DSpace at VNU: A method for mining top-rank-k frequent closed itemsets

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A method for mining top-rank-k
frequent closed itemsets
Loan T.T. Nguyena,b,∗ , Truc Trinhc , Ngoc-Thanh Nguyend and Bay Voe

a Division of Knowledge and System Engineering for ICT, Ton Duc Thang University,
Ho Chi Minh City, Vietnam
b Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam
c VOV College, Ho Chi Minh City, Vietnam
d Faculty of Computer Science and Management, Wroclaw University of Science and Technology,
Wrocław, Poland
e Faculty of Information Technology, Ho Chi Minh City University of Technology, Vietnam

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Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx
DOI:10.3233/JIFS-169128
IOS Press

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Keywords: DCI-Plus, dynamic bit vectors, frequent closed itemsets, top-rank-k frequent closed itemsets

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1. Introduction

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Data mining is the process of extracting interesting
knowledge from data. Various methods for discovering knowledge have been proposed, such as mining
traditional association rules [1–4, 6, 7, 23, 31, 36, 37],
mining non-redundant association rules [8, 41], mining minimal non-redundant association rules [26, 27],
mining most generalization association rules [38],

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Abstract. Mining frequent closed itemsets (FCIs) is important in mining non-redundant (minimal) association rules. Therefore, many algorithms have been developed for mining FCIs with reduced mining time and memory usage. For mining FCIs,
algorithms use the minimum support threshold, minSup, to prune itemsets. However, using a fixed minSup is not suitable for
mining top-rank-k FCIs. A large threshold will lead to a small number of generated FCIs, leading to insufficient FCIs to query
when k is large. On the other hand, a small minSup will generate a huge number of generated FCIs, leading to large runtimes
and high memory usage. In this paper, we propose a method for mining top-rank-k FCIs without using a fixed minimum
support threshold. A strategy is first used to eliminate 1-items that cannot generate FCIs belonging to top-rank-k FCIs. Next,
based on the set of candidate 1-items, we propose TRK-FCI, a DCI-Plus-based algorithm, for mining top-rank-k FCIs. In

the process of mining top-rank-k FCIs, TRK-FCI automatically increases minSup according to the mined FCIs, efficiently
pruning itemsets that cannot belong to top-rank-k FCIs. We also modify the dynamic bit vector (DBV) structure and apply
it to reduce memory usage and runtime in the TRK-FCI-DBV algorithm. Experimental results show that TRK-FCI-DBV is
more efficient than TRK-FCI for various databases.

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∗ Corresponding

author. Loan T.T. Nguyen, Division of
Knowledge and System Engineering for ICT, Ton Duc
Thang University, Ho Chi Minh City, Vietnam. E-mail:


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classification using decision trees [13, 20, 30] or
ILA [33], classification based on association rules
[13, 14, 20, 21], and clustering [22]. Mining association rules has many applications in practice [3,
23]. For mining association rules, frequent itemsets
[2, 11, 21, 42], frequent closed itemsets (FCIs) [15,
21, 26, 28, 31, 32, 37, 40–42], or maximal frequent
itemsets [12, 19] must be mined. Mining frequent
itemsets is often used for generating all association
rules that satisfy minimum support threshold (minSup) and minimum confidence threshold (minConf)
[1, 2, 35, 36] and mining FCIs is used for mining
(minimal) non-redundant association rules (i.e., rules

1064-1246/16/$35.00 © 2016 – IOS Press and the authors. All rights reserved

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belong to tabk . Because DCI-Plus uses fixed bit
vectors, it has high memory usage and runtime for
storing and computing the bit vector of a new itemset, checking subsets, and computing the supports
of itemsets. TRK-FCI-DBV, an improved version of
TRK-FCI, is then developed. TRK-FCI-DBV uses
the dynamic bit vector (DBV) structure instead of
the bit vector structure to reduce mining time and
memory usage.
The rest of this paper is organized as follows. Section 2 presents definitions of FCIs and top-rank-k
FCIs and states the problem of mining top-rank-k
FCIs. In Section 3, we review works related to the
problem of mining FCIs, top-k and top-rank-k frequent itemsets, and top-k FCIs. Section 4 describes
a method for mining top-rank-k based on the DCIPlus algorithm and an improved algorithm based on
DBVs. Experimental results on standard databases
for TRK-FCI and TRK-FCI-DBV are presented in
Section 5. Conclusions and suggestions for future
work are given in Section 6.

2. Definitions and problem statement
Let I = {i1 , i2 , . . . , im } be a set of items and
DB = {t1 , t2 , . . . , tn } be a set of transactions, where
each ti (1 ≤ i ≤ n) is a transaction labeled by a unique
identifier and contains a set of items in I.

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considered redundant based on certain criteria are
eliminated) [26, 27, 41]. For mining maximal frequent itemsets, all frequent itemsets or FCIs (for
which the database must be scanned to compute the
supports of itemsets) must be generated to mine above
kinds of rules.
Mining FCIs is important for pruning redundant
rules. The problem was first stated in 1999 by
Pasquier et al. [26]. Since then, many algorithms have
been developed to enhance the efficiency of mining FCIs, such those based on FP-tree [11, 28, 40],
IT-tree [32, 42], bit vectors [31, 37], and N-Lists
[15]. To mine FCIs, the minSup is set. The FCIs
that satisfy the minSup threshold are selected. It is
difficult to mine a sufficient number of top-rank-k
FCIs because an excessively high threshold will lead
to very few FCIs, not enough to query. Conversely,
a minSup that is too low will lead to a very large number of FCIs, requiring a lot of memory and time to
mine. Therefore, developing efficient algorithms for
mining top-rank-k FCIs is necessary.
Some algorithms have been developed for mining top-rank-k frequent itemsets. Deng et al. [5]
proposed the NTK algorithm and used a Node-list

to mine top-rank-k frequent itemsets. The iNTK
algorithm, an improved version of the NTK algorithm proposed by Le et al. [14], uses the subsume
concept and the N-list structure to fast mine top-rankk itemsets. After that, some algorithms have been
developed for mining top-k frequent itemsets [29],
top-k FCIs [39], and top-k non-redundant association
rules [8].
Mining top-rank-k FCIs is important for mining
non-redundant association rules. However, for our
best knowledge, there are no developed algorithms
for mining top-rank-k FCIs. Besides, algorithms
developed for mining top-rank-k frequent itemsets
or top-k FCIs cannot be applied to mine top-rankk FCIs. Therefore, in this paper, we propose the
TRK-FCI algorithm, which is based on DCI-Plus
[31], for mining top-rank-k FCIs. First, the algorithm finds a set of candidate items that may belong
to top-rank-k FCIs, where k is a given threshold.
Then, it uses the DCI-Plus algorithm to generate
FCIs based on these candidate items. When an FCI
is generated, it is directly inserted into a table named
tabk . FCIs with the same support are stored in the
same entry. The number of entries in tabk is below
the threshold k. In the process of mining top-rank-k
FCIs, the algorithm automatically increases minSup
to reduce the number of FCI candidates that do not

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Definition 1. (support of an itemset). Given a DB and
an itemset X (X ⊆ I), the support of X, denoted by
SUPX , is the number of transactions containing X in
DB.
Definition 2. (frequent itemset). Given a DB and
an itemset X (X ⊆ I), X is a frequent itemset if
SUPX ≥ min Sup.
Definition 3. (FCI). Given a DB and an itemset
X (X ⊆ I), X is called an FCI if no itemset Y exists
such that X ⊂ Y and SUPX = SUPY .
Definition 4. (rank of an FCI). Given a set of CI
including all closed itemsets from a transaction
database DB and an FCI X (X ∈ CI), the rank of
X in CI is the number of itemsets whose support values are no greater than the support of X. The rank of
X is defined as:
RX = |{SUPY |Y ∈ CI ∧ SUPY ≥ SUPX }|

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Definition 5. (a top-rank-k FCI). Given a set of
CI including all closed itemsets from a transaction
database DB and a threshold k, an itemset X ∈ CI is
called a top-rank-k FCI if and only if RX is no greater
than k, i.e., RX ≤ k.
Definition 6. (mining top-rank-k FCIs). Given CI
including all FCIs from transaction database DB and
a threshold k, the goal of mining top-rank-k FCIs is to
find a complete set of FCIs whose ranks are no greater
than k, i.e., top-rank-k FCIs are a set of itemsets for
which {X ∈ CI|RX ≤ k}.
From definition 6, the problem of mining top-rankk FCIs is stated as follows. Given a database BD and
a threshold k, mining top-rank-k FCIs is divided into
two steps:
Step 1: Mine all closed itemsets in DB, a set called
CI.
Step 2: Keep the closed itemsets that satisfy definition 6 in CI.

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3. Related works

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3.1. Mining frequent closed itemsets

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The above approach is simple but not feasible
because the number of closed itemsets in the database
is often large. Therefore, finding a direct solution for
mining top-rank-k FCIs without mining all closed
itemsets is a challenge.

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eliminate items at high levels that have the same
support as that of items at low levels. FPClose [11] is
an improved version of Closet+ that uses FP-array to
reduce the number of FP-tree scans when FP-tree is
projected. CHARM [42] is based on tidsets for fast
computing the supports of itemsets and uses subset
checking to fast prune non-closed itemsets. To check
whether a generated itemset is closed, CHARM uses
a hash table in which the key of each itemset is the sum
of its items. dCHARM, a diffset approach for mining
FCIs is also developed [42]. CloseMiner [32] uses
closed tidsets to check whether an itemset is closed.
Although CHARM, dCHARM and CloseMiner have

advantages over algorithms based on horizon data
format such as Close, A-Close, Closet, Closet+, and
FPClose, they must use hash tables to check whether
a candidate itemset is closed, and thus closed itemsets
must be stored in main memory for easy checking.
DCI-Closed [21] uses tidsets and a non-duplication
generation strategy for mining FCIs. DCI-Plus [31],
an improved version of DCI-Closed [21], generates
FCIs and minimal generators of each FCI. Because
DCI-Closed is based on tidsets, when the tidsets of
itemsets are long, a lot of memory is required to store
the tidsets and the runtime required to compute the
intersection with other tidsets is high. To reduce the
length of tidsets and reduce computation time, DCIPlus uses BitTable.

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3.2. Mining top-rank-k frequent itemsets

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Problem of mining FCIs was first proposed in
1999 [26]. Many algorithms for mining FCIs have
since been developed to reduce runtime and memory usage. Apriori-based algorithms for this purpose
include Close [26] and A-Close [27]. These algorithms generate candidates and compute their closure
to find FCIs. Algorithms based on the divide-andconquer technique have been developed. Closet [28]
uses FP-tree to compress the database and early pruning to prune non-closed itemsets. Closet+ [40], an
improved version of Closet (which uses a bottomup projection scheme for FP-tree), uses a hybrid

approach: bottom-up for dense databases and topdown for sparse databases. It uses item merging and
sub-itemset pruning, which are widely used in other
algorithms, and applies the subset checking strategy
to fast check closed itemsets and item skipping to

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Deng et al. proposed the NTK algorithm for mining top-rank-k frequent itemsets [5]. NTK uses the
Node-list data structure to represent itemsets and
uses a level-wise approach for mining top-rank-k
frequent itemsets, i.e., t-patterns are used to form
(t+1)-patterns. By using Node-lists, the algorithm
does not need to rescan the database to compute the
supports of itemsets. A dynamic minSup is used to
efficient prune candidates. Le et al. developed iNTK
[14], an improved version of NTK. iNTK uses the
subsume concept to reduce the number of generated
candidates compared to those for NTK, reducing the
time required to generate candidates.
3.3. Mining top-k frequent closed itemsets
Wang et al. [39] proposed the TFP algorithm for
mining top-k FCIs, where k is the number of FCIs
that need to be mined. TFP uses a divide-and-conquer
technique (like FP-Growth) and prunes candidates

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3.4. Mining top-k association rules

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In 2012, Fournier-Viger et al. [10] proposed the
TopKRules algorithm for mining top-k association
rules from datasets. This algorithm uses the minConf value during the mining process of top-k rules.
The change of the minSup value is dependent on
the lowest support of itemsets. The TopKRules algorithm is based on the principle of extending rules and
some methods for early eliminating rules that do not
belong to top-k rules. Fournier-Viger and Tseng also
extended TopKRules for mining top-k non-redundant
rules [8] and top-k sequential rules [9]. These algorithms are very efficient compared to post-processing
methods.

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3.5. Dynamic bit vectors

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In 2012, Vo et al. [37] proposed the concept of
dynamic bit vectors (DBV) and used it in mining frequent closed itemsets. DBV of an itemset is a bit
vector in which zero bits from the begin and the end
are removed. With this concept, we can save memory
to store bit vectors and time to compute the intersection of bit vectors. Tran et al. expanded this concept
to mine frequent closed sequences [34]. Le et al. also
used DBV to develop an efficient algorithm for mining frequent closed inter-sequence using DBV [16].

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4.1. TRK-FCI algorithm

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based on minSup (automatically increased in the
process of updating candidates). The authors also
used a threshold min l to eliminate itemsets whose
lengths are smaller than min l.

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In this section, we present the TRK-FCI algorithm

for mining top-rank-k FCIs based on BitTable. TRKFCI uses DCI Plus [31] to generate candidate closed
itemsets and apply some early pruning techniques to
prune candidates. First, the algorithm chooses a set
of candidate items that may belong to top-rank-k
FCIs, where k is a given threshold. Then, it uses
the DCI-Plus algorithm to generate FCIs based on
these candidate items. When an FCI is generated, it
is directly inserted into a table named tabk . FCIs with
the same support are stored in the same entry. The
number of entries in tabk is below the threshold k. In
the process of mining top-rank-k FCIs, the algorithm
automatically increases minSup to reduce the number
of FCI candidates that do not belong to tabk .

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4. Proposed algorithms

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Fig. 1. TRK-FCI algorithm for mining top-rank-k FCIs.

In the above algorithm, database D is first scanned
to compute the BitTable and determine single items

F1 . These items are sorted in descending order
according to their supports; if two items have the
same support, then they are sorted in increasing lexicographical order. Next, the algorithm creates F2
by inserting each item in F1 into F2 such that the
number of items (which are different in their BitTables) is equal to k. The items in F2 are sorted in
increasing order according to their supports; if two
items have the same support, then they are sorted in
increasing lexicographical order. POST SET is created by computing the closure of each item in F2 . The
procedure DCI CLOSED++ is called with the input
iCLOSED SET = ∅, PRE SET = ∅, POST SET, and
minSup, where minSup is the support of the first item

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4.2. Illustration

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Consider the database in Table 1, which includes
10 transactions and 10 items.
Assume that k = 5, the process of mining top-rankk FCIs is as follows. First, the BitTable and support
of each item are obtained, as shown in Table 2.
After F1 is sorted, we have F1 = {G, F, E, H, D, C,
B, A, J, I}. Next, we choose items from F1 that may
belong to top-rank-k FCIs and store them in F2 , i.e.,
F2 = {A, C, D, H, E, F, G} (after sorting). Because
A has the same BitTable as that of C and E has the
same BitTable as that of F, they are grouped into
two groups as (A, B) and (E, F), respectively. After
grouping, the algorithm computes the closure of each
item. The results are shown in Table 3. From Table 3,
we have POST SET = {ACEFG, CEFG, DEF, H,
EF, F, G}.

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DCI CLOSED++

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Table 1
Transaction database D

Transaction

Items

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2
3
4
5
6
7
8
9
10

A, C, E, F, G, H
D, E, F, H
G, H, I
B, D, E, F, G
D, E, F, G
G, H
A, C, E, F, G
B, E, F, H

D, E, F, G, H
D, E, F, G, H, J

Table 2
Items in D with their BitTable and support

Item

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in F2 if the number of items which are different in
their BitTables in F2 is equal to k; otherwise, minSup
is set to 0.

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Fig. 2. DCI CLOSED++ procedure.

BitTable

Support

520
68
520
355
879
879
763
919
128
1

0.2
0.2
0.2
0.5
0.8
0.8
0.8
0.7
0.1
0.1

A
B
C
D
E
F

G
H
I
J

Table 3
BitTable and Closure of items in D
Item
A
C
D
H
E
F
G

BitTable

Closure

Support

520
520
355
919
879
879
763

ACEFG
CEFG
DEF
H
EF
F
G

0.2
0.2
0.5
0.7
0.8
0.8
0.8

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4.3. Improved algorithm
The TRK-FCI algorithm is based on the DCI-Plus
algorithm. Because DCI-Plus uses bit vectors to represent the tidsets of items, it requires more memory
to store bit vectors and more time to compute the
intersection of bit vectors when the number of transactions in the database is large. To reduce the mining
time and memory usage, we develop an improved
algorithm that uses DBVs instead.
Table 5 is presented to show the process of using
DBVs for mining top-rank-k FCIs. It shows the details
of items, supports, closures, and DBVs of F2 .
Procedure DCI CLOSED++ is the same as that in
TRK-FCI but the operations for BitTable are replied
by operations for DBVs. The final results are the same
as those obtained with TRK-FCI.

5. Experiments

Table 4
Top-rank-k FCIs generated according to
TRK-FCI algorithm
k

key/sup

1
2
3
4
5

0.8
0.7
0.6
0.5
0.4

FCIs

{EF}, {G}
{H}
{EFG},
{DEF}, {HEF}, {HG}
{DEFG}

Table 5
DBVs, closures, and supports of items in F2
A

C
D
H
E
F
G

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The algorithms used in the experiments were

implemented in C# 2012 on a personal computer
with an i5-4200U 1.60-GHz CPU and 4 GB of
RAM running Windows 8.1. The experiments were
tested on three databases downloaded from the UCI
Machine Learning Repository ( />data). Table 6 shows the characteristics of the experimental databases.

Item

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removed and EF is inserted into tabk , and minSup
is set to 0.3 (the key of the last entry in tabk ). The
algorithm will continue to process other FCIs. The
results are shown in Table 4.

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Procedure DCI CLOSED++ is called with the
input PRE SET = ∅, POST SET, CLOSED SET = ∅,
and minSup = supp(A) = 0.2. The first element of
POST SET (ACEFG) is set to I. Because PRE
SET = ∅ and supp(ACEFG) = minSup, ACEFG
is an FCI, and it is put into tabk with its key,
which is its support (0.2), ACEFG is also inserted
into PRE SET. Next, itemset CEFG is processed.
Because supp(CEFG) = minSup and the BitTable
of CEFG is a subset of the BitTable of ACEFG
in PRE SET, CEFG is pruned. When DEF is
processed, because supp(DEF) > minSup and its
BitTable is not a subset of the BitTable of any
itemset in PRE SET, DEF is an FCI, and it is put into
tabk with its key, which is its support (0.5). After
that, procedure DCI CLOSED++ is called with
PRE SET = {ACEFG}, CLOSED SETnew = DEF,
and POST SETnew = {H, G}. EF and F do not
appear in POST SETnew because they belong to
CLOSED SET. Because CLOSED SET =
/ φ, DEF
is joined with H to create a newgen, which is
DEFH. Similarly, DEFH is an FCI, and is
inserted into tabk with its key (0.3). The procedure is called recursively with parameters PRE
SET = {ACEFG}, CLOSED SETnew = DEFH, and
POST SETnew = {G}. Because CLOSED SET =
/ φ
and its generator is DEFHG, and there is no itemset
X in PRE SET such that the BitTable of DEFHG is

a subset of the BitTable of X, and thus DEFGH is
an FCI, and is inserted into tabk with its key (0.2).
Now, POST SET = φ and thus DEF is added into
PRE SET. The process continues by joining DEF
with G to form DEFG. DEFG is also an FCI and it
is inserted into tabk with its key (0.4. The algorithm
then starts with a newgen H. H is an FCI and is
inserted into tabk with its key (0.7). Note that now
the number of entries in tabk is 5 and equal to k. The
algorithm will continue to insert generated FCIs into
tabk . They include HEF (key is 0.5), HEFG (key
is 0.3), and HG (key is 0.5). Consider the process
of inserting FCI EF (whose key is 0.8) into tabk .
Because the key of EF is greater than that of the
last entry (DEFGH) in tabk (key is 0.2), DEFGH is

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DBV

Closure

Support

{0,520}
{0,520}
{0,355}
{0,919}
{0,879}
{0,879}
{0,763}

ACEFG
CEFG
DEF
H
EF
F
G

0.2
0.2
0.5
0.7
0.8
0.8

0.8

Table 6
Characteristics of experimental databases
Database

# of transactions

# of items

Chess
Pumsb
Accidents

3196
49046
340183

76
7117
468

368
369
370
371
372
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382

The efficiency of applying BitTable and DBVs
for mining top-rank-k FCIs was evaluated. The

Fig. 3. Runtimes of TRK-FCI-DBV and TRK-FCI for Accidents
database.

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5.1. Execution time

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experiments were conducted with various values of
threshold k for the Accidents, Chess, and Pumsb
databases. With increasing threshold k, the number of
FCIs increased, increasing the time required to obtain
top-rank-k FCIs.

Figures 3 to 5 show that the time required for
mining top-rank-k FCIs from the three databases
increases with increasing k. TRK-FCI-DBV runs

Fig. 6. Memory usage of TRK-FCI-DBV and TRK-FCI for Chess
database.

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The experimental databases have different features. The Pumsb and Accidents databases have many
transactions (or records), whereas the Chess database
is small (3196 transactions).

Fig. 4. Runtimes of TRK-FCI-DBV and TRK-FCI for Chess
database.

Fig. 7. Memory usage of TRK-FCI-DBV and TRK-FCI for Accidents database.

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Fig. 5. Runtimes of TRK-FCI-DBV and TRK-FCI for Pumsb
database.

Fig. 8. Memory usage of TRK-FCI-DBV and TRK-FCI for Pumsb
database.

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faster than TRK-FCI. For example, consider the
Pumsb database with a threshold k of 200. The mining
time of TRK-FCI is 179.8 s and that of TRK-FCIDBV is 130.7 s. Most of the processing time for both
algorithms is in the itemset expansion stage. TRKFCI-DBV has a lower processing time because it uses
a better data format.

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5.2. Memory usage
Figures 6 to 8 show that the memory usage for
mining top-rank-k FCIs for the three experimental
databases increases with increasing threshold k. The

memory required by TRK-FCI-DBV is significantly
less than that required by TRK-FCI. Consider the
Pumsb database with a threshold k of 120. The memory usage values of the two algorithms are similar;
however, when the threshold k is increased to 200,
the memory used by TRK-FCI is nearly double that
used by TRK-FCI-DBV.

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6. Conclusion and future work

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This paper proposed a method for mining top-rankk FCIs based on DCI-Plus. Two efficient algorithms,
TRK-FCI and TRK-FCI-DBV, were proposed. These
two algorithms differ in the way they represent data
for each itemset, which gives them different mining
times and memory usage values. A strategy is used
to automatically change minSup to prune candidates
in the mining process. The mining time and memory
usage of the two algorithms were analyzed to compare the effectiveness of DBV compared to that of
BitTable.
In the future, we will study how to prune candidates

more efficiently. Moreover, we will try to use other
approaches for mining top-rank-k FCIs. We will also
expand our research to quantitative databases.

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