k-means Improvement by Dynamic Pre-aggregates
Nabil El malki
1,2
, Franck Ravat
1
and Olivier Teste
2
1
Université de Toulouse, UT2, IRIT(CNRS/UMR5505), Toulouse, France
2
Capgemini, 109 Avenue du Général Eisenhower, Toulouse, France
Keywords: k-means, Machine Learning, Data Aggregations.
Abstract: The k-means algorithm is one well-known of clustering algorithms. k-means requires iterative and repetitive
accesses to data up to performing the same calculations several times on the same data. However, intermediate
results that are difficult to predict at the beginning of the k-means process are not recorded to avoid
recalculating some data in subsequent iterations. These repeated calculations can be costly, especially when
it comes to clustering massive data. In this article, we propose to extend the k-means algorithm by introducing
pre-aggregates. These aggregates can then be reused to avoid redundant calculations during successive
iterations. We show the interest of the approach by several experiments. These last ones show that the more
the volume of data is important, the more the pre-aggregations speed up the algorithm.
1 INTRODUCTION
For years until now, data clustering also known as
cluster analysis has been one of the most important
tasks in exploratory data analysis. It is also applied in
a variety of applications, such as web page clustering,
pattern recognition, image segmentation, data
compression and nearest neighbor search (Zhao et al,
2018). Various clustering algorithms have been
available since the early 1950s. The goal of data
clustering is to classify a set of patterns, points or
objects into groups known as clusters. The data that
are in the same group are as similar as possible, in the
same way that the data belonging to different groups
are as dissimilar as possible (Jain et al., 1999).
Clustering algorithms can be divided into two main
groups: hierarchical and partitional (Celebi et al,
2013). Hierarchical algorithms recursively discover
nested groups either in a divisive way or
agglomerative way. On the other hand, partitional
algorithms find all groups simultaneously as a data
partition without hierarchical structures. One of the
most widely used data clustering and partitional
clustering is k-means. It is the subject of our study.
Due its simplicity and its linearity of its time
complexity it remains popular since it was proposed
over four decades (Jain, 2010). In this paper, we
consider the standard k-means; i.e. the Lloyd-Forgy
version (Forgy, 1965).
1.1 k-means
Let X = {x
1
, ..., x
n
} be a set of points in the d-
dimensional space ℝ
d
. The distance between x
i
and x
j
is denoted ||x
i
− x
j
||. Several distances could be
calculated using various formulas (Euclidean,
Manhattan, Canberra, Cosine…). In this study, the
standard Euclidean distance is used as the distance
measure.
Let k be a positive integer specifying the number
of clusters. Then C = {C
1
, ..., C
k
} is a set of non-
overlapping clusters that partition X into k clusters,
and G = {G
1
, ..., G
k
} is a set of centroids, each of
which is corresponding to the arithmetic mean of
points it contains. |C
j
| is the cardinality of C
j
.
If we consider t as the current iteration, we denote
the set of clusters and
the set of centroids
at the previous iteration, whereas
and
are
respectively the set of clusters and the set of centroids
of the current iteration.
In the initialization phase, each k centroid is
assigned a data point from X either randomly or via
an initialization method such as k-means++ (Arthur
and Vassilvitskii, 2007). Then, k-means algorithm
partitions the set of points X into k non-overlapping
clusters such that the sum of the distances between
points and the corresponding cluster centroid is
minimized. To do that, the k-means algorithm repeats
iteratively steps until convergence (i.e.,
≈
).
El malki, N., Ravat, F. and Teste, O.
k-means Improvement by Dynamic Pre-aggregates.
DOI: 10.5220/0007675201330140
In Proceedings of the 21st International Conference on Enterprise Information Systems (ICEIS 2019), pages 133-140
ISBN: 978-989-758-372-8
Copyright
c
2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
133
Algorithm 1: Classic k-means process.
In: X, k
Out: C
1: t ← 0 -- Initialization
2: for j [1…k] do
3:
← random(X);
← Ø
4: end do
5: repeat
6: foreach i [1…n] do -- assignment
7: j’ ←
;
←
8: end do
9: t ← t+1 --update
10: for j [1…k] do
11:
←
12: end do
13: until
14: return {
}
The algorithm of k-means is based on an iterative
and repetitive processing. The iterative aspect is
based on the fact that the different steps of the
algorithm must be performed several times before
converging towards a result; i.e., towards a stable
partitioning. For instance, the assignment phase (see
Algorithm 1) that calculates for each point x
i
, the
distance separating it from the k centroids and keeps
the index of the closest centroid j to x
i
via the argmin
function, is repeated until convergence; i.e. until there
is no longer any point that changes class. The
repetitive aspect lies in the fact that the same
calculation is potentially performed several times on
the same portions of data. For instance, in the phase
of updating the centroids (Algorithm 1), the centroid
of a class may be recalculated several times even if
this class remains the same in the following iterations.
If the points belong to the data space of dimension d
then the centroid
corresponds to the arithmetic
average of the data values of the class C
j
for each
dimension.
Even if the k-means algorithm is based on an
iterative and repetitive processing, the results of one
iteration are not stored for the next iteration. Each
calculation is done on original dataset. This limit can
cause k-means execution times to slow down.
Moreover, the impact of this limit is all the more
important if we consider two following notions:
The notion of dimension is very important in the
case of clustering algorithms. The dimension of
a point is the number of the attributes set that
characterizes it. The attributes are all numeric in
the case of using k-means. A point may have
from one to millions of dimensions.
The notion of massive data corresponds to the
fact that large volumes of data can be used for
each attribute, thus leading to clustering on a
large data set, large but also of varying density.
1.2 Contribution
In order to allow k-means to offer better performance,
especially on large volumes of data and numerous
dimensions, we provide an extension of the k-means
algorithm based on the idea of pre-calculations (pre-
aggregations). More precisely, we propose an
extension to optimize the repetitive access to data
performed during the different iterations, based on a
strategy whose principles are as follows:
Pre-calculate and store the different calculations
performed during the successive iterations.
Reuse the stored pre-calculations to accelerate
future iterations (unlike the traditional algorithm
that recalculates each iteration from the initial
data).
We show through experiments that our approach
is effective when it comes to exploratory analyses on
massive data with numerous dimensions. In section 2,
we discuss the state of the art. In sections 3 and 4, we
present our model and our experimentations.
2 RELATED WORK
k-means Approaches: Using the standard version of
k-means requires an execution time proportional to
the product of the number of classes and the number
of points per iteration. This total execution time is
relatively expensive in terms of calculation,
especially for large data sets (Alsabti et al., 1997). As
a result, the k-means algorithm cannot satisfy the need
for fast response time for some applications (Hung et
al., 2005). Several extensions of the standard version
of the k-means have been proposed to accelerate
execution times:
Acceleration by parallelizing the algorithm,
particularly based on MapReduce or MPI
paradigms (Zhang et al., 2013; Zhao et al.,
2009) , which are programming models
designed to process large volumes of data in a
parallelized and distributed way;
Acceleration by reducing the number of
calculations to be performed. The algorithms of
(Elkan, 2003) and (Hamerly, 2010) are based on
the property of triangular inequality to avoid
ICEIS 2019 - 21st International Conference on Enterprise Information Systems
134
calculating at each iteration the distance
between a given point and all other centroids of
the classes.
Acceleration by restructuring the dataset. In
(Hung et al., 2005) the authors propose an
algorithm accelerating k-means by splitting the
dataset into equal non-empty unit blocks. Then
k-means is unrolled on the centroids of these
blocks and not on the contained points.
These extensions accelerate the execution time of
standard k-means but do not use a pre-calculation
approach of intermediate results. These results, which
are referred to as dynamic, are likely to make these
calculations more efficient.
Pre-aggregation Approaches: Pre-calculation of
aggregates has already been used in several areas,
including multidimensional data warehouses (OLAP)
and statistical analysis.
Pre-calculations are commonly used in
multidimensional data warehouses to effectively
support OLAP analyses. Indeed, the
multidimensional structure allows anticipating
analytical calculations, which are materialized,
constituting data cubes (Gray et al., 1996). In
(Deshpande et al., 1998) the authors score data in
uniform "chunks" blocks. Each of these chunks is
reused individually in subsequent calculations.
Calculation operations are decomposed in order to
focus on the chunks in memory. The ability to
memory is not sufficient in the context of a single
machine, the computation operations can also fetch
the complementary chunks from disk.
Statistical data mining allows knowledge
extraction by repeating the same operations several
times, most often on detailed data (such as calculating
averages, correlations or measures of similarities
between data). Works improving exploration by
integrating the pre-calculation of aggregates has been
proposed by (Wasay et al., 2017) where the authors
describe a system called Data canopy. It is based on
a memory cache for exploratory statistical analysis.
This system pre-aggregates statistical calculations to
minimize repetitive access to original data. To do this,
it breaks down the calculations into elementary
operations and the original data into unit blocks. The
latter are stored in a binary tree that recursively
aggregates the calculations from the leaves to the
root. The construction must respect a precise
scheduling; this requirement requires Data canopy to
support only calculations that respect this scheduling.
All the approaches mentioned above have been
developed on static data. In the context of the
clustering we process algorithms that operate on very
dynamic data. The dynamic aspect lies in the fact that
it is not possible to know the calculations to be pre-
aggregated. Our approach is therefore based on a
principle of "hot storage" as the calculations.
3 OUR EXTENDED k-means
ALGORITHM
Algorithm 2: Extended k-means(EKM) process.
In: X, k
Out:
1: t ← 0 -- Initialization
2: for j [1…k] do
3:
← random(X);
← Ø
4: end do
5: repeat
6: foreach i [1…n] do -- assignment
7: j’ ←
;
←
8: end do
9: t ← t+1 --update
10: for j [1…k] do
10:
← sort(
); key ← concatenate(
)
11: If
then --aggregation
12:
←
13: end if
←
14: end do
15: until
--convergence
16: return {
}
The algorithm below aims to accelerate the standard
k-means algorithm using a pre-aggregation approach.
Our approach is mainly focused on the part of
updating the calculated centroids.
We note that a class is associated with a key (also
called class index) that allows it to be uniquely
identified. This key will be used to identify the
centroid of a given class along the successive
iterations. Knowing that each data has a unique
numerical index, then the key (or class index) is
formed by the concatenation of the indexes of the data
points contained in this class.
Let I = {I
1
, ..., I
k
} be a set of ordered set of keys.
From each cluster C
j
, we determine its corresponding
key denoted I
j
from each index of data points within
the cluster.
Let M = {m
1
, ..., m
A
} be a set of aggregates (i.e.
centroids), which are calculated from each cluster C
j
.
k-means Improvement by Dynamic Pre-aggregates
135
These aggregates are identified by keys that are
formed as iterations from the classes built.
In the assignment part, we identify the index j' of
the closest centroid to point x
i
. Then the index i is
added to the set
. In the update step, the function
sort makes the set of indexes
sorted in ascending
order. Subsequently, a key is created in which the
value returned by the concatenate function is
assigned. The latter concatenates the indices of the
data contained in
. The symbol "-" is placed
between each index pair, e.g. 4-7-9. This key allows
identifying the centroid of each class, if it exists in M,
otherwise the average is calculated and stored in M.
4 EXPERIMENTATIONS
The purpose of the following experiments is to
evaluate our approach in comparison with the
standard approach (Forgy, 1965). For these
experiments, we use a computing platform composed
of a cluster of 24 nodes; each of them hosts 8
processors. The usable memory of a processor
reaches a maximum of 7.5 GB.
4.1 Experimental Framework
Data Set: We considered two types of synthetic
datasets.
spherical data (SD): several separate groups of
homogeneous data;
homogeneous data (HD): a single compact
group of data.
Each data represents decimal values within the
range [-10; 10]. The spherical data have been
generated according to an isotropic Gaussian
distribution; the scikit-learn python library offers
functionalities that allow us to generate well
separated data sets. The library also allows us to
generate homogeneous data (data compacted into a
single group) using the Gaussian mixture.
Experimental Protocol: The datasets are generated
using the following parameters:
the number of classes k [4...20] incremented
with a step of 4,
the dimension d [1 ;2000] with a step of 100
or d [2000; 97000] with a step of 5000,
the number of data points n [2000; 202000],
for each of the two types of distributions (SD
and HD).
2798 synthetic datasets were generated including
1987 spherical datasets and 811 homogeneous
datasets reaching up to 62 GB.
An experiment consists first of all in generating
homogeneous or spherical datasets, then initializing
the centroids with k-means++ (Arthur and
Vassilvitskii, 2007) and finally applying the both
unsupervised classification standard k-means and the
extended k-means. The experiment is repeated 10
times with the same parameters (number of classes k,
number of data points n, type of data distribution i.e.
HD or SD). The average of the execution times of
each of the both methods is kept.
Note that the experiments produce always the
same partitioning of the data for the two k-means
(standard and extended versions) since both versions
use the same set of initialization centroids.
4.2 Comparisons between Extended
and Standard k-means Applied to
Spherical and Homogeneous
Datasets
We evaluate, for spherical data and homogeneous data,
the difference in execution time between the extended
k-means version and the standard version. This
difference will be called later in this document DETES.
DETES is calculated as the difference between the time
taken by the extended k-means and the standard k-
means; it is positive when k-means extended is
favourable compared to k-means standard, otherwise
negative in the unfavourable case.
The Table 1. summarizes all the experiments
performed on spherical and homogeneous data. The
extended version shows better results in terms of
execution time: out of 1987 experiments on spherical
data (d [1; 97000]), the extended version shows 1353
favourable cases (68%) where the DETES can reach
1758 seconds. We can also see that the performance of
our extended version exceeds the standard version by
more than 100 seconds in 25% of cases. The number
of favourable cases is even more important when it
comes to spherical data with dimensions d 2000,
since a favourable case rate of 97% is reached and with
55% having a DETES greater than 100 seconds.
Table 1: Favourable cases (%) to extended version k-means
applied to SD and HD according to k (number of classes).
k
SD ; d>=2000
HD; d>=2000
4
100%
67%
8
99%
71%
12
96%
68%
16
97%
66%
20
95%
55%
ICEIS 2019 - 21st International Conference on Enterprise Information Systems
136
In Figures 1 to 4, we can observe the distribution
of favourable and unfavourable cases. The horizontal
(red) line separates favourable cases (green crosses)
from unfavourable cases (orange circles) to extended
k-means.
Figure 1 shows the percent of execution time
gained by the extended version compared to the
standard k-means: from 3.15 GB of data, the extended
k-means is almost always favourable up to 30 %
faster. In Figure 2, we can note that the extended
version of k-means is clearly favourable from data of
dimension greater than 2000 (green vertical line)
where DETES reaches up to 1758 seconds (97% of
favourable cases) whereas the average execution time
of k-means standard is 3473 seconds. Unfavourable
cases are mainly concentrated in the subspace of
dimensions less than 2000.
In Figures 3 and 4, we evaluate the behaviour of
the both versions with homogeneous data. In Figure
3, the percentage of execution time gained by the
extended version compared to standard k-means is
approximately from 0 to 21% when the homogeneous
data dimension is at least equivalent to 42000. As
shown in Figure 4, the extended version is
advantageous from 42000 dimensions. From this
dimension, the average execution time of extended k-
means is 1755 seconds. The gain can reach up to
nearly 2156 seconds.
In Figures 3 and 4, we evaluate the behaviour of
the both versions with homogeneous data. In Figure
3, the percentage of execution time gained by the
extended version compared to standard k-means is
approximately from 0 to 21% when the homogeneous
data dimension is at least equivalent to 42000. As
shown in Figure 4, the extended version is
advantageous from 42000 dimensions. From this
dimension, the average execution time of extended k-
means is 1755 seconds. The gain can reach up to
nearly 2156 seconds.
In the performed experiments, the influence of the
parameter k (number of classes) was evaluated. We
consider in Table 2 that the HD and SD datasets have
dimension larger than 2000. The extended version
unrolled on the spherical data gains with a very good
rate clearly higher than 95% for classes ranging from
4 to 20. We can see that for k = 4 it gains 100%.
Concerning homogeneous data, the favourable case
rate is correct and is higher than 55% for classes 4 to
20. There is no decline or increase in favourable case
rates as a function of k for either type of data
distribution. The number of classes k does not alone
influence the behaviour of the extended version of k-
means. But favourable case rates in SD is largely
favourable comparing to a distribution of HD data.
Table 2: Results of the executions of the both k-means versions on SD and HD.
Data
Number of
experiments
cases favourable
to EKM
cases favourable to
EKM (DETES100s)
Max DETES
(sec)
SD (d [1;97000])
1987
1353 (68%)
497 (25%)
1758
SD (d<=2000)
1155
542 (46%)
39 (2%)
532
SD (d>=2000)
832
811 (97%)
458 (55%)
1758
HD (d [1;97000])
937
462 (49%)
110 (11%)
2156
Figure 2: Evolution of the time differences (DETES) of
the both versions of k-means according to the dimension.
Figure 1: Time gained (%) by the EKM compared to the
standard version as a function of the SD volume (GB).
k-means Improvement by Dynamic Pre-aggregates
137
Table 3: Percentage of centroids instances reused.
Figure 3: The time gained (%) by the EKM compared to the
standard version as a function of the SD volume (GB).
Figure 4: Time gained (%) by the EKM compared to the
standard version as a function of HD dimension.
Figure 5: DETES as a function of HD dimension.
4.3 The Proportions of the Reuse of the
Centroids According to the
Distribution HD and SD
The extended version does not obtain the same
favourable case rate in both types of data
distributions. It is more favourable on spherical data
than on homogeneous data.
We consider two definitions of reuse of centroids.
The first defines reuse as the number of reused
centroids in a k-means execution. The second is the
number of instances of centroids reused, i.e. if a
centroid is reused p times it is said that there are p
times reuses of the instance of this centroid.
Table 3 and Figures 6-7 show results of
experiment on reuse of centroids according to the data
distribution (HD or SD), the number of classes k and
the number of data points n (data size). Table 3 and
figure 7 discuss the reuse of centroids instances while
figure 6 discuss reuse of centroids without
considering their instances. The experiment was
performed under the same conditions as the previous
one, but the dimension d is defined from 2 to 1802
with a step of 200, the size n [20000 ;100000] with
a step of 20000 and k [4 ;16] with a step of 4.
In Figure 6, the number of centroids used SD is on
average 6 times smaller compared to those in the case
of HD. Similarly, the number of centroids reused
among the centroids used remains marginal in SD and
HD. It is practically 4% for each of the both
distributions. Moreover, the more the number of
classes k increases, the more the number of centroids
used and reused also increases. This increase is more
pronounced in HD.
In Figure 7, in each of the different k-means
executions, the centroid whose instances were most
reused was identified. The number of reuses of this
centroid corresponds to "max reuse". Thus, "avg
reuse" is the average of the number of the instances
of all the centroids used in a k-means execution. "Max
reuse" is about 2.5 times higher than the average reuse
of the instances of the centroids (avg reuse) in the case
of SD. It increases from two to nine times when k
k = 4
k = 8
k = 12
k = 16
Dataset
size
SD
HD
SD
HD
SD
HD
SD
HD
20000
33%
5%
41%
15%
45%
19%
51%
27%
40000
32%
3%
44%
10%
50%
12%
55%
16%
60000
43%
3%
43%
7%
38%
12%
50%
19%
80000
40%
4%
53%
6%
50%
11%
53%
14%
100000
40%
3%
32%
4%
40%
7%
45%
12%
AVG
37.6%
3.6%
42.6%
8.4%
44.6%
12.2%
50.8%
17.6%
ICEIS 2019 - 21st International Conference on Enterprise Information Systems
138
increases in the case of HD. Moreover, when k
increases "avg reuse" remains stable for both data
distributions, it is approximately 2.5 and 18
respectively for DH and SD. This shows that the
larger k is, the more the reuse of the centroids
instances of a small set of centroids is increased but
the stability of the average reuse of centroids
instances is ensured when k increases.
Table 3 completes Figure 7. For each data
distribution the proportion (%) of reused centroids
instances is a function of the size of the data n and the
number of classes k. This proportion is the average of
the reused centroids for each dimension. The
proportions of centroids reused in the extended
version are superior in the case where it is applied on
spherical data than in homogeneous data. For k = 4,
the proportion is on average 37.6% for SD against
3.6% for HD that to say a difference of 30%. This
difference is roughly similar for the other values of k,
but the proportions in both distributions of data
increase as k increases. In short, even if a subset of
centroids is small compared to all of the centroids
used through iterations in k-means, their instances are
reused several times in order to exceed 50% of the
reuses of centroids instances in the case of SD and
27% for HD. Also, experiments have shown that if a
centroid is reused, it is reused successively over
iterations. In other words, if the centroid is not reused
in the iteration following the one where it was reused,
it will no longer appear in all subsequent iterations.
4.4 Extended Gini Index
We found that our approach works better in spherical
data than in homogeneous data. We used the gini
index (Gini, 1921) to characterize a spherical dataset
and predict in advance if our approach could be used.
The gini index is a static measure used to measure
the level of dispersion of a dataset.
(1)
(2)
EG = VAR(G*)
(3)
Equation (1) defines G with S as a series of
ordered positive numerical values. It is defined for
univariate datasets. In our case, the data are
multivariate. Let X be a dataset with d dimensions.
The gini index is calculated for each dimension
resulting in the list G* (equation 2). Finally, the
variance of G* is calculated to obtain a scalar
considered as the dispersion level of X (equation 3).
In practice, the cste constant of (equation 2) has been
set to 10, which allows for a good separation of the
two distributions.
In Figure 8, spherical datasets (green crosses) are
distinguished from homogeneous data sets (orange
circles). Each data set has a size between 20 000 and
100 000. There is a clear separation (horizontal line)
between the both types of data according to the gini
index EG. The limit is about EG =300.
Figure 6: Figures on reuse of centroids.
Figure 7: Figures on reuse of centroids instances.
Figure 8: Gini index EG as a function of dimension.
5 DISCUSSION
The extended k-means algorithm is advantageous on
homogeneous data with numerous dimensions (if the
101
214
348
480
4
9
16
21
654
1457
1740
2074
11
36
73
123
0
1000
2000
3000
4 8 12 16
number of resuse
k
centroids used (SD) Centroids reused (SD)
centroids used (HD) Centroids reused (HD)
34
44
46
40
11
18
19 19
4
9
29
26
2 2
3 3
0
10
20
30
40
50
4 8 12 16
number of reuse
k
Max reuse(SD) Avg reuse (SD)
Max reuse (HD) Avg reuse(HD)
k-means Improvement by Dynamic Pre-aggregates
139
dimension is at least equal to 42000) while it is
advantageous on spherical data only from dimension
2000. In addition, the execution of both versions of k-
means on spherical data is much faster than on
homogeneous data. The k-means processes applied to
homogeneous data calculate a significantly higher
number of different centroids than those applied to
spherical data if we consider that these processes
adopt the same parameters (number of classes k,
dimension d and size of data n). This difference
therefore generates more iterations on homogeneous
data than on spherical data and convergence is also
slower. Among these different centroids, very few are
used in k-means applied to homogeneous data. On the
contrary, they are much reused in the case of k-means
unrolled on spherical data, which explains why the
extended k-means is much more favourable on these
data than the homogeneous data.
If EKM is applied several times on the same
dataset with different initializations of the centroids.
It can be seen that the proportions of the reused
centroids are not the same from one EKM execution
to another. This observation is valid for both data
distributions. So, the initialization of the centroids
coupled with the choice of the distribution have an
influence on our approach.
6 CONCLUSION
In this paper, we provided an approach to accelerate
the process of the unsupervised data learning
algorithm, called k-means. This approach is based on
an algorithm that pre-calculates and stores
intermediate results, called dynamic pre-aggregates,
to be reused in subsequent iterations. Our
experiments compare our extended k-means version
with the standard version using two types of data
(spherical and homogeneous data). 2798 synthetic
datasets that have been generated reaching up to
62GB. We demonstrate that our approach is
advantageous for partitioning large datasets from
dimensions 2000 and 42000 respectively for spherical
data and homogeneous data.
Several perspectives are planned. We are ongoing
to experiment our extended version of k-means with
even more massive data to better evaluate the cost of
calculating pre-aggregates. In addition, it is proposed
to study situations where it might be more effective
to start from an archived centroid corresponding to a
class almost similar to the new class encountered
rather than recalculate it entirely on the pretext that
the class is not exactly identical.
REFERENCES
Alsabti, K., Ranka, S. and Singh, V. (1997) ‘An efficient k-
means clustering algorithm’, EECS
Arthur, D. and Vassilvitskii, S. (2007) ‘k-means++: the
advantages of careful seeding’, SODA, pp. 1027–1025
Celebi, M. E., Kingravi, H. A. and Vela, P. A. (2013) ‘A
comparative study of efficient initialization methods for
the k-means clustering algorithm’, ESA, pp. 200–210.
Deshpande, P. M. and al. (1998) ‘Caching
multidimensional queries using chunks’, in
Proceedings of the 1998 ACM SIGMOD, pp. 259–270.
Elkan, C. (2003) ‘Using the Triangle Inequality to
Accelerate k-Means’, Proc.Twent. ICML, pp. 147–153.
.
Forgy, E. W. (1965) ‘Cluster analysis of multivariate data:
efficiency versus interpretability of classifications’,
Biometrics.
Gini, C. (1921) ‘Measurement of Inequality of Incomes’,
The Economic Journal. .
Gray, J. and al. (1996) ‘Data cube: A relational aggregation
operator generalizing group-by, cross-tab, and sub-
totals’, DMKD, 1(1), pp. 29–53.
Hamerly, G. (2010) ‘Making k -means even faster’, SDM
2010, pp. 130–140.
Hung, M.-C., Wu, J. and Chang, J.-H. (2005) ‘An Efficient
k-Means Clustering Algorithm Using Simple
Partitioning’, JISE 21, 1177, pp. 1157–1177.
Jain, A. K. (2010) ‘Data clustering: 50 years beyond K-
means’, PRL. North-Holland, 31(8), pp. 651–666.
Jain, A. K., Murty, M. N. and Flynn, P. J. (1999) ‘Data
clustering: a review’, CSUR, 31(3), pp. 264–323.
Wasay, A. and al. (2017) ‘Data Canopy’, Proceedings of
the 2017 ACM SIGMOD ’17, pp. 557–572.
Zhang, J. and al. (2013) ‘A Parallel Clustering Algorithm
with MPI – MKmeans’, JCP, 8(1), pp. 10–17.
Zhao, Weizhong; Ma, Huifang; He, Q. (2009) ‘Parallel K -
Means Clustering Based on MapReduce’, CLOUD
Zhao, W. L., Deng, C. H. and Ngo, C. W. (2018) ‘k-means:
A revisit’, Neurocomputing, 291, pp. 195–206.
ICEIS 2019 - 21st International Conference on Enterprise Information Systems
140
