volume 27, Number 1 
Table 8 
The combination of RIDF and MI is better in a word extraction task than either by itself, 
which is better than neither. Each cell reports performance over a sample of 100 substrings. 
Substrings were subjectively judged to be "good" (the substring is a word), "bad" (the 
substring is not a word), or "gray" (the judge is not sure). Two performance values are 
reported, indicating what percentage of the 100 substrings are words. The larger performance 
values count the "gray" substrings as words; the smaller performance values count the "gray" 
substrings as nonwords. 
All MI MI(high 10%) MI(low 10%) 
All RIDF -- 20-44% 2-11% 
RIDF(high 10%) 29-51% 38-55% 11-35% 
RIDF(low 10%) 3-18% 4-13% 0-8% 
Street Journal news articles and a Japanese corpus of 216 million characters of Mainichi 
Shimbun news articles. 
The method works by grouping substrings into classes so that the computation 
of tf and df over order N 2 substrings can be reduced to a computation over order N 
classes. The reduction makes use of four properties: 
Properties 1-2: all substrings in a class have the same statistics (at least 
for the statistics of interest, namely tf and dJ), 
Property 3: the set of all substrings are partitioned by the classes, and 
Property 4: there are many fewer classes (order N) than substrings 
(order N2). 
The second half of the paper used the results of computing tf and df for all n-grams 
in the two large corpora mentioned above. We compared and contrasted RIDF and 
MI, statistics that are motivated by work in lexicography and information retrieval. 
Both statistics compare the frequency of an n-gram to chance, but they use different 
notions of chance. RIDF looks for n-grams whose distributions pick out a relatively 
small number documents, unlike a random (Poisson) distribution. These n-grams tend 
to be good keywords for information retrieval, such as technical terms and names. MI 
looks for n-grams whose internal structure cannot be attributed to compositionality. MI 
tends to pick out general vocabulary--words and phrases that appear in dictionaries. 
We believe that both statistics are useful, but in different and complementary ways. 
In a Japanese word extraction task, the combination of MI and RIDF performed better 
than either by itself. 
Appendix: C Code 
The following code is also available at http://www.milab.is.tsukuba.ac.jp/~myama/ 
tfdf. 
A: C Code to Print All LCP-Delimited Intervals using C Language's Stack 
The function output (below) is called 2N - 1 times. It will output an interval if the 
interval is lcp-delimited (LBL < SIL). Trivial intervals are always lcp-delimited. Non- 
trivial intervals are lcp-delimited if the bounding lcps are smaller than the SIL=lcp\[k\], 
26 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
where k is the representative. 
void output(int i, int j, int k){ 
int LBL = (Icp\[i\] > icp\[j+l\]) ? icp\[i\] : icp\[j+l\]; 
int SIL = icp\[k\]; 
if(i==j) printf("trivial <~d,~d>, tf=l\n ", i, j); 
else if(LBL < SIL) printf("non-trivial <~d, ~d>, rep=~d, tf=~d\n", 
i, j, k, j-i+i); 
int print_LDIs(int i, int k){ 
int j = k; 
output(k,k,O); /* trivial intervals */ 
while(icp\[k\] <= icp\[j+l\] && j+l < N) j = print_LDIs(k, j+l); 
output(i,j,k); /~ non-trivial intervals */ 
return j;} 
B: C Code to Print All LCP-Delimited Intervals Using an Own Stack 
print_LDIs_stack is similar to print_LDIs, but uses its own stack. It takes the corpus 
size, N, as an argument. 
#define STACK_SIZE i00000 
#define Top_i (stack\[sp-l\].i) 
#define Top_k (stack\[sp-l\].k) 
struct STACK {int i; int k;} stack\[STACK_SIZE\]; 
int sp = O; /~ stack pointer ~/ 
void push(int i, int k) { 
if(sp >= STACK_SIZE) { 
fprintf(stderr, "stack overflow\n"); 
exit(2);} 
stack\[sp\].i = i; 
stack\[sp++\].k = k;} 
void pop() {sp--;} 
void print_LDIs_stack(int N) { 
int j; 
push(O,O); 
for(j = O; j < N; j++) { 
output(j, j, 0); 
while(icp\[j+l\] < Icp\[Top_k\]) { 
output(Top_i, j, Top_k); 
pop();} 
push(Top_k, j+l);}} 
27 
Computational Linguistics Volume 27, Number 1 
C: C Code to Print All LCP-Delimited Intervals with tf and df 
The steps 5 and 6 of the algorithm in Section 2.6 are implemented as the function 
dec df using the binary search. 
#define STACK_SIZE I00000 
#define Top_i (stack\[sp-l\] .i) 
#define Top_k (stack\[sp-l\] .k) 
#define Top_dr (stack\[sp-l\] .dr) 
struct STACK {int i; int k; int dr;} stack\[STACK_SIZE\]; 
int sp = O; /* stack pointer */ 
void push(int i, int k, int dr) { 
if(sp >= STACK_SIZE){ 
fprintf(stderr, "stack overflowkn"); 
exit (2) ;} 
stack\[sp\] .i = i; 
stack\[sp\] .k = k; 
stack \[sp++\] . df = df ; } 
void pop() {sp--;} 
void output(int i, int j, int k, int dr) { 
int LBL; 
if(icp\[i\] > icp\[j+l\]) LBL = icp\[i\]; 
else LBL = icp\[3+l\]; 
if(i==j) printf("trivial <~d,~d>, tf=l\n '', i, j); 
else if(LBL < Icp\[k\]) 
printf("non-trivial <~d, ~d>, rep=~d, tf=~d, df=~d\n", 
i, j, k, j-i+l, df); 
* Print_LDIs with_dr does not only print tf, but also df. 
* It takes the corpus size, N, and the number of documents, D. 
*doc() returns the document number of the suffix array's index. 
* dec_dr() decrease a dr-counter in the stack when duplicate 
* counting occurs. 
*/ 
void dec_df(int docid) { 
int beg=0, end=sp, mid=sp/2; 
while(beg != mid) { 
if(doclink\[docid\] >= stack\[mid\].i) beg = mid; 
else end = mid; 
mid = (beg + end) / 2; 
} 
stack\[mid\].df--; 
28 
Yamamoto and Church Term Frequency and Document Frequency for All Substrings 
print_LDIs_with_df(int N, int D) { 
int i, j, dr; 
doclink = (int ~)malloc(sizeof(int) • D); 
for(i = O; i < D; i++) doclink\[i\] = -I; 
push(O,O, I) ; 
for(j = O; j < N; j++) { 
output (j ,j ,0, I) ; 
if(doclink\[doc(j)\] != -I) dec_df(doc(j)); 
doclink\[doc(j)\] = j ; 
df = i; 
while (icp\[j+l\] < icp\[Top_k\]) { 
df = Top_dr + dr; 
output (Top_i, j, Top_k, dr) ; 
pop() ; 
} 
push(Top_k, j+l, dr); 
} 
Acknowledgments 
We would like to thank the anonymous 
reviewers for Computational Linguistics who 
made insightful comments on an earlier 
draft. 

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