SUMMARIZING DOCUMENTS USING FRACTAL TECHNIQUES

M. Dolores Ruiz and Antonio B. Bail

Department of Computer Science and Artiﬁcial Intelligence, University of Granada

C/Daniel Saucedo Aranda s/n 18071, Granada, Spain

Keywords:

Fractal summarization, fractal dimension, summarization.

Abstract:

Every day we search new information in the web, and we found a lot of documents which contain pages with

a great amount of information. There is a big demand for automatic summarization in a rapid and precise

way. Many methods have been used in automatic extraction but most of them do not take into account the

hierarchical structure of the documents. A novel method using the structure of the document was introduced

by Yang and Wang in 2004. It is based in a fractal view method for controlling the information displayed. We

explain its drawbacks and we solve them using the new concept of fractal dimension of a text document to

achieve a better diversiﬁcation of the extracted sentences improving the performance of the method.

1 INTRODUCTION

Nowadays, a lot of information is accesible in the

web, and many people can use this huge source to re-

trieve what they need. For example, users can check

e-mail, read news, buy products, etc. using a com-

puter, but at present the use of mobile and handheld

devices is growing signiﬁcantly. However, the re-

duced dimensions of handheld devices limit the handy

visualization of large documents.

The visualization of summarized documents in a

portable device can facilitate a fast review of their

main content in just a few seconds. Organizations

need to make decisions as fast as possible, access to

large text documents or to other information sources

during decision making, and as a result, there is an

urgent need of a tool that summarizes the information

in a accurate and precise way.

Many methods have been used in automatic sum-

marization (Buyukkokten O., 2001).We can ﬁnd in

the literature that those methods employ mainly three

techniques for the automatic single-document sum-

marization (Daume III H., 2005): sentence extrac-

tion, bag-of-words headline generation, and docu-

ment compression.

Research in sentence extraction started with the

works made by Luhn (Luhn, 1958) and Edmundson

(Edmundson, 1969). But recent techniques based in

the structure of the document had been proposed. A

novel model named fractal summarization has been

applied (Yang C. C., 2003b),(Yang C. C., 2003c)

based on the idea of fractal view of H. Koike (Koike,

1995) and on the techniques of fractal image com-

pression (Yang C. C., 2003a). The main idea is to

take into account the structure of the document and

the inheritance of the importance to the substructures,

but this method does not use all the information that

fractal structure gives.

The calculation of dimensions is a useful tool to

quantify structural information of artiﬁcial and natu-

ral objects. There are some types of dimension: the

Euclidean one, the Hausdorff-Besicovitch dimension,

and so on. We are going to work with the fractal di-

mension in the special case of text documents.

Fractal dimension of many objects cannot be de-

termined analytically but there are some good estima-

tors. We do a brief overview of some of them and

choose the most suitable for the calculation of fractal

dimension of text documents.

The paper follows with a description of the fea-

tures used in traditional summarization techniques.

We present their adaptations to structured documents.

Then we describe the fractal summarization method

Dolores Ruiz M. and B. Bailón A. (2007).

SUMMARIZING DOCUMENTS USING FRACTAL TECHNIQUES.

In Proceedings of the Ninth International Conference on Enterprise Information Systems - SAIC, pages 26-33

DOI: 10.5220/0002363300260033

 SciTePress

given in (Yang C. C., 2003b) to comment its short-

comings, proposing some modiﬁcations to the propa-

gation formula. We follow with a review of how to

calculate the fractal dimension of text documents, ﬁ-

nishing with the results of some experiments summa-

rizing web pages. We conclude with the conclusions

and the bibliography used.

2 STRUCTURE IN

SUMMARIZATION

TECHNIQUES

Many traditional summarization models involve the

extraction of sentences from the source document

based on some of its salient features. But the docu-

ment is considered as a sequence of sentences instead

of considering its hierarchical structure. The hierar-

chical structure of the document give to us more in-

formation about the importance of some paragraphs

depending on their location.

2.1 Hierarchical Structure of a Text

Document

In most of the cases we work with structured infor-

mation. When humans write a document they use an

important source of information: the structure. Using

a structured text we achieve a better division of infor-

mation and we give more importance to some parts

allocating them in special places. We are going to

use this property to transform the text document into

a tree.

Documents are usually hierarchical and can be

viewed as compositions of simpler constituents. A

document consists of chapters; a chapter consists of

sections; a section consists of subsections, a section

or subsection contains paragraphs; a paragraph con-

sists of sentences, and a sentence is a group of terms.

The smallest units which contain information are the

terms (words with signiﬁcance for themselves; words

that are deemed irrelevant are eliminated). This is the

highest division we have considered. We will refer to

each level in the compositional structure of a docu-

ment depending of its position with regard to the title

of the document. Then, the original document is re-

presented as a tree according to its structure. As

shown in ﬁgure 1, in the tree there is a inclusion rela-

tion between a node and its children. Every node in

the tree will contain all of its descendants. In this way,

the root or the tree represents the whole document, not

only the title; and the terms will be the leaves of the

tree.

Figure 1: Document hierarchical structure.

Figure 2: Tree associated to a text document.

2.2 Adaptation of Traditional Features

The most widely used summarization characteristics

are the thematic, location, heading and cue features.

These features have been modiﬁed according to the

structure of the document. We have used the adap-

tation of these features using the tree structure of the

document (Yang C. C., 2004).

• The thematic feature was ﬁrst identiﬁed by Luhn

(Luhn, 1958) and then modiﬁed by Edmundson

(Edmundson, 1969). He proposed a thematic

SUMMARIZING DOCUMENTS USING FRACTAL TECHNIQUES

weight of keywords based on the term frequency.

The tﬁdf (Term Frequency, Inverse Document

Frecuency) score is the most used. We consider

the i

term in a document as t

. In our case, the

tﬁdf of the term t

in a node is deﬁned as the term

frequency within the node inverse the numbers of

nodes at the same level that contains the term, i.e.,

= t f

× log





, (1)

where t f

is the frequency of the term t

in the

node x, N

is the number of nodes in the docu-

ment at the same level as x and n

is the number

of nodes that contain the term t

in the document

at the same level. The Sentence Thematic Score,

, of the k

sentence s

in the node x is calcu-

lated as the sum of the w

of the constituent terms

of the sentence s

, i.e.,

,x) =

∑

∈s

. (2)

The Thematic Score TS of the node x is again

calculated as the sum of the Sentence Thematic

Score of all the sentences in the node, i.e.,

T S(x) =

∑

∈x

,x). (3)

• Edmundson also considers the location feature

(Edmundson, 1969) as an indicator of the signiﬁ-

cance of a sentence based on the hypotheses that

topic sentences tend to occur at the beginning or

the end of a document or paragraph. The Location

Score LS of a node x is calculated as the inverse

of the minimal distance of the node x to the ﬁrst

node and the last node at the same level with the

same parent as x, i.e.,

LS(x) =

min{d(x, f nb(x)),d(x,lnb(x))}

(4)

where d(x,y) is the distance function that calcu-

lates the number of nodes at the same level under

the same parent between x and y, f nb(x) is the

ﬁrst node brother of x, and lnb(x) is the last node

brother of x. See ﬁgure 3 as an example of this

feature.

• The third feature we have used is the heading fea-

ture. It is based on the hypotheses that the heading

contains the subject of the document summarized

in its words.

The Sentence Heading Score SS

of a sentence

has been modiﬁed to consider the structure of

the document. For example, at each level we have

to consider the previous headings giving a higher

weight when the heading is closer to the node we

are considering.

Figure 3: Location feature.

Let s

be the sentence into consideration that it

is included in the node x. Its Sentence Heading

Score would be the sum of the weights of the

terms t

that appears in it and in the heading of

other nodes z in the path from the root to the node

This weight is computed as the sum of the weights

seen in (1) of terms t

in the sentence s

divided

by the product of the numbers of children m

node

the nodes in the path from node z to node x, i.e.,

,x) =

∑

z∈ path from root to x

∑

∈z∩s

∏

i∈ path form z to x

(5)

The Heading Score HS of node x is calculated as

the sum of the Sentence Heading Score of all sen-

tences in the node, i.e.,

HS(x) =

∑

∈x

,x). (6)

• The last feature we have considered is the cue fea-

ture. Humans pay more attention to the sections

which has bonus words such as conclusion. In the

tree we examine the headings predecessors to the

node and sum the cue weights of the terms t

being

in the cue dictionary and in the headings. The Cue

Score is

CS(x) =

∑

∈heading(x)

cue(t

) (7)

where cue(t

) is the cue weight of the term in the

dictionary.

All these features must be normalized because

they have different range of values. Therefore, we di-

vide each feature by its maximum score in the whole

document, so we have the normalized scores associ-

ated to the four features, NTS, NLS, NHS and NCS.

To calculate the total weight associated to a node we

only have to do a weighted sum of the normalized the-

matic, location, heading and cue score of the node.

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We call this total punctuation the Node Signiﬁcance

Score, NSS, i.e., NSS(x) is equal to

NT S(x)+a

NLS(x) + a

NHS(x)+a

NCS(x) (8)

where a

are positive reals that sum 1. They

are chosen to adjust the weighting of the different

summarization features according to our preferences.

3 FRACTAL SUMMARIZATION

Many studies of human abstraction process has shown

that humans extract the sentences according to the

document structure. However, most traditional auto-

matic summarization techniques consider a document

only as a set of sentences ignoring the structure of

the document. Some advanced summarization models

take into account the structure to compute the proba-

bility of a sentence to be included in the summary.

Fractal summarization model was proposed by Yang

and Wang to generate a summary based on document

structure.

Fractal summarization is developed based on the

idea of fractal view (Koike, 1995) and adapting the

traditional models of automatic extraction. The im-

portant information is captured from the source text

by exploring the hierarchical structure and salient fea-

tures of the document. Therefore they adapted the

features considering the structure of the document.

3.1 Fractal View & First

Approximation to Fractal

Summarization

Fractal summarization was developed by Yang and

Wang (Yang C. C., 2004) based on the idea of frac-

tal view of Koike (Koike, 1995) and adapting the tra-

ditional models of automatic extraction. They use the

inheritance of values to transmit the importance of the

node inside the whole document and to give out the

number of sentences we have to extract from each of

its children according to their importance. But they

do it using ﬁxed values for some parameters like the

fractal dimension. They also do a division into range

blocks (Yang C. C., 2003b),(Yang C. C., 2003a) of the

document.

Next subsections show the process to convert the

fractal view method for controlling information in a

fractal automatic summarization method.

3.1.1 Fractal View

Fractal view is a fractal-based method that provides a

mechanism to control the amount of information dis-

played (Koike, 1995). The idea is based on the pro-

perty of self-similarity of a fractal tree. A tree is made

of a lot of sub-trees; each of them is also a tree. They

represent the degree of importance of each node by

its fractal value and they propagate the importance to

other nodes with the following expression:

(

root

= 1

= Fv

p(x)

×C N

−1/D

p(x)

(9)

where Fv

is the fractal value of node x, p(x) de-

notes the parent node of x, N

p(x)

is the number of child

nodes of p(x), D is the fractal dimension, and C is a

constant value satisfying 0 < C ≤ 1. This constant is

used to distinguish when we have a single tree, in the

sense that each node has a unique branch, and in con-

clusion we have node-branch-node-branch-node etc.

In this special case we have to choose C 6= 1. In other

cases, the majority of them, we choose C = 1. Notice

that when D = 1 = C, the above formula at the ﬁrst

level divide the fractal value of the root in N

root

equal

parts. Then, divide 1/N

root

in n

equal parts, where

is the number of children that node x has. So if

we sum the fractal values of all nodes with the same

parent, they sum the fractal value of the parent; and if

we sum Fv of all nodes at the same level, we have as a

result 1. That is, the formula share out in an equitable

form when D = 1.

3.1.2 First Approximation to Fractal

Summarization

In (Yang C. C., 2004) the authors adapt the techniques

of fractal view to propagate the values from parents

nodes to their child nodes using the fractal dimension

D = 1 in the equation 9. They ﬁrst divide the docu-

ment into range blocks. Then they transform the do-

cument into a tree and for the propagation of values

they use the following formula:











root

= 1

= Fv

p(x)







NSS (x)

∑

y∈p(x)

NSS (y)







−1

(10)

where NSS(x) is the Signiﬁcance Score associated to

the node x using a division into range blocks, p(x)

is the parent node of x, and the expression y ∈ p(x)

denotes that y is a child of p(x).

The above formula doesn’t take into account

the fractal dimension of the document proposed by

Koike. We are going to solve this problem computing

the fractal dimension of a text document.

SUMMARIZING DOCUMENTS USING FRACTAL TECHNIQUES

Other shortcoming is that they change in the frac-

tal view formula the number of child nodes N

p(x)

(an

integer number ≥ 1) for a rational number between 0

and 1. For this reason the formula does not work well.

We do some important changes in the formula

and introduce the calculation of the fractal dimension

associated to a text document. This diversiﬁes the ex-

traction of sentences according to the content of the

different sections.

4 ENHANCED FRACTAL

SUMMARIZATION

To solve all problems, we propose some modiﬁcations

for the last formula. We adapt it for a well working

and we introduce the use of fractal dimension D as a

parameter and not as a ﬁxed value.

So the hierarchical inheritance formula between

nodes we take into consideration is











root

= 1

= Fv

p(x)







NSS (x)

∑

y∈p(x)

NSS (y)







1/D(x)

(11)

where D(x) is the fractal dimension of the node x. The

value of D(x) can change depending of the amount of

information contained in the node x, and D(root) is

the fractal dimension of the document (see next sec-

tion 6). We have used the NSS(x) measure seen in

section 2 to measure the importance of the node using

the hierarchical structure of the document. For a more

complete comprehension of the formula we explain in

next sections the computation of the fractal dimension

of text documents.

The propagation formula (11) has the following

advantages:

• It takes into account the structure of the document,

because it is based on the fractal tree representa-

tion associated.

• Inheritance of values. When a parent node is very

important because it has a lot of information, we

want to extract many sentences from this node. To

do that the child node inherits the quota of sen-

tences (see 22) from its parent using the propaga-

tion formula (11).

5 FRACTAL DIMENSION

The calculation of dimensions is a useful tool to quan-

tify structural information of artiﬁcial and natural ob-

jects. There are some types of dimension: the Eu-

clidean one, the Hausdorff-Besicovitch dimension,

and so on (Kraft, 1995).

The dimension that everybody knows is the Eu-

clidean one. A point has dimension 0, a line has

dimension 1, a plane has dimension 2 and when we

work in the space we say that it has dimension 3. For

example we are three-dimensional entities. But not all

the objects in this world have integer dimensions. The

matematicians since the nineteenth century have met

with other objects like the Cantor Set and the Koch

Curve whose dimensions are real numbers instead of

integers.

Mandelbrot told that a fractal is a shape made of

parts similar to the whole (Mandelbrot, 1986). This

deﬁnition uses the concept of self-similarity. We say

a set is strictly self-similar if we broke it into arbitrary

small pieces and each of them is a replica of the entire

set. When this happens the calculation of the fractal

dimension is easier.

Imagine we have a segment of length 1. We can

put two segments of length 6/10 each one at the end of

the branch. We can repeat this process till the inﬁnity

(see ﬁgure 4). In this case the fractal dimension is

very easy to calculate because of the self-similarity

property of the tree that we have built, and is

D = − log

2 ≈ 1.36. (12)

. . .

Figure 4: Fractal tree.

If we generalize this construction when the tree

has N children at each node (cross of two or more

branches) and any branch is r times longer than that

of the previous branch, the dimension of the tree can

be calculated as

D = − log

N. (13)

This type of special tree is called a fractal tree and

we have seen that its fractal dimension is uniquely de-

ﬁned by the above formula.

But exact fractal dimension can only be calculated

for ideal mathematical objects. For the rest we need to

use methods to approximate the fractal dimension. In

the literature there are different methods to calculate

the fractal dimension.

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One of them is the Box-counting method. It basis

on the idea of covering the object with a grid of boxes

of smaller size in each iteration, and then compute

the frequency which data points fall into each box.

But the box-counting can be computed only for low-

dimensional sets because the algorithmic complexity

grows exponentially with the set dimension. Besides

this, it is too difﬁcult to select the best suited grid to

the object. Notice that this method uses the coordi-

nates of the points to allocate them into their corre-

sponding box. See (Liebovitch, 1989) for more de-

tails.

If we estimate the dimension of a curve (e.g. cell

membrane, coastline, landscape edge) we can use the

Compass method (Kraft, 1995). The procedure is

analogous to moving a compass with a ﬁxed length δ

along the curve. The estimated length of the curve is

the product of the number of rulers required to ‘cover’

the object and the scale factor δ. The relationship

between the length L, δ, and the fractal dimension D

is L = kδ

1−D

where k is a constant. The fractal di-

mension is estimated by measuring the length of the

curve at various scale values δ. This method is only

for curves and it is exact for self-similar curves.

Another method used to calculate the correlation

dimension (Grasberger P., 1983) is a good substitute

of the box-counting method due to its computational

simplicity. In each iteration we choose circles of

a ﬁxed ratio in decreasing order and then count the

number of data points that fall in each circle.

There are other methods in the literature but many

of them are used with special types of objects using

in each case their particular characteristics (spatial or

temporal series, point patterns information theory and

diversity, topographic surfaces, etc.). For a more com-

plete view of this topic consult (Camastra F., 2002),

(Kraft, 1995) and (Grasberger P., 1983).

The disadvantages of the box-counting method

and the inappropriate use of coordinates in a text do-

cument to designate the sentences or words in it, are

some of the reasons to reject the method. The com-

pass method is used only for curves and it is inappro-

priate when curves have intersections. However the

correlation dimension uses a distance function that we

are going to adapt to our document using its structure.

This is the reason to choose the Grasberger and Pro-

caccia method to calculate the fractal dimension of a

text document.

6 FRACTAL DIMENSION OF A

TEXT DOCUMENT

We are going to work with the fractal dimension in

the special case of text documents. Fractal dimension

of many objects cannot be determined analytically, in

those cases we can use some estimators of the fractal

dimension. For text documents we are going to use

the Grasberger and Procaccia method (Grasberger P.,

1983).

To calculate the fractal dimension we need the tree

associated to the document, and we have to deﬁne a

distance function between any two nodes. Let N be

the number of nodes in the tree, R the maximum num-

ber of levels of the document (it will be the maximum

number of branches between the root node and the last

node in all directions), and I the indicator function de-

ﬁned as 1 if it is true and 0 in the rest of cases. To de-

ﬁne the distance between two arbitrary nodes we ﬁrst

need a function that tell us how similar are two nodes.

For this purpose we take the cosine measure (Daume

III H., 2005),

cos(x,y) =

∑

∈x

∑

∈y

∑

∈x

∑

∈y

(14)

where we have used the t f id f weighting in equation

(1) associated to the node. Another similarity mea-

sures can be founded in (Guerrini G., 2006).

With this deﬁnition, the cosine takes values

between 0 and 1, having a value very near to 1 when

nodes have similar contents. For that reason we take

the distance between nodes x and y as

dist(x,y) = 1 − cos(x, y). (15)

Let x

, x

two arbitrary nodes, there is always a

path joining them. Suppose x

= x

,...,x

= x

the shortest path between the two nodes. Then the

distance between x

and x



− x



n−1

∑

l=1

dist(x

l+1

)

. (16)

Next, we calculate the correlation integral C

)

(see the Grasberguer and Procaccia method in (Gras-

berger P., 1983) and (Ruiz M. D., 2006)) for each

= (2R − k)/(2R) (17)

with k = 1,...,2R − 1, that is, we do

) =

N(N − 1)

∑

i=1

∑

j=i+1



− x



≤ r

) (18)

SUMMARIZING DOCUMENTS USING FRACTAL TECHNIQUES

where N is the number of nodes in the tree and R is

the maximum number of levels of the document, and

I is the indicator function:

I(λ) =

(

1 if λ is true

0 rest of cases.

(19)

The correlation dimension is then deﬁned with the

formula

D = lim

→0

ln(C

))

ln(r

)

. (20)

But in practice we can calculate it computing

the slope of the regression line that the points

(ln(r

),ln(C

))) form.

This approximation to the calculation of the frac-

tal dimension of a text document indicates the distri-

bution of the information in the document. The above

distance function tells how similar are the contents of

the node x

and the node x

. If they are very far, the

distance will be small and therefore it won’t count in

the sum.

Some experiments with web pages for computing

the fractal dimension of texts can be found in (Ruiz

M. D., 2006).

7 FRACTAL SUMMARIZATION

ALGORITHM

Now we propose the algorithm for the process of au-

tomatic summarization using the propagation formula

in (11) and then we explain how it works with more

details.

We deﬁne the compression ratio, r, of summariza-

tion as the number of sentences we would like to be

in the summary divided by the number of sentences

of the document, i.e.,

r =

# sentences of summary

# sentences of document

. (21)

This is the ideal deﬁnition, but the number of sen-

tences we are going to extract depends on the size of

the document. It has been proved that extraction of

20% sentences can be as informative as the full text

of the source document (Morris G., 1992). But the

users can also choose the number of sentences they

want to extract and change it if they are interested or

not in the content of the document.

The quota of the summary is the compression ra-

tio times the number of sentences of the document,

and we propagate the quota to the child nodes by the

formula

quota

= Fv

p(x)

× quota

p(x)

(22)

Algorithm 1 : Fractal Summarization.

1. Choose a compression ratio.

2. Choose a threshold value.

3. Calculate the sentence number quota of the sum-

mary.

4. Transform the document into a fractal tree.

5. Calculate the fractal dimension of each node of the

document.

6. Set the current node to the root of the fractal tree.

7. Repeat

7.1 For each child node under current node, cal-

culate the fractal value of child node.

7.2 Allocate quota to child nodes in proportion

to fractal values.

7.3 For each child nodes.

If the quota is less than threshold value

Select the sentences in the node by ex-

traction.

Else

Set the current node to the child node

Repeat steps 7.1, 7.2, 7.3.

8. Until all the child nodes under current node are

processed.

where x is the child node and p(x) denotes de parent

node of x.

The threshold value is the maximum number of

sentences that we want to extract from the same node.

The optimum value is between 3 and 5 (Goldstein J.,

1999). This threshold value is going to prevent the

appearance of overlapped sentences in the summary

because we will extract few sentences of each para-

graph.

8 EXPERIMENTS AND RESULTS

We have used this automatic summarization method

in web pages with a large content of text and with a

good structure in the sense that for the title they use

the markup tags < H1 >, for the sections < H2 >,

< H3 > for the subsections, etc. with their corre-

sponding close markup tags < /H1 >, < /H2 >, etc.

We do a fractal tree using the above structure and

when we are at the last level we choose only the terms

with signiﬁcance, that is we drop all the articles, ordi-

nal and cardinal numbers, the verb to be, prepositions,

pronouns, conjunctions, etc. We can see an example

of tree associated to a document in the ﬁgure 2. For

the cue feature we take the page keywords to form the

dictionary of bonus words. And we have considered

the coefﬁcients in (8) all of them equal to 1/4, that is,

we give the same importance to the four salient fea-

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tures of the document.

We have run some experiments with a wide range

type of web pages and we observe the following facts:

• The method achieve a good performance with

documents that have several levels of granularity,

in other words, when the tree associated to the

document has many levels and the nodes have a

lot of branches, the calculation of the fractal di-

mension helps to get a summary with information

more diversiﬁed according to the document struc-

ture.

• With wrong structured web pages the method ob-

tains bad results since the calculation of fractal di-

mension doesn’t give information in those cases.

In conclusion, we have seen in our experiments

that traditional summarization extracts most of the

sentences from few chapters, fractal summarization

with D = 1 extracts the sentences distributively from

each section, and with our new approximation using

the fractal dimension of the document, the method

share out the sentences according to their content and

their position.

9 CONCLUSIONS

In this paper, we present an improvement to the frac-

tal summarization method. The propagation formula

have been modiﬁed according to the fractal view

method, and it uses the novel concept of fractal di-

mension of text documents presented in (Ruiz M. D.,

2006).

We have used this automatic summarization

method in web pages with a large content of text and

with a good structure as in ﬁgure 2, giving very good

results and showing the good performance of the pro-

posed method.

In the future, we are going to use a similarity mea-

sure taking into account the semantic of words giving

a more complete solution to the problem of summa-

rizing documents. Moreover, we are working about

the problem of summarizing the document according

the preferences of the user, giving more importance

to those sections that the user wants to spread out

using the fractal dimension. We also want to adapt our

method in the case of summarizing a group of docu-

ments with similar contents.

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SUMMARIZING DOCUMENTS USING FRACTAL TECHNIQUES