# On an identity of Delange and its application to Cohen-Ramanujan expansions

Vinod Sivadasan  Department of Mathematics, University College,
Thiruvananthapuram (Research Centre of the University of Kerala), Kerala -
695034, India  Department of Mathematics, College of Engineering Trivandrum,
Thiruvananthapuram, Kerala - 695034, India  Department of Collegiate
Education, Government of Kerala, India<EMAIL_ADDRESS>and  K Vishnu
Namboothiri  Department of Mathematics, Baby John Memorial Government College,
Chavara, Sankaramangalam, Kollam, Kerala - 691583, India  Department of
Collegiate Education, Government of Kerala, India<EMAIL_ADDRESS>
###### Abstract.

Srinivasa Ramanujan provided Fourier series expansions of certain arithmetical
functions in terms of the exponential sum defined by
$c_{q}(n)=\sum\limits_{\begin{subarray}{c}{m=1}\\\
(m,q)=1\end{subarray}}^{q}e^{\frac{2\pi imn}{q}}$. Later, H. Delange derived
the bound $\sum\limits_{q|k}|c_{q}(n)|\leq n\,2^{\omega(k)}$ and gave a
sufficient condition for such expansions to exist. A. Grytczuk gave an exact
value for this bound, and derived a converse implication of the absolute
convergence stated by H. Delange. We here show that these results have natural
generalizations in terms of the Cohen-Ramanujan sum $c_{q}^{(s)}(n)$ defined
by E. Cohen in [_Duke Mathematical Journal, 16(85-90):2, 1949_]. We derive a
bound as well as exact value for $\sum\limits_{q|k}|c_{q}^{(s)}(n)|$ and
provide a sufficient condition for the Cohen-Ramanujan expansions to exist.

###### Key words and phrases:

Ramanujan sum, Ramanujan expansions, Cohen-Ramanujan sum, Cohen-Ramanujan
expansions, Jordan totient function, Möbius inversion formula

###### 2010 Mathematics Subject Classification:

11A25, 11L03

##  1\. Introduction

In [11], Srinivasa Ramanujan gave Fourier series like expansions for certain
arithmetical functions. These pointwise expansions were of the form
$f(n)=\sum\limits_{q=1}^{\infty}a_{q}c_{q}(n)$ where $a_{q}$ are complex
numbers and $c_{q}(n)$ denotes the _Ramanujan sum_ defined as

$\displaystyle c_{q}(n):=\sum\limits_{\begin{subarray}{c}{m=1}\\\
(m,q)=1\end{subarray}}^{q}e^{\frac{2\pi imn}{q}}.$

Later, A. Wintner [12] proved that if

(1) $\displaystyle\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k}<\infty,$

where

(2) $\displaystyle
f^{\prime}(k)=\sum\limits_{d|k}\mu(d)f\left(\frac{k}{d}\right)$

and $\mu$ is the Möbius function, then with
$a_{q}=\sum\limits_{m=1}^{\infty}\frac{f(mq)}{mq}$,
$\sum\limits_{q=1}^{\infty}a_{q}c_{q}(n)$ converges absolutely to $f(n)$.

H. Delange improved the inequality (1) in [4]. He proved that the absolute
convergence of $\sum\limits_{q=1}^{\infty}a_{q}c_{q}(n)$ holds with the weaker
condition
$\sum\limits_{k=1}^{\infty}2^{\omega(k)}\frac{|f^{\prime}(k)|}{k}<\infty$
where $\omega(k)$ is the number of distinct prime divisors of $k$. The most
important part of his proof to establish this was the inequality
$\sum\limits_{q|k}|c_{q}(n)|\leq n\,2^{\omega(k)}$. Later, A. Grytczuk gave an
exact equality for the sum $\sum\limits_{q|k}|c_{q}(n)|$ in [7]. He used this
equality to prove that the absolute convergence implication given by H.
Delange has an easy converse ([5, Theorem 1]).

Ramanujan sum was generalized by many later. E. Cohen gave a generalization of
the Ramanujan sum in [2]. He defined the sum

(3) $\displaystyle c_{q}^{s}(n):=\sum\limits_{\begin{subarray}{c}{h=1}\\\
(h,q^{s})_{s}=1\end{subarray}}^{q^{s}}e^{\frac{2\pi inh}{q^{s}}},$

where $(a,b)_{s}$ is the generalized gcd of $a$ and $b$. A. Chandran and K. V.
Namboothiri derived some conditions for the existence of Ramanujan series like
expansions for certain arithmetical functions in terms of the generalized
Ramanujan sum (3) (henceforth called as the _Cohen-Ramanujan sum_) in [1]. The
method of arguments they used were those employed by L. Lucht in [9] and [10].
We will call the expansions in terms of the Cohen-Ramanujan sums by the name
_Cohen-Ramanujan expansions_.

The aim of this paper is to show that the bounds and other results mentioned
above proved by H. Delange and A. Grytczuk have natural generalizations
involving Cohen-Ramanujan sums. Hence we will give a bound and an exact value
for the sum $\sum\limits_{q|k}|c_{q}^{(s)}(n)|$ and use them to give a
sufficient condition for the Cohen-Ramanujan expansions to exist.

##  2\. Notations and basic results

By $(a,b)_{s}$, we mean the _generalized GCD_ of $a$ and $b$ defined to be the
largest $d^{s}\in\mathbb{N}$ ($d\in\mathbb{N}$) such that $d^{s}|a$ and
$d^{s}|b$. Hence $(a,b)_{1}$ is the usual GCD $(a,b)$. By $J_{s}$ we mean the
_Jordan totient function_ defined as
$J_{s}(n):=n^{s}\prod\limits_{\begin{subarray}{c}{p\mid
n}\end{subarray}}(1-\frac{1}{p^{s}})$. Note that $J_{1}(n)=\varphi(n)$ where
$\varphi$ is the Euler totient function.

_The Cohen-Ramanujan sum_ (or _generalized Ramanujan sum_) given by E. Cohen
in [2] is the sum given in equation (3). By ([2, Theorem 1]), $c_{q}^{(s)}(n)$
is multiplicative in $q$. When $q$ is a prime power, the values of the Cohen-
Ramanujan sum are as folows ([2, Theorem 3]).

(4) $\displaystyle
c_{p^{j}}^{(s)}(n)=\begin{cases}p^{sj}-p^{s(j-1)}&\quad\text{if }p^{sj}|n\\\
-p^{s(j-1)}&\quad\text{if }p^{s(j-1)}||n\\\ 0&\quad\text{if }p^{s(j-1)}\nmid
n.\end{cases}$

Cohen-Ramanujan sum satisfies the following relation with the Möbius function
$\mu$ ([2, Theorem 3]):

(5) $\displaystyle c_{r}^{s}(n)=\sum\limits_{\begin{subarray}{c}d|r\\\
d^{s}|n\end{subarray}}\mu(r/d)d^{s}.$

Also, with the Jordan totient function, it holds the relation ([3, Theorem 2])

(6) $\displaystyle c_{r}^{(s)}(n)=\frac{J_{s}(n)\mu(m)}{J_{s}(m)}\quad\text{
where }m=\frac{n}{(r,n)}.$

Sometimes we club the conditions $d|r$ and $d^{s}|n$ into the single condition
$d^{s}|(n,r^{s})_{s}$.

By the notation $d^{m}||n$ we mean that $d^{m}|n$ and $d^{m+1}\nmid n$. For a
prime $p$, the exponent of $p$ in the prime factorization of $k$ will be
denoted by $e_{p}(k)$. Hence $p^{e_{p}(k)}||k$. Analogously, we write
$a:=e_{p}^{(s)}(n)$ if $p^{as}|n$ and $p^{(a+1)s}\nmid n$. In this case, we
also use the notation $p^{as}||_{s}n$. We use the Möbius inversion formula for
the Dirichlet product of two arithmetical functions which states that

(7) $\displaystyle
g(k)=(\mu*f)(k):=\sum\limits_{d|k}\mu(d)f\left(\frac{k}{d}\right)\Longleftrightarrow
f(k)=(g*u)(k)=\sum\limits_{d|k}g(k)$

where $u\equiv 1$ is the unit function defined by $u(n)=1$ for all
$n\in\mathbb{N}$ .

##  3\. Main Results and proofs

We start this section giving a generalization of the bound
$\sum\limits_{q|k}|c_{q}(n)|\leq n\,2^{\omega(k)}$ given by H. Delange in [4].

######  Theorem 3.1.

The bound $\sum\limits_{q|k}|c_{q}^{(s)}(n)|\leq n.2^{\omega(k)}$ holds for
all positive integers $k,n$, and $s$.

###### Proof.

Since $c_{q}^{(s)}(n)$ is multiplicative in $q$, $|c_{q}^{(s)}(n)|$ is also
multiplicative in $q$. Then
$h_{n}(k)=(c_{\bullet}^{(s)}(n)*u)(k)=\sum\limits_{q|k}|c_{q}^{(s)}(n)|u(k/q)=\sum\limits_{q|k}|c_{q}^{(s)}(n)|$
is multiplicative in $k$ it being the Dirichlet product two multiplicative
functions. Hence we may write
$h_{n}(k)=h_{n}(\prod\limits_{p|k}p^{e_{p}(k)})=\prod\limits_{p|k}h_{n}(p^{e_{p}(k)})$.
Now for any positive integer $r$, we have
$h_{n}(p^{r})=\sum\limits_{q|p^{r}}|c_{q}^{(s)}(n)|=\sum\limits_{j=0}^{r}|c_{p^{j}}^{(s)}(n)|$.
If $p^{s}\nmid n$, then by property 4, $c_{p}^{(s)}(n)=-1$ and
$c_{p^{j}}^{(s)}(n)=0$ for any $j>1$. By identity 5, $c_{1}^{(s)}(n)=1$.
Therefore, if $p^{s}\nmid n$, then
$h_{n}(p^{r})=|c_{1}^{(s)}(n)|+|c_{p}^{(s)}(n)|=2$.

Now suppose that $p^{s}|n$. If $p^{rs}|n$, that is, $r\leq e_{p}^{(s)}(n)$,
then

$\displaystyle h_{n}(p^{r})=$
$\displaystyle|c_{1}^{(s)}(n)|+|c_{p}^{(s)}(n)|+\ldots+|c_{p^{r}}^{(s)}(n)|$
$\displaystyle=$ $\displaystyle
1+|p^{s}-1|+|p^{2s}-p^{s}|+\ldots+|p^{rs}-p^{(r-1)s)}|$ $\displaystyle=$
$\displaystyle p^{rs}$ $\displaystyle\leq$ $\displaystyle
2p^{e_{p}^{(s)}(n)s}.$

On the other hand, if $e_{p}^{(s)}(n)=r-m$ for some positive integer $m$, then

$\displaystyle h_{n}(p^{r})=$
$\displaystyle|c_{1}^{(s)}(n)|+|c_{p}^{(s)}(n)|+\ldots+|c_{p^{r-m}}^{(s)}(n)|+|c_{p^{r-m+1}}^{(s)}(n)|+\ldots+|c_{p^{r}}^{(s)}(n)|$
$\displaystyle=$ $\displaystyle
1+|p^{s}-1|+\ldots+|p^{(r-m)s}-p^{(r-m+1)s}|+|-p^{(r-m)s}|+0+0+\ldots$
$\displaystyle=$ $\displaystyle 2p^{(r-m)s}$ $\displaystyle=$ $\displaystyle
2p^{e_{p}^{(s)}(n)s}.$

Therefore $h_{n}(p^{r})=2$ if $p^{s}\nmid n$ and $h_{n}(p^{r})\leq
2p^{e_{p}^{(s)}(n)s}$ if $p^{s}|n$. Hence for distinct primes $p$, we get

$\displaystyle h_{n}(k)$
$\displaystyle=\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}|n\end{subarray}}h_{n}(p^{e_{p}(k)})\times\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}\nmid n\end{subarray}}h_{n}(p^{e_{p}(k)})$
$\displaystyle\leq\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}|n\end{subarray}}2p^{e_{p}^{(s)}(n)s}\times\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}\nmid n\end{subarray}}2$
$\displaystyle\leq\prod\limits_{\begin{subarray}{c}p^{s}|n\end{subarray}}p^{e_{p}^{(s)}(n)s}\times\prod\limits_{\begin{subarray}{c}p|k\end{subarray}}2$
$\displaystyle\leq n\,2^{\omega(k)}$

∎

######  Remark 3.2.

For equality to hold in the above, it should be true that $p|k\Rightarrow
p^{s}|n$ and $p^{r}||k\Rightarrow r>e_{p}^{(s)}(n)$. In short, the bound is an
equality when $n=m^{s}$ and $k$ is a multiple of $m\prod\limits_{p|m}p$. So
the bound in the theorem is the best possible.

Let $f^{\prime}$ be defined by equation 2. We now derive a sufficient
condition for the Cohen-Ramanujan series expansions to exist for $f$.

######  Theorem 3.3.

Let $a_{q}=\sum\limits_{m=1}^{\infty}\frac{f^{\prime}(mq)}{(mq)^{s}}$. If
$\sum\limits_{k=1}^{\infty}2^{\omega(k)}\frac{|f^{\prime}(k)|}{k^{s}}<\infty$,
then $\sum\limits_{q=1}^{\infty}a_{q}c_{q}^{(s)}(n^{s})$ converges absolutely
to $f(n)$.

###### Proof.

We will first prove the absolute convergence of
$\sum\limits_{q=1}^{\infty}a_{q}c_{q}^{(s)}(n^{s})$ assuming the bound given.
We have

$\displaystyle|\sum\limits_{q=1}^{\infty}a_{q}c_{q}^{(s)}(n^{s})|$
$\displaystyle=\sum\limits_{m,q\geq
1}^{\infty}\frac{|f^{\prime}(mq)|}{(mq)^{s}}|c_{q}^{(s)}(n^{s})|$
$\displaystyle=\sum\limits_{k=1}^{\infty}\sum\limits_{mq=k}\frac{|f^{\prime}(k)|}{k^{s}}|c_{q}^{(s)}(n^{s})|$
$\displaystyle=\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k^{s}}\sum\limits_{q|k}|c_{q}^{(s)}(n^{s})|$
$\displaystyle\leq
n^{s}\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k^{s}}2^{\omega(k)}$

which is bounded by the assumption. Now we will show that the series converges
to $f(n)$.

$\displaystyle\sum\limits_{q=1}^{\infty}a_{q}c_{q}^{(s)}(n^{s})$
$\displaystyle=\sum\limits_{m,q\geq
1}^{\infty}\frac{f^{\prime}(mq)}{(mq)^{s}}c_{q}^{(s)}(n^{s})$
$\displaystyle=\sum\limits_{k=1}^{\infty}\frac{f^{\prime}(k)}{k^{s}}\sum\limits_{\begin{subarray}{c}mq=k\end{subarray}}c_{q}^{(s)}(n^{s}).$

Fix $k$. With thee notation $e(x)=exp(2\pi ix)$, we have

$\displaystyle\sum\limits_{\begin{subarray}{c}mq=k\\\
m=1\end{subarray}}^{k}c_{q}^{(s)}(n^{s})$
$\displaystyle=\sum\limits_{m|k}\,\sum\limits_{\begin{subarray}{c}(h,k^{s}/m^{s})_{s}=1\\\
h=1\end{subarray}}^{k^{s}/m^{s}}e(\frac{hn^{s}}{k^{s}/m^{s}})$
$\displaystyle=\sum\limits_{m|k}\,\sum\limits_{\begin{subarray}{c}(h,k^{s})_{s}=m^{s}\\\
h=1\end{subarray}}^{k^{s}}e(\frac{hn^{s}}{k^{s}})$
$\displaystyle=\sum\limits_{h=1}^{k^{s}}e(\frac{hn^{s}}{k^{s}})$
$\displaystyle=\begin{cases}k^{s}\quad\text{if }k^{s}|n^{s}\text{ or
equivalently, if }k|n\\\ 0\quad\text{otherwise}.\end{cases}$

Therefore

$\displaystyle\sum\limits_{q=1}^{\infty}a_{q}c_{q}^{(s)}(n^{s})$
$\displaystyle=\sum\limits_{\begin{subarray}{c}k=1\\\
k|n\end{subarray}}^{\infty}\frac{f^{\prime}(k)}{k^{s}}k^{s}=f(n)\text{ by the
Mobius inversion formula}.$

∎

For the sum discussed in Theorem 3.1, A. Grytczuk ([7]) gave the equality

$\sum\limits_{q|k}|c_{q}(n)|=2^{\omega(\frac{k}{(k,n)})}(k,n).$

We will derive an analogous equality in the next theorem and use it to give a
converse for the above theorem, as Grytczuk did in [5, Section 2].

######  Theorem 3.4.

For positive integers $k,s$, and $n$, we have

(8)
$\displaystyle\sum\limits_{d|k}|c_{d}^{(s)}(n)|=2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}(k^{s},n)_{s}.$

###### Proof.

Note that $(a,b)_{s}$ is multiplicative in both $a$ and $b$ as a single
variable function. Now if $(k,l)=1$, then

$\displaystyle\frac{k^{s}l^{s}}{(k^{s}l^{s},n)_{s}}=\frac{k^{s}}{(k^{s},n)_{s}}\frac{l^{s}}{(l^{s},n)_{s}}$

so that

$\displaystyle
2^{\omega\left(\frac{k^{s}l^{s}}{(k^{s}l^{s},n)_{s}}\right)}=2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\frac{l^{s}}{(l^{s},n)_{s}}\right)}.$

Since $(k,l)=1$, so is $\frac{k^{s}}{(k^{s},n)_{s}}$ and
$\frac{l^{s}}{(l^{s},n)_{s}}$ and so the above is equal to

$\displaystyle
2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)+\omega\left(\frac{l^{s}}{(l^{s},n)_{s}}\right)}=2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}\,2^{\omega\left(\frac{l^{s}}{(l^{s},n)_{s}}\right)}$

so that the right hand side of equation (8) is multiplicative. Now the left
hand side of this equation is $h_{n}(k)$ in the notation used in the proof of
Theorem 3.1 and it multiplicative. Hence it enough to prove our claim when $k$
is a prime power.

Let $k=p^{r}$ for some positive integer $r$. We have to prove that
$h_{n}(p^{r})=2^{\omega\left(\frac{p^{rs}}{(p^{rs},n)_{s}}\right)}(p^{rs},n)_{s}$
for this $k$. We have already seen that if $p^{s}\nmid n$, then
$h_{n}(p^{r})=2$. In this case, $(p^{rs},n)_{s}=1$ and so
$2^{\omega\left(\frac{p^{rs}}{(p^{rs},n)_{s}}\right)}(p^{rs},n)_{s}=2=h_{n}(p^{r})$.
Now let $p^{s}|n$. If $r\leq e_{p}^{(s)}(n)$, then

$\displaystyle h_{n}(p^{r})$ $\displaystyle=p^{rs}$

and

$\displaystyle
2^{\omega\left(\frac{p^{rs}}{(p^{rs},n)_{s}}\right)}(p^{rs},n)_{s}=2^{\omega\left(\frac{p^{rs}}{p^{rs}}\right)}p^{rs}=p^{rs}$

so that the claimed equality holds. On the other hand, if $r>e_{p}^{(s)}(n)$,
then we may take $r-m=e_{p}^{(s)}(n)$ for some positive integer $m$. So
$p^{(r-m)s}||_{s}n$. Then, as we have seen previously,

$\displaystyle h_{n}(p^{r})$ $\displaystyle=2p^{(r-m)s}.$

Now $(k^{s},n)_{s}=p^{(r-m)s}$ and $\frac{p^{rs}}{(p^{rs},n)_{s}}=p^{ms}$ so
that
$2^{\omega\left(\frac{p^{rs}}{(p^{rs},n)_{s}}\right)}(p^{rs},n)_{s}=2p^{(r-m)s}$.
This completes our proof. ∎

It was observed previously in Remark 3.2 that equality in Theorem 3.1 holds
when $n=m^{s}$ and $k$ is a multiple of $m\prod\limits_{p|m}p$. So the next
statement follows easily.

######  Corollary 3.5.

When $n=m^{s}$, where $m$ is a positive integer, and $k$ is a multiple of
$m\prod\limits_{p|m}p$, we have
$n.2^{\omega(k)}=2^{\omega(\frac{k^{s}}{(k^{s},n)_{s}})}(k^{s},n)_{s}$.

In [6], A. Grytczuk used the sum $\sum\limits_{q|k}|c_{q}(n)|$ to evaluate
$S(k,n):=\sum\limits_{d|k}2^{\omega\left(\frac{k}{(k,n)}\right)}(k,n)$. We
will now show how to evaluate an analogous sum.

######  Theorem 3.6.

Let
$S(k,n)=\sum\limits_{d|k}2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}(k^{s},n)_{s}\,\mu(k/d)$,
then we have

(9) $\displaystyle
S(k,n)=\begin{cases}\frac{J_{s}(k)}{J_{s}\left(\frac{k}{(k,n)}\right)}\quad&\text{if
}\frac{k}{(k,n)}\text{ is square-free}\\\ 0\quad&\text{ otherwise}\end{cases}$

###### Proof.

From (8), we have
$\sum\limits_{d|k}|c_{d}^{(s)}(n)|=2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}(k^{s},n)_{s}$.
Then by the Möbius inversion formula,
$|c_{k}^{(s)}(n)|=\sum\limits_{d|k}2^{\omega\left(\frac{d^{s}}{(d^{s},n)_{s}}\right)}(d^{s},n)_{s}\,\mu(k/d)$.
Using equation 6, we get

$\displaystyle\left|\frac{J_{s}(n)\mu(\frac{k}{(k,n)})}{J_{s}(\frac{k}{(k,n)})}\right|=\sum\limits_{d|k}2^{\omega\left(\frac{d^{s}}{(d^{s},n)_{s}}\right)}(d^{s},n)_{s}\mu(k/d).$

Since $\mu$ is 0 on positive integers that are not square-free and $\pm 1$ on
other positive integers, the statement in the theorem follows.

∎

Now we give the converse of the absolute convergence in Theorem 3.3 to
conclude this paper.

A. Grytczuk [5, Section 2] proved that the absolute convergence of the second
series in Theorem 3.3 (with $s=1)$ is sufficient for the absolute convergence
of the first series. This is true for any $s$.

######  Theorem 3.7.

If
$\sum\limits_{q=1}^{\infty}\sum\limits_{m=1}^{\infty}\frac{|f^{\prime}(mq)|}{(mq)^{s}}|c_{q}^{(s)}(n^{s})|<\infty$
then
$\sum\limits_{k=1}^{\infty}2^{\omega(k)}\frac{|f^{\prime}(k)|}{k^{s}}<\infty$.

###### Proof.

Suppose
$\sum\limits_{q=1}^{\infty}\sum\limits_{m=1}^{\infty}\frac{|f^{\prime}(mq)|}{(mq)^{s}}|c_{q}^{(s)}(n^{s})|<\infty$.

$\displaystyle\sum\limits_{q=1}^{\infty}\sum\limits_{m=1}^{\infty}\frac{|f^{\prime}(mq)|}{(mq)^{s}}|c_{q}^{(s)}(n^{s})|$
$\displaystyle=\sum\limits_{k=1}^{\infty}\sum\limits_{mq=k}\frac{|f^{\prime}(mq)|}{(mq)^{s}}|c_{q}^{(s)}(n^{s})|$
$\displaystyle=\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k^{s}}\sum\limits_{q|k}|c_{q}^{(s)}(n^{s})|$
$\displaystyle=\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k^{s}}2^{\omega(\frac{k^{s}}{(k^{s},n)_{s}})}(k^{s},n)_{s}.$

To arrive at the last step above, we used identity (8). Therefore
$\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{(k)^{s}}2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}(k^{s},n)_{s}<\infty$.
Since $\omega(m/n)\geq\omega(m)-\omega(n)$ with equality holding if and only
if $(m,n)=1$, we have

$\displaystyle
2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}\geq\frac{2^{\omega(k^{s})}}{2^{\omega\left((k^{s},n)_{s}\right)}}\geq\frac{2^{\omega(k^{s})}}{(k^{s},n)_{s}}.$

Hence

$\displaystyle\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k^{s}}2^{\omega\left(\frac{k^{s}}{(k^{s},n)_{s}}\right)}(k^{s},n)_{s}\geq\sum\limits_{k=1}^{\infty}\frac{|f^{\prime}(k)|}{k^{s}}2^{\omega(k^{s})}.$

Since $\omega(k^{s})=\omega(k)$, the theorem follows. ∎

######  Remark 3.8.

We would like to note here that K. R. Johnson evaluated the sum
$\sum\limits_{q|k}|c^{(s)}_{q}(n)|$ (with $s=1$) in the other variable of the
Ramanujan sum in [8]. Precisely, he evaluated $\sum\limits_{q|n}|c_{q}(n)|$.
It is possible to evaluate such a sum in our case also.

## References

  * [1] Arya Chandran and K Vishnu Namboothiri.  On a Ramanujan type expansion of arithmetical functions.  The Ramanujan Journal, 62:177–188, 2023.
  * [2] Eckford Cohen.  An extension of Ramanujan’s sum.  Duke Mathematical Journal, 16(2):85–90, 1949.
  * [3] Eckford Cohen.  Trigonometric sums in elementary number theory.  The American Mathematical Monthly, 66(2):105–117, 1959.
  * [4] Hubert Delange.  On Ramanujan expansions of certain arithmetical functions.  Acta Arith, 31(3):259–270, 1976.
  * [5] Aleksander Grytczuk.  On Ramanujan sums on arithmetical semigroups.  Tsukuba journal of mathematics, 16(2):315–319, 1992.
  * [6] Aleksander Grytczuk.  Note on Ramanujan sums.  Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 29:41–46, 2002.
  * [7] Aleksandr Grytczuk.  An identity involving Ramanujan’s sum.  Elemente der Mathematik, 36:16–24, 1981.
  * [8] Kenneth R Johnson.  A result for the’other’variable of Ramanujan’s sum.  Elemente der Mathematik, 38:122–124, 1983.
  * [9] Lutz Lucht.  Ramanujan expansions revisited.  Archiv der Mathematik, 64(2):121–128, 1995.
  * [10] Lutz G Lucht.  A survey of Ramanujan expansions.  International Journal of Number Theory, 6(08):1785–1799, 2010.
  * [11] Srinivasa Ramanujan.  On certain trigonometrical sums and their applications in the theory of numbers.  Trans. Cambridge Philos. Soc, 22(13):259–276, 1918.
  * [12] Aurel Wintner.  Eratosthenian averages.  Waverly Press, Baltimore, MD, 1943.


15 | 40 | 58500 | 2 | 1.0 | 2.5E+07 | 58500 | 14 | 1.0 | 6.8E+07 | 58500 | 4 | 1.0 | 2.2E+08 | 58500 | 3 | 1.0 | 2.1E+08 | 58500 | 4 | 0.4 | 0.6 | 2.2E+08
20 | 40 | 58500 | 1 | 1.0 | 3.1E+07 | 58500 | 10 | 1.0 | 7.9E+07 | 58500 | 6 | 1.0 | 2.7E+08 | 58500 | 4 | 1.0 | 2.7E+08 | 58500 | 6 | 0.533 | 0.467 | 2.8E+08
25 | 40 | 58500 | 1 | 1.0 | 4.4E+07 | 58500 | 9 | 1.0 | 8.6E+07 | 58500 | 4 | 1.0 | 3.5E+08 | 58500 | 6 | 1.0 | 3.5E+08 | 58500 | 6 | 0.767 | 0.233 | 3.7E+08

###  A.12 Sensorless Drive Diagnosis (normalized)

Dimensions: $m$ = 58509, $n$ = 48.

Description: a data set for sensorless drive diagnosis with features extracted
from motor current. Min-max scaling was used for normalization of data set
values for better clusterization.

Table 27: Summary of the results with Sensorless Drive Diagnosis (normalized)
($\times 10^{3}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 |  15.64798∗ | 0.89545 | 0.084 | 0.042 | 0.09 | 0.09 | 0.116 | 0.093 | 0.073 | 0.041 | 0.11 | 0.064 | 0.143 | 0.093 | 0.077 | 0.038 | 0.211 | 0.089 | 0.255 | 0.082
3 |  12.19375∗ | 0.70582 | 3.481 | 1.721 | 0.106 | 0.078 | 0.156 | 0.08 | 3.47 | 1.646 | 0.096 | 0.093 | 0.15 | 0.092 | 0.215 | 1.377 | 0.162 | 0.099 | 0.188 | 0.095
5 |  7.85054∗ | 0.45297 | 0.281 | 2.801 | 0.086 | 0.076 | 0.133 | 0.095 | 0.258 | 2.357 | 0.108 | 0.084 | 0.186 | 0.082 | 0.313 | 0.248 | 0.225 | 0.076 | 0.245 | 0.069
10 |  4.71275∗ | 0.28291 | 4.093 | 2.632 | 0.079 | 0.09 | 0.205 | 0.08 | 3.383 | 2.494 | 0.046 | 0.059 | 0.193 | 0.081 | 1.897 | 1.619 | 0.253 | 0.082 | 0.318 | 0.059
15 |  3.62541∗ | 0.21583 | 3.586 | 2.228 | 0.18 | 0.074 | 0.221 | 0.075 | 4.465 | 2.532 | 0.067 | 0.071 | 0.215 | 0.083 | 2.831 | 1.249 | 0.352 | 0.096 | 0.352 | 0.077
20 |  2.971∗ | 0.17839 | 4.911 | 2.442 | 0.112 | 0.079 | 0.212 | 0.077 | 4.191 | 2.521 | 0.109 | 0.067 | 0.186 | 0.067 | 2.91 | 1.366 | 0.439 | 0.059 | 0.441 | 0.06
25 |  2.60929∗ | 0.15589 | 4.5 | 2.541 | 0.147 | 0.061 | 0.192 | 0.078 | 4.278 | 2.633 | 0.059 | 0.066 | 0.214 | 0.076 | 3.205 | 1.528 | 0.448 | 0.088 | 0.486 | 0.085
Mean: | 2.991 |  | 0.114 |  | 0.176 |  | 2.874 |  | 0.085 |  | 0.184 |  | 1.635 |  | 0.299 |  | 0.326 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 |  15.64798∗ | 0.89545 | 0.107 | 0.066 | 0.143 | 0.077 | 0.232 | 0.081 | 0.1 | 0.053 | 0.237 | 0.088 | 0.244 | 0.089
3 |  12.19375∗ | 0.70582 | 0.216 | 1.381 | 0.132 | 0.087 | 0.246 | 0.07 | 0.221 | 1.335 | 0.239 | 0.08 | 0.248 | 0.09
5 |  7.85054∗ | 0.45297 | 0.327 | 0.23 | 0.165 | 0.087 | 0.243 | 0.08 | 0.465 | 0.287 | 0.28 | 0.042 | 0.28 | 0.055
10 |  4.71275∗ | 0.28291 | 2.465 | 2.031 | 0.277 | 0.087 | 0.31 | 0.049 | 2.639 | 1.214 | 0.247 | 0.088 | 0.353 | 0.076
15 |  3.62541∗ | 0.21583 | 2.643 | 1.421 | 0.324 | 0.083 | 0.353 | 0.068 | 2.655 | 1.615 | 0.375 | 0.077 | 0.399 | 0.079
20 |  2.971∗ | 0.17839 | 2.78 | 1.288 | 0.45 | 0.096 | 0.46 | 0.093 | 3.66 | 1.296 | 0.416 | 0.082 | 0.447 | 0.074
25 |  2.60929∗ | 0.15589 | 3.456 | 1.401 | 0.482 | 0.08 | 0.494 | 0.08 | 3.968 | 1.673 | 0.471 | 0.087 | 0.432 | 0.098
Mean: | 1.713 |  | 0.282 |  | 0.334 |  | 1.958 |  | 0.324 |  | 0.343 |

Table 28: Clustering details with Sensorless Drive Diagnosis (normalized)

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 40 | 3500 | 34 | 0.3 | 2.7E+06 | 3500 | 42 | 0.3 | 2.1E+06 | 3500 | 84 | 0.3 | 3.5E+06 | 3500 | 114 | 0.3 | 5.6E+06 | 3500 | 73 | 0.16 | 0.14 | 3.8E+06
3 | 40 | 3500 | 45 | 0.3 | 3.6E+06 | 3500 | 46 | 0.3 | 3.9E+06 | 3500 | 25 | 0.3 | 4.6E+06 | 3500 | 88 | 0.3 | 6.4E+06 | 3500 | 45 | 0.13 | 0.17 | 3.9E+06
5 | 40 | 3500 | 27 | 0.3 | 4.6E+06 | 3500 | 33 | 0.3 | 5.0E+06 | 3500 | 30 | 0.3 | 5.6E+06 | 3500 | 30 | 0.3 | 6.8E+06 | 3500 | 18 | 0.01 | 0.29 | 3.9E+06
10 | 40 | 3500 | 20 | 0.3 | 6.5E+06 | 3500 | 31 | 0.3 | 8.8E+06 | 3500 | 12 | 0.3 | 7.4E+06 | 3500 | 13 | 0.3 | 7.4E+06 | 3500 | 10 | 0.23 | 0.07 | 7.1E+06
15 | 40 | 3500 | 10 | 0.3 | 6.8E+06 | 3500 | 28 | 0.3 | 1.5E+07 | 3500 | 6 | 0.3 | 9.3E+06 | 3500 | 8 | 0.3 | 9.3E+06 | 3500 | 7 | 0.26 | 0.04 | 9.5E+06
20 | 40 | 3500 | 7 | 0.3 | 8.2E+06 | 3500 | 18 | 0.3 | 1.8E+07 | 3500 | 6 | 0.3 | 1.2E+07 | 3500 | 6 | 0.3 | 1.2E+07 | 3500 | 6 | 0.1 | 0.2 | 1.2E+07
25 | 40 | 3500 | 4 | 0.3 | 8.1E+06 | 3500 | 22 | 0.3 | 2.3E+07 | 3500 | 6 | 0.3 | 1.5E+07 | 3500 | 4 | 0.3 | 1.4E+07 | 3500 | 4 | 0.1 | 0.2 | 1.5E+07

###  A.13 Online News Popularity

Dimensions: $m$ = 39644, $n$ = 58.

Description: this dataset summarizes a heterogeneous set of features about
articles published by Mashable in a period of two years for predicting the
number of shares in social networks (popularity).

Table 29: Summary of the results with Online News Popularity ($\times
10^{14}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 9.53913 | 2.23423 | 0.011 | 0.012 | 0.257 | 0.182 | 0.328 | 0.172 | 0.012 | 0.01 | 0.361 | 0.153 | 0.464 | 0.194 | 0.023 | 0.013 | 0.484 | 0.124 | 0.551 | 0.131
3 | 5.91077 | 1.36838 | 0.054 | 7.214 | 0.262 | 0.091 | 0.324 | 0.163 | 0.042 | 5.241 | 0.283 | 0.166 | 0.316 | 0.2 | 0.068 | 0.035 | 0.399 | 0.147 | 0.538 | 0.153
5 | 3.09885 | 0.70088 | 0.096 | 8.792 | 0.436 | 0.177 | 0.444 | 0.194 | 0.068 | 7.88 | 0.231 | 0.215 | 0.328 | 0.222 | 0.077 | 0.025 | 0.46 | 0.153 | 0.554 | 0.109
10 | 1.17247 | 0.28511 | 3.18 | 4.776 | 0.362 | 0.132 | 0.545 | 0.17 | 2.573 | 5.959 | 0.094 | 0.063 | 0.323 | 0.221 | 0.865 | 0.893 | 0.469 | 0.124 | 0.651 | 0.129
15 | 0.77637 | 0.19493 | 2.957 | 3.473 | 0.347 | 0.178 | 0.529 | 0.175 | 3.679 | 2.729 | 0.124 | 0.21 | 0.536 | 0.192 | 2.352 | 1.067 | 0.581 | 0.081 | 0.659 | 0.071
20 | 0.59809 | 0.14827 | 4.145 | 2.843 | 0.294 | 0.211 | 0.489 | 0.16 | 4.676 | 4.8 | 0.169 | 0.163 | 0.331 | 0.206 | 3.428 | 2.203 | 0.801 | 0.181 | 0.788 | 0.135
25 | 0.49616 | 0.12234 | 5.389 | 2.55 | 0.634 | 0.242 | 0.6 | 0.195 | 5.109 | 2.286 | 0.319 | 0.163 | 0.533 | 0.193 | 4.623 | 1.757 | 0.923 | 0.127 | 1.029 | 0.204
Mean: | 2.262 |  | 0.37 |  | 0.466 |  | 2.309 |  | 0.226 |  | 0.404 |  | 1.634 |  | 0.588 |  | 0.681 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 9.53913 | 2.23423 | 0.024 | 0.017 | 0.342 | 0.166 | 0.478 | 0.163 | 0.015 | 0.01 | 0.386 | 0.127 | 0.426 | 0.159
3 | 5.91077 | 1.36838 | 0.062 | 0.043 | 0.353 | 0.112 | 0.477 | 0.148 | 0.068 | 0.042 | 0.475 | 0.137 | 0.564 | 0.083
5 | 3.09885 | 0.70088 | 0.087 | 0.022 | 0.448 | 0.093 | 0.543 | 0.091 | 0.079 | 0.029 | 0.438 | 0.106 | 0.564 | 0.104
10 | 1.17247 | 0.28511 | 0.968 | 1.82 | 0.494 | 0.112 | 0.586 | 0.097 | 1.525 | 1.74 | 0.475 | 0.097 | 0.699 | 0.111
15 | 0.77637 | 0.19493 | 1.934 | 3.257 | 0.582 | 0.085 | 0.721 | 0.102 | 2.887 | 5.434 | 0.622 | 0.194 | 0.782 | 0.222
20 | 0.59809 | 0.14827 | 3.349 | 3.874 | 0.824 | 0.164 | 0.884 | 0.17 | 4.016 | 1.626 | 0.834 | 0.177 | 0.907 | 0.175
25 | 0.49616 | 0.12234 | 4.646 | 1.986 | 0.996 | 0.198 | 1.025 | 0.184 | 4.784 | 1.902 | 0.95 | 0.221 | 0.988 | 0.201
Mean: | 1.581 |  | 0.577 |  | 0.673 |  | 1.911 |  | 0.597 |  | 0.704 |

Table 30: Clustering details with Online News Popularity

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 20 | 10000 | 88 | 0.7 | 8.3E+06 | 10000 | 97 | 0.7 | 6.2E+06 | 10000 | 558 | 0.7 | 3.3E+07 | 10000 | 466 | 0.7 | 3.5E+07 | 10000 | 226 | 0.63 | 0.07 | 3.3E+07
3 | 20 | 10000 | 38 | 0.7 | 1.1E+07 | 10000 | 56 | 0.7 | 1.5E+07 | 10000 | 176 | 0.7 | 4.1E+07 | 10000 | 139 | 0.7 | 4.5E+07 | 10000 | 152 | 0.14 | 0.56 | 3.8E+07
5 | 20 | 10000 | 40 | 0.7 | 1.2E+07 | 10000 | 43 | 0.7 | 2.0E+07 | 10000 | 72 | 0.7 | 3.8E+07 | 10000 | 108 | 0.7 | 4.1E+07 | 10000 | 84 | 0.467 | 0.233 | 3.9E+07
10 | 20 | 10000 | 13 | 0.7 | 1.3E+07 | 10000 | 32 | 0.7 | 3.5E+07 | 10000 | 31 | 0.7 | 4.0E+07 | 10000 | 22 | 0.7 | 3.8E+07 | 10000 | 24 | 0.49 | 0.21 | 3.0E+07
15 | 20 | 10000 | 6 | 0.7 | 1.5E+07 | 10000 | 32 | 0.7 | 4.5E+07 | 10000 | 10 | 0.7 | 4.6E+07 | 10000 | 12 | 0.7 | 4.2E+07 | 10000 | 8 | 0.047 | 0.653 | 3.3E+07
20 | 20 | 10000 | 4 | 0.7 | 1.5E+07 | 10000 | 10 | 0.7 | 5.0E+07 | 10000 | 6 | 0.7 | 4.1E+07 | 10000 | 7 | 0.7 | 4.1E+07 | 10000 | 5 | 0.63 | 0.07 | 3.8E+07
25 | 20 | 10000 | 2 | 0.7 | 1.6E+07 | 10000 | 12 | 0.7 | 5.2E+07 | 10000 | 5 | 0.7 | 5.0E+07 | 10000 | 5 | 0.7 | 4.7E+07 | 10000 | 6 | 0.653 | 0.047 | 5.1E+07

###  A.14 Gas Sensor Array Drift

Dimensions: $m$ = 13910, $n$ = 128.

Description: this data set contains measurements from chemical sensors
utilized in simulations for drift compensation in a discrimination task of
different gases at various levels of concentrations.

Table 31: Summary of the results with Gas Sensor Array Drift ($\times
10^{13}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 7.91186 | 4.79755 | 0.098 | 0.081 | 0.981 | 0.426 | 1.218 | 0.565 | 0.11 | 0.084 | 0.784 | 0.533 | 0.83 | 0.585 | 0.104 | 0.087 | 0.717 | 0.33 | 0.851 | 0.436
3 | 5.02412 | 3.0044 | 0.091 | 5.919 | 1.001 | 0.338 | 1.162 | 0.509 | 0.105 | 5.91 | 1.063 | 0.633 | 1.152 | 0.491 | 0.126 | 0.083 | 0.735 | 0.549 | 1.505 | 0.526
5 | 3.22394 | 2.03336 | 6.94 | 3.582 | 0.644 | 0.613 | 1.154 | 0.569 | 0.468 | 3.662 | 0.548 | 0.009 | 0.895 | 0.576 | 0.317 | 3.454 | 1.236 | 0.439 | 1.493 | 0.429
10 | 1.65524 | 1.07115 | 2.826 | 3.166 | 0.418 | 0.204 | 1.164 | 0.51 | 3.524 | 2.646 | 0.294 | 0.429 | 0.948 | 0.652 | 0.153 | 1.071 | 0.94 | 0.284 | 1.746 | 0.297
15 | 1.13801 | 0.73549 | 4.94 | 3.712 | 0.655 | 0.455 | 1.288 | 0.533 | 5.162 | 3.489 | 0.361 | 0.339 | 1.469 | 0.539 | -0.251 | 1.344 | 1.256 | 0.295 | 1.793 | 0.366
20 | 0.87916 | 0.57809 | 2.81 | 2.405 | 0.737 | 0.459 | 1.458 | 0.446 | 3.418 | 2.788 | 0.29 | 0.516 | 1.169 | 0.533 | 2.143 | 0.875 | 1.694 | 0.33 | 1.946 | 0.205
25 | 0.72274 | 0.47133 | 3.845 | 1.711 | 1.131 | 0.537 | 1.397 | 0.513 | 3.983 | 2.45 | 0.357 | 0.337 | 1.227 | 0.591 | 2.69 | 1.513 | 1.785 | 0.202 | 1.888 | 0.23
Mean: | 3.079 |  | 0.795 |  | 1.263 |  | 2.396 |  | 0.528 |  | 1.099 |  | 0.755 |  | 1.195 |  | 1.603 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 7.91186 | 4.79755 | 0.134 | 0.078 | 0.744 | 0.446 | 1.113 | 0.441 | 0.122 | 0.092 | 0.568 | 0.399 | 1.018 | 0.44
3 | 5.02412 | 3.0044 | 0.274 | 0.12 | 0.919 | 0.442 | 1.279 | 0.466 | 0.13 | 0.096 | 1.122 | 0.444 | 1.244 | 0.459
5 | 3.22394 | 2.03336 | 0.169 | 2.495 | 1.088 | 0.347 | 1.294 | 0.423 | 0.2 | 1.915 | 1.069 | 0.359 | 1.405 | 0.436
10 | 1.65524 | 1.07115 | 0.4 | 1.623 | 0.867 | 0.236 | 1.729 | 0.364 | 0.132 | 0.871 | 0.776 | 0.228 | 1.69 | 0.299
15 | 1.13801 | 0.73549 | 0.789 | 2.112 | 1.447 | 0.326 | 1.888 | 0.273 | -0.224 | 1.534 | 1.23 | 0.301 | 1.908 | 0.323
20 | 0.87916 | 0.57809 | 2.06 | 2.176 | 1.682 | 0.242 | 1.976 | 0.175 | 1.818 | 0.992 | 1.623 | 0.32 | 2.093 | 0.353
25 | 0.72274 | 0.47133 | 3.439 | 1.048 | 2.022 | 0.43 | 2.13 | 0.253 | 2.283 | 1.372 | 1.82 | 0.406 | 2.205 | 0.394
Mean: | 1.038 |  | 1.253 |  | 1.63 |  | 0.637 |  | 1.173 |  | 1.652 |

Table 32: Clustering details with Gas Sensor Array Drift

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 30 | 9000 | 121 | 2.0 | 1.5E+07 | 9000 | 80 | 2.0 | 1.5E+07 | 9000 | 394 | 2.0 | 8.4E+07 | 9000 | 461 | 2.0 | 8.7E+07 | 9000 | 482 | 1.0 | 1.0 | 8.4E+07
3 | 30 | 9000 | 88 | 2.0 | 1.5E+07 | 9000 | 122 | 2.0 | 1.8E+07 | 9000 | 533 | 2.0 | 8.4E+07 | 9000 | 397 | 2.0 | 8.5E+07 | 9000 | 432 | 0.867 | 1.133 | 8.1E+07
5 | 30 | 9000 | 48 | 2.0 | 1.6E+07 | 9000 | 20 | 2.0 | 9.8E+06 | 9000 | 217 | 2.0 | 7.5E+07 | 9000 | 172 | 2.0 | 7.5E+07 | 9000 | 154 | 0.867 | 1.133 | 7.5E+07
10 | 30 | 9000 | 17 | 2.0 | 1.7E+07 | 9000 | 51 | 2.0 | 5.1E+07 | 9000 | 92 | 2.0 | 7.1E+07 | 9000 | 108 | 2.0 | 8.4E+07 | 9000 | 94 | 1.4 | 0.6 | 7.5E+07
15 | 30 | 9000 | 10 | 2.0 | 1.7E+07 | 9000 | 50 | 2.0 | 5.7E+07 | 9000 | 46 | 2.0 | 7.0E+07 | 9000 | 31 | 2.0 | 5.6E+07 | 9000 | 46 | 1.867 | 0.133 | 6.6E+07
20 | 30 | 9000 | 9 | 2.0 | 1.8E+07 | 9000 | 35 | 2.0 | 6.1E+07 | 9000 | 13 | 2.0 | 5.6E+07 | 9000 | 19 | 2.0 | 5.4E+07 | 9000 | 14 | 0.2 | 1.8 | 4.3E+07
25 | 30 | 9000 | 4 | 2.0 | 1.8E+07 | 9000 | 26 | 2.0 | 6.6E+07 | 9000 | 9 | 2.0 | 6.1E+07 | 9000 | 10 | 2.0 | 5.5E+07 | 9000 | 11 | 0.8 | 1.2 | 5.2E+07

###  A.15 3D Road Network

Dimensions: $m$ = 434874, $n$ = 3.

Description: 3D road network from Denmark with highly accurate elevation
information which contains longitude, latitude and altitude for each road
segment or edge in the graph. Usually this data set used in eco-routing and
fuel/Co2-estimation routing algorithms.

Table 33: Summary of the results with 3D Road Network ($\times 10^{6}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 49.13298 | 11.16871 | 0.004 | 0.009 | 0.22 | 0.128 | 0.239 | 0.139 | 0.003 | 0.005 | 0.185 | 0.141 | 0.265 | 0.161 | 0.005 | 0.006 | 0.336 | 0.135 | 0.403 | 0.122
3 | 22.77818 | 5.17279 | 0.01 | 0.01 | 0.225 | 0.122 | 0.276 | 0.139 | 0.006 | 0.017 | 0.251 | 0.149 | 0.304 | 0.14 | 0.008 | 0.015 | 0.392 | 0.099 | 0.439 | 0.093
5 | 8.82574 | 2.00137 | 0.017 | 0.02 | 0.324 | 0.149 | 0.307 | 0.149 | 0.02 | 0.027 | 0.252 | 0.129 | 0.278 | 0.137 | 0.021 | 0.031 | 0.471 | 0.095 | 0.467 | 0.089
10 | 2.56661 | 0.58486 | 0.243 | 0.204 | 0.32 | 0.102 | 0.362 | 0.122 | 0.189 | 0.176 | 0.285 | 0.142 | 0.407 | 0.131 | 0.193 | 0.192 | 0.766 | 0.191 | 0.755 | 0.196
15 | 1.27069 | 0.29096 | 0.553 | 0.476 | 0.348 | 0.126 | 0.508 | 0.275 | 0.241 | 0.47 | 0.308 | 0.099 | 0.391 | 0.104 | 0.305 | 0.383 | 1.133 | 0.234 | 1.175 | 0.3
20 | 0.80865 | 0.18552 | 0.77 | 0.911 | 0.535 | 0.176 | 0.551 | 0.199 | 1.244 | 0.902 | 0.382 | 0.124 | 0.443 | 0.095 | 0.446 | 0.714 | 1.305 | 0.25 | 1.358 | 0.267
25 | 0.59259 | 0.137 | 1.071 | 1.407 | 0.462 | 0.253 | 0.567 | 0.32 | 1.72 | 1.069 | 0.335 | 0.155 | 0.487 | 0.116 | 0.701 | 0.488 | 1.367 | 0.324 | 1.46 | 0.408
Mean: | 0.381 |  | 0.348 |  | 0.402 |  | 0.489 |  | 0.285 |  | 0.368 |  | 0.24 |  | 0.824 |  | 0.866 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 49.13298 | 11.16871 | 0.007 | 0.005 | 0.27 | 0.134 | 0.352 | 0.117 | 0.006 | 0.016 | 0.375 | 0.11 | 0.424 | 0.111
3 | 22.77818 | 5.17279 | 0.009 | 0.015 | 0.321 | 0.124 | 0.398 | 0.085 | 0.008 | 0.012 | 0.392 | 0.123 | 0.444 | 0.116
5 | 8.82574 | 2.00137 | 0.03 | 0.038 | 0.47 | 0.115 | 0.508 | 0.091 | 0.037 | 0.037 | 0.494 | 0.148 | 0.573 | 0.109
10 | 2.56661 | 0.58486 | 0.214 | 0.27 | 0.762 | 0.165 | 0.794 | 0.197 | 0.265 | 0.209 | 0.749 | 0.19 | 0.775 | 0.172
15 | 1.27069 | 0.29096 | 0.46 | 0.303 | 1.113 | 0.346 | 1.232 | 0.322 | 0.405 | 0.386 | 1.0 | 0.287 | 1.141 | 0.328
20 | 0.80865 | 0.18552 | 0.515 | 0.646 | 1.349 | 0.269 | 1.425 | 0.266 | 0.536 | 0.696 | 1.316 | 0.253 | 1.424 | 0.31
25 | 0.59259 | 0.137 | 0.579 | 0.539 | 1.25 | 0.202 | 1.389 | 0.375 | 0.503 | 0.402 | 1.351 | 0.295 | 1.449 | 0.358
Mean: | 0.259 |  | 0.791 |  | 0.871 |  | 0.252 |  | 0.811 |  | 0.89 |

Table 34: Clustering details with 3D Road Network

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 40 | 100000 | 14 | 0.5 | 1.7E+07 | 100000 | 15 | 0.5 | 1.7E+07 | 100000 | 54 | 0.5 | 6.7E+07 | 100000 | 66 | 0.5 | 7.5E+07 | 100000 | 45 | 0.033 | 0.467 | 6.0E+07
3 | 40 | 100000 | 16 | 0.5 | 2.9E+07 | 100000 | 17 | 0.5 | 2.9E+07 | 100000 | 32 | 0.5 | 7.8E+07 | 100000 | 45 | 0.5 | 9.8E+07 | 100000 | 35 | 0.467 | 0.033 | 8.2E+07
5 | 40 | 100000 | 14 | 0.5 | 4.9E+07 | 100000 | 10 | 0.5 | 5.2E+07 | 100000 | 8 | 0.5 | 1.1E+08 | 100000 | 12 | 0.5 | 1.1E+08 | 100000 | 7 | 0.117 | 0.383 | 1.1E+08
10 | 40 | 100000 | 5 | 0.5 | 1.0E+08 | 100000 | 6 | 0.5 | 1.2E+08 | 100000 | 4 | 0.5 | 3.3E+08 | 100000 | 4 | 0.5 | 3.4E+08 | 100000 | 4 | 0.15 | 0.35 | 3.5E+08
15 | 40 | 100000 | 2 | 0.5 | 1.3E+08 | 100000 | 4 | 0.5 | 2.3E+08 | 100000 | 5 | 0.5 | 6.6E+08 | 100000 | 4 | 0.5 | 6.7E+08 | 100000 | 4 | 0.45 | 0.05 | 6.7E+08
20 | 40 | 100000 | 1 | 0.5 | 1.4E+08 | 100000 | 4 | 0.5 | 2.2E+08 | 100000 | 4 | 0.5 | 9.1E+08 | 100000 | 5 | 0.5 | 8.6E+08 | 100000 | 4 | 0.333 | 0.167 | 8.9E+08
25 | 40 | 100000 | 1 | 0.5 | 1.7E+08 | 100000 | 4 | 0.5 | 3.1E+08 | 100000 | 3 | 0.5 | 1.3E+09 | 100000 | 4 | 0.5 | 1.3E+09 | 100000 | 4 | 0.317 | 0.183 | 1.3E+09

###  A.16 Skin Segmentation

Dimensions: $m$ = 245057, $n$ = 3.

Description: Skin and Nonskin dataset is generated using skin textures from
face images of diversity of age, gender, and race people and constructed over
B, G, R color space.

Table 35: Summary of the results with Skin Segmentation ($\times 10^{9}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 1.32236 | 0.04208 | 0.031 | 0.02 | 0.106 | 0.062 | 0.115 | 0.058 | 0.025 | 0.019 | 0.056 | 0.071 | 0.118 | 0.059 | 0.031 | 0.027 | 0.178 | 0.057 | 0.187 | 0.047
3 | 0.89362 | 0.02832 | 0.045 | 0.032 | 0.067 | 0.061 | 0.076 | 0.063 | 0.03 | 0.023 | 0.067 | 0.057 | 0.115 | 0.061 | 0.051 | 0.031 | 0.115 | 0.073 | 0.155 | 0.068
5 | 0.50203 | 0.01591 | 1.74 | 2.778 | 0.052 | 0.052 | 0.112 | 0.053 | 0.148 | 3.377 | 0.086 | 0.054 | 0.138 | 0.054 | 0.086 | 0.29 | 0.16 | 0.088 | 0.188 | 0.081
10 | 0.25121 | 0.00828 | 5.715 | 4.337 | 0.026 | 0.058 | 0.088 | 0.054 | 6.426 | 5.524 | 0.023 | 0.04 | 0.102 | 0.055 | 0.196 | 2.128 | 0.15 | 0.066 | 0.211 | 0.067
15 | 0.16964 | 0.00548 | 5.075 | 3.389 | 0.084 | 0.067 | 0.126 | 0.054 | 3.952 | 3.838 | 0.074 | 0.037 | 0.124 | 0.045 | 3.054 | 2.076 | 0.215 | 0.089 | 0.229 | 0.086
20 | 0.12615 | 0.00408 | 4.829 | 4.32 | 0.087 | 0.045 | 0.123 | 0.051 | 4.798 | 3.504 | 0.086 | 0.059 | 0.115 | 0.057 | 3.781 | 2.197 | 0.249 | 0.072 | 0.256 | 0.078
25 | 0.10228 | 0.00338 | 5.64 | 2.485 | 0.076 | 0.041 | 0.124 | 0.057 | 5.008 | 2.327 | 0.058 | 0.05 | 0.144 | 0.058 | 4.735 | 2.103 | 0.273 | 0.093 | 0.336 | 0.082
Mean: | 3.297 |  | 0.071 |  | 0.109 |  | 2.912 |  | 0.065 |  | 0.122 |  | 1.705 |  | 0.191 |  | 0.223 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 1.32236 | 0.04208 | 0.026 | 0.021 | 0.071 | 0.067 | 0.124 | 0.061 | 0.028 | 0.026 | 0.088 | 0.059 | 0.096 | 0.075
3 | 0.89362 | 0.02832 | 0.056 | 0.036 | 0.092 | 0.056 | 0.15 | 0.054 | 0.04 | 0.18 | 0.154 | 0.062 | 0.155 | 0.077
5 | 0.50203 | 0.01591 | 0.093 | 0.974 | 0.064 | 0.047 | 0.158 | 0.061 | 0.098 | 0.641 | 0.17 | 0.087 | 0.214 | 0.082
10 | 0.25121 | 0.00828 | 0.227 | 2.879 | 0.103 | 0.05 | 0.18 | 0.061 | 0.23 | 2.093 | 0.118 | 0.077 | 0.242 | 0.083
15 | 0.16964 | 0.00548 | 3.089 | 2.814 | 0.157 | 0.077 | 0.224 | 0.059 | 2.97 | 2.515 | 0.281 | 0.087 | 0.291 | 0.068
20 | 0.12615 | 0.00408 | 4.897 | 2.213 | 0.222 | 0.081 | 0.237 | 0.077 | 3.348 | 2.498 | 0.312 | 0.076 | 0.303 | 0.078
25 | 0.10228 | 0.00338 | 4.356 | 2.171 | 0.244 | 0.102 | 0.282 | 0.083 | 4.631 | 2.313 | 0.268 | 0.087 | 0.32 | 0.072
Mean: | 1.821 |  | 0.136 |  | 0.194 |  | 1.621 |  | 0.199 |  | 0.231 |

Table 36: Clustering details with Skin Segmentation

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 30 | 8000 | 18 | 0.2 | 2.1E+06 | 8000 | 14 | 0.2 | 2.0E+06 | 8000 | 16 | 0.2 | 2.6E+06 | 8000 | 46 | 0.2 | 4.8E+06 | 8000 | 10 | 0.047 | 0.153 | 2.3E+06
3 | 30 | 8000 | 11 | 0.2 | 3.8E+06 | 8000 | 10 | 0.2 | 2.7E+06 | 8000 | 12 | 0.2 | 4.9E+06 | 8000 | 31 | 0.2 | 6.8E+06 | 8000 | 10 | 0.033 | 0.167 | 5.1E+06
5 | 30 | 8000 | 16 | 0.2 | 6.0E+06 | 8000 | 12 | 0.2 | 5.5E+06 | 8000 | 12 | 0.2 | 8.1E+06 | 8000 | 26 | 0.2 | 1.1E+07 | 8000 | 16 | 0.153 | 0.047 | 8.3E+06
10 | 30 | 8000 | 10 | 0.2 | 1.4E+07 | 8000 | 10 | 0.2 | 1.2E+07 | 8000 | 8 | 0.2 | 1.7E+07 | 8000 | 21 | 0.2 | 2.2E+07 | 8000 | 12 | 0.127 | 0.073 | 1.7E+07
15 | 30 | 8000 | 12 | 0.2 | 2.3E+07 | 8000 | 9 | 0.2 | 1.9E+07 | 8000 | 6 | 0.2 | 2.5E+07 | 8000 | 11 | 0.2 | 2.9E+07 | 8000 | 9 | 0.153 | 0.047 | 2.7E+07
20 | 30 | 8000 | 6 | 0.2 | 2.6E+07 | 8000 | 10 | 0.2 | 2.9E+07 | 8000 | 7 | 0.2 | 3.6E+07 | 8000 | 8 | 0.2 | 3.6E+07 | 8000 | 8 | 0.053 | 0.147 | 3.3E+07
25 | 30 | 8000 | 6 | 0.2 | 2.8E+07 | 8000 | 12 | 0.2 | 4.3E+07 | 8000 | 6 | 0.2 | 3.8E+07 | 8000 | 6 | 0.2 | 4.0E+07 | 8000 | 6 | 0.16 | 0.04 | 4.1E+07

###  A.17 KEGG Metabolic Relation Network (Directed)

Dimensions: $m$ = 53413, $n$ = 20.

Description:

Table 37: Summary of the results with KEGG Metabolic Relation Network
(Directed) ($\times 10^{8}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 11.3853 | 11.32433 | 0.0 | 7.528 | 0.199 | 0.24 | 0.585 | 0.269 | 0.0 | 8.995 | 0.308 | 0.143 | 0.367 | 0.276 | 0.0 | 0.104 | 0.357 | 0.13 | 0.713 | 0.199
3 | 4.9006 | 4.85767 | 0.001 | 27.211 | 0.39 | 0.204 | 0.564 | 0.239 | 0.001 | 27.187 | 0.47 | 0.255 | 0.467 | 0.294 | 0.559 | 0.273 | 0.438 | 0.12 | 0.659 | 0.228
5 | 1.88367 | 1.87146 | 0.004 | 10.819 | 0.514 | 0.276 | 0.498 | 0.246 | 0.005 | 0.434 | 0.359 | 0.28 | 0.686 | 0.326 | 0.009 | 0.016 | 0.617 | 0.168 | 0.86 | 0.148
10 | 0.60513 | 0.63478 | 2.549 | 3.238 | 0.297 | 0.112 | 0.641 | 0.221 | 5.128 | 7.621 | 0.116 | 0.04 | 0.544 | 0.229 | 0.036 | 1.156 | 0.701 | 0.14 | 0.96 | 0.108
15 | 0.35393 | 0.3655 | 2.731 | 4.373 | 0.263 | 0.174 | 0.68 | 0.223 | 4.4 | 6.429 | 0.116 | 0.173 | 0.494 | 0.274 | -0.003 | 3.446 | 0.916 | 0.105 | 1.076 | 0.152
20 | 0.25027 | 0.25564 | 3.027 | 4.652 | 0.378 | 0.155 | 0.826 | 0.166 | 2.609 | 6.138 | 0.179 | 0.088 | 0.651 | 0.226 | 0.237 | 0.627 | 1.238 | 0.153 | 1.387 | 0.291
25 | 0.19289 | 0.19904 | 3.391 | 1.992 | 0.559 | 0.265 | 0.971 | 0.213 | 4.706 | 2.172 | 0.211 | 0.15 | 0.591 | 0.262 | 1.649 | 34.397 | 1.618 | 0.178 | 1.702 | 0.237
Mean: | 1.672 |  | 0.371 |  | 0.681 |  | 2.407 |  | 0.251 |  | 0.543 |  | 0.355 |  | 0.841 |  | 1.051 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 11.3853 | 11.32433 | 0.24 | 0.11 | 0.293 | 0.093 | 0.683 | 0.154 | 0.0 | 0.118 | 0.375 | 0.138 | 0.816 | 0.173
3 | 4.9006 | 4.85767 | 0.002 | 0.273 | 0.591 | 0.181 | 0.598 | 0.193 | 0.001 | 0.266 | 0.5 | 0.157 | 0.735 | 0.173
5 | 1.88367 | 1.87146 | 0.009 | 0.626 | 0.644 | 0.091 | 0.749 | 0.16 | 0.007 | 0.316 | 0.697 | 0.113 | 0.88 | 0.133
10 | 0.60513 | 0.63478 | 0.038 | 0.019 | 0.674 | 0.096 | 0.927 | 0.09 | 0.03 | 0.022 | 0.792 | 0.101 | 1.067 | 0.264
15 | 0.35393 | 0.3655 | -0.277 | 1.591 | 0.902 | 0.095 | 1.028 | 0.089 | -0.224 | 3.386 | 0.975 | 0.169 | 1.22 | 0.166
20 | 0.25027 | 0.25564 | 0.032 | 0.835 | 1.258 | 0.15 | 1.413 | 0.314 | 0.561 | 0.704 | 1.301 | 0.186 | 1.58 | 0.413
25 | 0.19289 | 0.19904 | 1.671 | 3.601 | 1.525 | 0.164 | 1.726 | 0.164 | 1.131 | 6.496 | 1.51 | 0.224 | 1.71 | 0.428
Mean: | 0.245 |  | 0.841 |  | 1.018 |  | 0.215 |  | 0.879 |  | 1.144 |

Table 38: Clustering details with KEGG Metabolic Relation Network (Directed)

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 20 | 53350 | 80 | 1.0 | 3.1E+07 | 53350 | 30 | 1.0 | 2.4E+07 | 53350 | 388 | 1.0 | 1.4E+08 | 53350 | 377 | 1.0 | 1.4E+08 | 53350 | 504 | 0.767 | 0.233 | 1.5E+08
3 | 20 | 53350 | 62 | 1.0 | 4.0E+07 | 53350 | 49 | 1.0 | 3.8E+07 | 53350 | 140 | 1.0 | 1.6E+08 | 53350 | 154 | 1.0 | 1.7E+08 | 53350 | 243 | 0.967 | 0.033 | 1.6E+08
5 | 20 | 53350 | 34 | 1.0 | 4.7E+07 | 53350 | 34 | 1.0 | 5.0E+07 | 53350 | 133 | 1.0 | 1.7E+08 | 53350 | 88 | 1.0 | 1.8E+08 | 53350 | 100 | 0.333 | 0.667 | 1.6E+08
10 | 20 | 53350 | 21 | 1.0 | 5.4E+07 | 53350 | 40 | 1.0 | 1.1E+08 | 53350 | 28 | 1.0 | 1.5E+08 | 53350 | 37 | 1.0 | 1.7E+08 | 53350 | 14 | 0.867 | 0.133 | 1.3E+08
15 | 20 | 53350 | 10 | 1.0 | 5.6E+07 | 53350 | 32 | 1.0 | 1.6E+08 | 53350 | 8 | 1.0 | 1.4E+08 | 53350 | 7 | 1.0 | 1.5E+08 | 53350 | 9 | 0.6 | 0.4 | 1.5E+08
20 | 20 | 53350 | 8 | 1.0 | 5.8E+07 | 53350 | 31 | 1.0 | 1.7E+08 | 53350 | 4 | 1.0 | 2.1E+08 | 53350 | 3 | 1.0 | 2.1E+08 | 53350 | 6 | 0.1 | 0.9 | 2.0E+08
25 | 20 | 53350 | 6 | 1.0 | 5.9E+07 | 53350 | 24 | 1.0 | 1.9E+08 | 53350 | 5 | 1.0 | 2.9E+08 | 53350 | 6 | 1.0 | 2.9E+08 | 53350 | 4 | 0.033 | 0.967 | 2.8E+08

###  A.18 Shuttle Control

Dimensions: $m$ = 58000, $n$ = 9.

Description: each entity in the dataset contains several shuttle control
attributes.

Table 39: Summary of the results with Shuttle Control ($\times 10^{8}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 21.34329 | 19.96233 | 1.86 | 2.186 | 1.065 | 0.323 | 1.065 | 0.35 | 5.043 | 12.354 | 0.217 | 0.343 | 0.765 | 0.494 | 1.86 | 2.266 | 0.371 | 0.275 | 0.812 | 0.333
3 | 10.85415 | 10.59526 | 0.0 | 19.391 | 0.988 | 0.094 | 0.869 | 0.351 | 0.0 | 28.656 | 0.535 | 0.351 | 0.943 | 0.325 | 0.666 | 1.515 | 0.44 | 0.179 | 1.046 | 0.356
4 | 8.8691 | 8.64099 | 4.196 | 5.23 | 0.502 | 0.297 | 0.515 | 0.413 | 4.137 | 4.043 | 0.516 | 0.456 | 1.007 | 0.478 | 8.689 | 6.287 | 0.598 | 0.37 | 1.037 | 0.397
5 | 7.24479 | 7.06445 | 10.824 | 8.874 | 0.384 | 0.436 | 0.468 | 0.494 | 4.67 | 10.318 | 0.507 | 0.199 | 0.778 | 0.434 | 0.505 | 9.041 | 0.606 | 0.285 | 0.841 | 0.347
10 | 2.83216 | 2.97155 | 9.787 | 15.739 | 0.059 | 0.393 | 0.614 | 0.501 | 11.264 | 6.069 | 0.059 | 0.034 | 0.795 | 0.444 | 0.96 | 2.686 | 0.409 | 0.234 | 1.129 | 0.243
15 | 1.53154 | 1.7471 | 12.966 | 7.817 | 0.117 | 0.223 | 0.358 | 0.422 | 21.015 | 11.428 | 0.119 | 0.075 | 1.094 | 0.384 | 3.815 | 2.809 | 0.615 | 0.113 | 1.296 | 0.245
20 | 1.06012 | 1.12972 | 3.699 | 7.561 | 0.166 | 0.13 | 1.002 | 0.454 | 4.941 | 7.216 | 0.128 | 0.054 | 0.962 | 0.341 | -1.647 | 2.623 | 0.919 | 0.172 | 1.436 | 0.22
25 | 0.77978 | 0.83166 | 5.865 | 4.094 | 0.478 | 0.233 | 0.891 | 0.405 | 6.171 | 5.422 | 0.225 | 0.212 | 0.73 | 0.315 | 3.17 | 2.067 | 0.93 | 0.263 | 1.48 | 0.217
Mean: | 6.15 |  | 0.47 |  | 0.723 |  | 7.155 |  | 0.288 |  | 0.884 |  | 2.252 |  | 0.611 |  | 1.135 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 21.34329 | 19.96233 | 0.0 | 1.309 | 0.289 | 0.135 | 0.758 | 0.357 | 0.0 | 1.992 | 0.294 | 0.142 | 0.868 | 0.366
3 | 10.85415 | 10.59526 | 3.658 | 1.552 | 0.453 | 0.125 | 0.88 | 0.331 | 3.658 | 1.447 | 0.531 | 0.273 | 0.868 | 0.383
4 | 8.8691 | 8.64099 | 0.0 | 0.117 | 0.449 | 0.073 | 0.862 | 0.373 | 0.0 | 2.926 | 0.617 | 0.166 | 0.847 | 0.33
5 | 7.24479 | 7.06445 | 0.178 | 5.982 | 0.497 | 0.196 | 1.147 | 0.347 | 0.178 | 0.1 | 0.623 | 0.212 | 1.025 | 0.25
10 | 2.83216 | 2.97155 | 0.79 | 1.819 | 0.381 | 0.099 | 1.1 | 0.265 | 0.794 | 0.546 | 0.394 | 0.13 | 1.085 | 0.277
15 | 1.53154 | 1.7471 | 5.052 | 36.931 | 0.584 | 0.133 | 1.152 | 0.243 | 2.995 | 2.968 | 0.602 | 0.139 | 1.299 | 0.141
20 | 1.06012 | 1.12972 | -0.211 | 1.88 | 0.777 | 0.142 | 1.277 | 0.172 | -1.54 | 42.218 | 0.875 | 0.15 | 1.453 | 0.176
25 | 0.77978 | 0.83166 | 3.646 | 1.646 | 0.928 | 0.21 | 1.413 | 0.226 | 2.629 | 1.194 | 1.001 | 0.25 | 1.511 | 0.22
Mean: | 1.639 |  | 0.545 |  | 1.074 |  | 1.089 |  | 0.617 |  | 1.119 |

Table 40: Clustering details with Shuttle Control

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 15 | 57950 | 216 | 1.5 | 8.1E+07 | 57950 | 83 | 1.5 | 9.5E+07 | 57950 | 824 | 1.5 | 4.9E+08 | 57950 | 838 | 1.5 | 4.8E+08 | 57950 | 928 | 0.75 | 0.75 | 4.7E+08
3 | 15 | 57950 | 160 | 1.5 | 9.8E+07 | 57950 | 253 | 1.5 | 1.4E+08 | 57950 | 843 | 1.5 | 5.4E+08 | 57950 | 740 | 1.5 | 5.5E+08 | 57950 | 756 | 0.65 | 0.85 | 5.3E+08
4 | 15 | 57950 | 78 | 1.5 | 1.2E+08 | 57950 | 258 | 1.5 | 1.8E+08 | 57950 | 805 | 1.5 | 6.8E+08 | 57950 | 625 | 1.5 | 6.2E+08 | 57950 | 492 | 0.6 | 0.9 | 6.1E+08
5 | 15 | 57950 | 66 | 1.5 | 1.3E+08 | 57950 | 141 | 1.5 | 1.7E+08 | 57950 | 502 | 1.5 | 6.6E+08 | 57950 | 672 | 1.5 | 6.5E+08 | 57950 | 653 | 0.6 | 0.9 | 6.4E+08
10 | 15 | 57950 | 48 | 1.5 | 1.5E+08 | 57950 | 52 | 1.5 | 9.4E+07 | 57950 | 322 | 1.5 | 7.5E+08 | 57950 | 359 | 1.5 | 7.3E+08 | 57950 | 275 | 0.1 | 1.4 | 7.0E+08
15 | 15 | 57950 | 16 | 1.5 | 1.7E+08 | 57950 | 68 | 1.5 | 3.2E+08 | 57950 | 131 | 1.5 | 6.0E+08 | 57950 | 128 | 1.5 | 6.2E+08 | 57950 | 166 | 1.1 | 0.4 | 5.8E+08
20 | 15 | 57950 | 21 | 1.5 | 1.7E+08 | 57950 | 75 | 1.5 | 4.1E+08 | 57950 | 77 | 1.5 | 4.4E+08 | 57950 | 85 | 1.5 | 5.4E+08 | 57950 | 24 | 0.6 | 0.9 | 3.6E+08
25 | 15 | 57950 | 11 | 1.5 | 1.5E+08 | 57950 | 20 | 1.5 | 2.0E+08 | 57950 | 29 | 1.5 | 4.6E+08 | 57950 | 44 | 1.5 | 4.4E+08 | 57950 | 13 | 0.35 | 1.15 | 2.5E+08

###  A.19 Shuttle Control (normalized)

Dimensions: $m$ = 58000, $n$ = 9.

Description: each entity in the dataset contains several shuttle control
attributes. Min-max scaling was used for normalization of data set values for
better clusterization.

Table 41: Summary of the results with Shuttle Control (normalized) ($\times
10^{1}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 104.41601 | 3.34565 | 0.132 | 0.242 | 0.212 | 0.123 | 0.194 | 0.129 | 0.123 | 0.135 | 0.126 | 0.132 | 0.246 | 0.119 | 0.202 | 0.093 | 0.309 | 0.06 | 0.334 | 0.061
3 | 73.28769 | 2.33238 | 1.202 | 6.538 | 0.149 | 0.128 | 0.187 | 0.116 | 0.576 | 0.723 | 0.157 | 0.107 | 0.252 | 0.1 | 0.308 | 0.149 | 0.275 | 0.128 | 0.353 | 0.088
4 | 50.076 | 1.55903 | 0.586 | 5.581 | 0.162 | 0.083 | 0.224 | 0.076 | 0.646 | 12.125 | 0.135 | 0.091 | 0.119 | 0.109 | 0.712 | 0.441 | 0.211 | 0.1 | 0.309 | 0.105
5 | 39.78043 | 1.23673 | 2.132 | 3.267 | 0.151 | 0.103 | 0.198 | 0.109 | 1.56 | 3.337 | 0.139 | 0.081 | 0.152 | 0.08 | 1.159 | 0.663 | 0.246 | 0.101 | 0.312 | 0.093
10 | 15.04997 | 0.44698 | 2.058 | 2.201 | 0.177 | 0.093 | 0.238 | 0.099 | 2.385 | 3.624 | 0.173 | 0.115 | 0.248 | 0.129 | 1.838 | 0.844 | 0.295 | 0.121 | 0.348 | 0.089
15 | 9.81804 | 0.28173 | 4.05 | 3.188 | 0.076 | 0.087 | 0.216 | 0.121 | 4.278 | 4.489 | 0.19 | 0.081 | 0.236 | 0.085 | 2.494 | 1.66 | 0.228 | 0.099 | 0.272 | 0.112
20 | 7.233 | 0.19737 | 6.012 | 3.684 | 0.131 | 0.059 | 0.252 | 0.112 | 6.38 | 4.123 | 0.169 | 0.051 | 0.184 | 0.088 | 3.319 | 2.099 | 0.311 | 0.082 | 0.356 | 0.058
25 | 5.86461 | 0.15321 | 5.715 | 2.884 | 0.224 | 0.068 | 0.227 | 0.109 | 7.671 | 3.788 | 0.164 | 0.068 | 0.223 | 0.109 | 4.882 | 1.523 | 0.326 | 0.082 | 0.355 | 0.105
Mean: | 2.736 |  | 0.16 |  | 0.217 |  | 2.952 |  | 0.157 |  | 0.208 |  | 1.864 |  | 0.275 |  | 0.33 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 104.41601 | 3.34565 | 0.237 | 0.172 | 0.187 | 0.112 | 0.282 | 0.117 | 0.226 | 0.301 | 0.273 | 0.102 | 0.314 | 0.111
3 | 73.28769 | 2.33238 | 0.564 | 0.635 | 0.171 | 0.109 | 0.285 | 0.102 | 0.383 | 0.488 | 0.295 | 0.124 | 0.372 | 0.046
4 | 50.076 | 1.55903 | 0.629 | 0.399 | 0.238 | 0.102 | 0.349 | 0.081 | 0.478 | 0.525 | 0.193 | 0.121 | 0.312 | 0.143
5 | 39.78043 | 1.23673 | 1.072 | 1.049 | 0.147 | 0.095 | 0.288 | 0.095 | 0.778 | 0.46 | 0.244 | 0.1 | 0.361 | 0.086
10 | 15.04997 | 0.44698 | 2.032 | 1.438 | 0.125 | 0.125 | 0.332 | 0.076 | 1.97 | 1.447 | 0.298 | 0.119 | 0.36 | 0.087
15 | 9.81804 | 0.28173 | 5.013 | 1.974 | 0.198 | 0.1 | 0.308 | 0.092 | 2.639 | 2.774 | 0.275 | 0.095 | 0.375 | 0.1
20 | 7.233 | 0.19737 | 3.652 | 3.773 | 0.176 | 0.091 | 0.303 | 0.099 | 3.628 | 3.228 | 0.265 | 0.086 | 0.329 | 0.079
25 | 5.86461 | 0.15321 | 5.687 | 2.858 | 0.217 | 0.103 | 0.364 | 0.09 | 6.256 | 2.401 | 0.315 | 0.062 | 0.331 | 0.075
Mean: | 2.361 |  | 0.182 |  | 0.314 |  | 2.045 |  | 0.27 |  | 0.344 |

Table 42: Clustering details with Shuttle Control (normalized)

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 20 | 2000 | 88 | 0.4 | 1.8E+06 | 2000 | 80 | 0.4 | 1.4E+06 | 2000 | 403 | 0.4 | 7.0E+06 | 2000 | 436 | 0.4 | 8.3E+06 | 2000 | 310 | 0.04 | 0.36 | 6.4E+06
3 | 20 | 2000 | 100 | 0.4 | 5.0E+06 | 2000 | 112 | 0.4 | 5.0E+06 | 2000 | 312 | 0.4 | 1.4E+07 | 2000 | 314 | 0.4 | 2.0E+07 | 2000 | 390 | 0.28 | 0.12 | 1.5E+07
4 | 20 | 2000 | 128 | 0.4 | 6.8E+06 | 2000 | 52 | 0.4 | 5.8E+06 | 2000 | 256 | 0.4 | 1.9E+07 | 2000 | 610 | 0.4 | 2.3E+07 | 2000 | 220 | 0.24 | 0.16 | 1.8E+07
5 | 20 | 2000 | 114 | 0.4 | 9.6E+06 | 2000 | 46 | 0.4 | 6.6E+06 | 2000 | 214 | 0.4 | 2.4E+07 | 2000 | 292 | 0.4 | 3.1E+07 | 2000 | 262 | 0.187 | 0.213 | 2.3E+07
10 | 20 | 2000 | 89 | 0.4 | 1.6E+07 | 2000 | 70 | 0.4 | 1.2E+07 | 2000 | 216 | 0.4 | 4.4E+07 | 2000 | 350 | 0.4 | 5.9E+07 | 2000 | 242 | 0.16 | 0.24 | 3.9E+07
15 | 20 | 2000 | 62 | 0.4 | 2.7E+07 | 2000 | 71 | 0.4 | 2.5E+07 | 2000 | 60 | 0.4 | 5.5E+07 | 2000 | 136 | 0.4 | 7.9E+07 | 2000 | 131 | 0.32 | 0.08 | 4.9E+07
20 | 20 | 2000 | 56 | 0.4 | 3.2E+07 | 2000 | 48 | 0.4 | 3.8E+07 | 2000 | 98 | 0.4 | 5.8E+07 | 2000 | 106 | 0.4 | 1.0E+08 | 2000 | 72 | 0.253 | 0.147 | 5.6E+07
25 | 20 | 2000 | 38 | 0.4 | 3.5E+07 | 2000 | 43 | 0.4 | 4.0E+07 | 2000 | 47 | 0.4 | 4.5E+07 | 2000 | 150 | 0.4 | 1.0E+08 | 2000 | 67 | 0.08 | 0.32 | 7.6E+07

###  A.20 EEG Eye State

Dimensions: $m$ = 14980, $n$ = 14.

Description: the data set consists of 14 electroencephalogram (EEG) values for
predicting the corresponding eye state.

Table 43: Summary of the results with EEG Eye State ($\times 10^{8}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 7845.09934 | 8178.13653 | 4.245 | 4.729 | 0.365 | 0.429 | 0.856 | 0.443 | 4.245 | 3.973 | 0.337 | 0.308 | 0.878 | 0.438 | 4.247 | 5.735 | 0.639 | 0.284 | 0.895 | 0.358
3 | 1833.88058 | 1833.87893 | 0.0 | 0.002 | 0.819 | 0.503 | 1.026 | 0.469 | 0.0 | 0.002 | 0.728 | 0.41 | 0.737 | 0.383 | 0.0 | 0.003 | 0.475 | 0.218 | 0.839 | 0.322
4 | 2.23605 | 2.23429 | 0.0 | 0.001 | 0.715 | 0.5 | 0.735 | 0.501 | 0.0 | 0.001 | 0.741 | 0.46 | 0.804 | 0.428 | 0.0 | 0.001 | 0.6 | 0.308 | 0.838 | 0.336
5 | 1.33858 | 1.33704 | 29.907 | 14.651 | 0.595 | 0.308 | 0.777 | 0.427 | -0.0 | 14.651 | 0.534 | 0.436 | 0.749 | 0.476 | -0.0 | 8.974 | 0.527 | 0.227 | 0.89 | 0.354
10 | 0.4531 | 0.45273 | 0.004 | 0.78 | 0.653 | 0.344 | 0.859 | 0.407 | 0.002 | 0.567 | 0.812 | 0.391 | 0.811 | 0.413 | -0.002 | 0.006 | 0.738 | 0.281 | 1.125 | 0.311
15 | 0.34653 | 0.34861 | 0.527 | 0.55 | 0.099 | 0.287 | 0.703 | 0.421 | 0.65 | 0.741 | 0.043 | 0.455 | 0.718 | 0.394 | 0.136 | 0.13 | 0.435 | 0.089 | 0.994 | 0.274
20 | 0.28986 | 0.29254 | 0.856 | 0.689 | 0.155 | 0.057 | 0.943 | 0.443 | 0.72 | 0.668 | 0.048 | 0.361 | 0.812 | 0.416 | 0.035 | 0.115 | 0.484 | 0.071 | 1.241 | 0.257
25 | 0.25989 | 0.26108 | 0.618 | 0.575 | 0.141 | 0.099 | 1.025 | 0.347 | 0.479 | 0.498 | 0.06 | 0.068 | 0.63 | 0.4 | 0.091 | 0.084 | 0.582 | 0.084 | 1.168 | 0.214
Mean: | 4.52 |  | 0.443 |  | 0.865 |  | 0.762 |  | 0.413 |  | 0.767 |  | 0.563 |  | 0.56 |  | 0.999 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 7845.09934 | 8178.13653 | 4.247 | 0.002 | 0.432 | 0.348 | 0.821 | 0.329 | 4.247 | 2.887 | 0.545 | 0.321 | 0.874 | 0.393
3 | 1833.88058 | 1833.87893 | 0.0 | 0.003 | 0.45 | 0.211 | 0.874 | 0.359 | 0.0 | 0.003 | 0.551 | 0.332 | 0.783 | 0.391
4 | 2.23605 | 2.23429 | 0.0 | 0.001 | 0.628 | 0.324 | 1.039 | 0.399 | 0.0 | 0.001 | 0.566 | 0.355 | 0.992 | 0.373
5 | 1.33858 | 1.33704 | -0.0 | 0.0 | 0.375 | 0.119 | 0.712 | 0.343 | -0.0 | 0.0 | 0.508 | 0.328 | 1.007 | 0.36
10 | 0.4531 | 0.45273 | -0.005 | 0.007 | 0.668 | 0.158 | 1.024 | 0.305 | -0.004 | 0.005 | 0.617 | 0.261 | 1.035 | 0.335
15 | 0.34653 | 0.34861 | 0.124 | 0.12 | 0.43 | 0.095 | 1.018 | 0.235 | 0.136 | 0.104 | 0.43 | 0.121 | 1.061 | 0.26
20 | 0.28986 | 0.29254 | 0.038 | 0.048 | 0.466 | 0.093 | 1.258 | 0.273 | 0.027 | 0.055 | 0.458 | 0.077 | 1.409 | 0.27
25 | 0.25989 | 0.26108 | 0.196 | 0.117 | 0.527 | 0.106 | 1.138 | 0.22 | 0.137 | 0.071 | 0.593 | 0.076 | 1.176 | 0.19
Mean: | 0.575 |  | 0.497 |  | 0.985 |  | 0.568 |  | 0.533 |  | 1.042 |

Table 44: Clustering details with EEG Eye State

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 20 | 14979 | 527 | 1.5 | 6.7E+07 | 14979 | 142 | 1.5 | 5.0E+07 | 14979 | 3056 | 1.5 | 3.6E+08 | 14979 | 2710 | 1.5 | 3.8E+08 | 14979 | 2764 | 0.7 | 0.8 | 3.7E+08
3 | 20 | 14979 | 457 | 1.5 | 7.5E+07 | 14979 | 388 | 1.5 | 7.5E+07 | 14979 | 2192 | 1.5 | 4.4E+08 | 14979 | 2342 | 1.5 | 4.6E+08 | 14979 | 1832 | 0.05 | 1.45 | 4.0E+08
4 | 20 | 14979 | 357 | 1.5 | 9.0E+07 | 14979 | 456 | 1.5 | 9.7E+07 | 14979 | 1948 | 1.5 | 5.0E+08 | 14979 | 2646 | 1.5 | 4.9E+08 | 14979 | 2172 | 0.3 | 1.2 | 4.7E+08
5 | 20 | 14979 | 328 | 1.5 | 9.5E+07 | 14979 | 412 | 1.5 | 1.3E+08 | 14979 | 1666 | 1.5 | 5.3E+08 | 14979 | 1112 | 1.5 | 4.9E+08 | 14979 | 1825 | 0.85 | 0.65 | 4.8E+08
10 | 20 | 14979 | 163 | 1.5 | 1.1E+08 | 14979 | 414 | 1.5 | 2.3E+08 | 14979 | 1326 | 1.5 | 5.9E+08 | 14979 | 1090 | 1.5 | 5.9E+08 | 14979 | 1066 | 0.85 | 0.65 | 6.1E+08
15 | 20 | 14979 | 108 | 1.5 | 1.2E+08 | 14979 | 242 | 1.5 | 2.5E+08 | 14979 | 536 | 1.5 | 5.7E+08 | 14979 | 608 | 1.5 | 5.9E+08 | 14979 | 622 | 0.35 | 1.15 | 5.3E+08
20 | 20 | 14979 | 104 | 1.5 | 1.2E+08 | 14979 | 254 | 1.5 | 3.1E+08 | 14979 | 554 | 1.5 | 5.7E+08 | 14979 | 684 | 1.5 | 6.6E+08 | 14979 | 682 | 1.4 | 0.1 | 6.1E+08
25 | 20 | 14979 | 96 | 1.5 | 1.2E+08 | 14979 | 190 | 1.5 | 3.6E+08 | 14979 | 374 | 1.5 | 5.8E+08 | 14979 | 394 | 1.5 | 5.9E+08 | 14979 | 211 | 0.2 | 1.3 | 4.5E+08

###  A.21 EEG Eye State (normalized)

Dimensions: $m$ = 14980, $n$ = 14.

Description: the data set consists of 14 electroencephalogram (EEG) values for
predicting the corresponding eye state. Min-max scaling was used for
normalization of data set values for better clusterization.

Table 45: Summary of the results with EEG Eye State (normalized) ($\times
10^{1}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 1.15267 | 1.2225 | 0.002 | 8.665 | 0.003 | 0.253 | 0.45 | 0.298 | 0.002 | 8.347 | 0.002 | 0.157 | 0.581 | 0.279 | 0.002 | 5.167 | 0.016 | 0.014 | 0.581 | 0.256
3 | 0.82423 | 0.88495 | 0.003 | 11.909 | 0.005 | 0.071 | 0.421 | 0.281 | 5.716 | 9.522 | 0.005 | 0.002 | 0.565 | 0.288 | 0.001 | 9.213 | 0.019 | 0.032 | 0.663 | 0.203
4 | 0.5429 | 0.57032 | 2.576 | 16.367 | 0.008 | 0.002 | 0.635 | 0.271 | 5.15 | 18.456 | 0.006 | 0.276 | 0.502 | 0.32 | 0.001 | 15.698 | 0.027 | 0.034 | 0.697 | 0.172
5 | 0.28952 | 0.44493 | 0.002 | 26.626 | 0.008 | 0.214 | 0.402 | 0.261 | 0.002 | 39.819 | 0.006 | 0.154 | 0.445 | 0.252 | 0.002 | 21.08 | 0.032 | 0.034 | 0.636 | 0.261
10 | 0.10269 | 0.10299 | 0.765 | 0.595 | 0.053 | 0.01 | 0.386 | 0.274 | 0.713 | 0.47 | 0.033 | 0.008 | 0.611 | 0.259 | -0.004 | 0.006 | 0.324 | 0.08 | 0.715 | 0.177
15 | 0.07469 | 0.07482 | 0.221 | 0.406 | 0.151 | 0.154 | 0.703 | 0.252 | 0.207 | 0.536 | 0.056 | 0.127 | 0.588 | 0.277 | 0.025 | 0.048 | 0.513 | 0.15 | 0.789 | 0.132
20 | 0.06125 | 0.06157 | 0.625 | 0.96 | 0.171 | 0.063 | 0.71 | 0.232 | 0.581 | 1.106 | 0.056 | 0.139 | 0.606 | 0.24 | 0.023 | 0.154 | 0.499 | 0.085 | 0.81 | 0.143
25 | 0.05385 | 0.05438 | 0.883 | 0.743 | 0.233 | 0.147 | 0.749 | 0.274 | 0.278 | 0.686 | 0.059 | 0.023 | 0.596 | 0.221 | -0.121 | 0.183 | 0.519 | 0.103 | 0.927 | 0.12
Mean: | 0.635 |  | 0.079 |  | 0.557 |  | 1.581 |  | 0.028 |  | 0.562 |  | -0.009 |  | 0.244 |  | 0.727 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 1.15267 | 1.2225 | 0.002 | 0.001 | 0.014 | 0.035 | 0.624 | 0.226 | 0.002 | 0.001 | 0.014 | 0.019 | 0.667 | 0.181
3 | 0.82423 | 0.88495 | 0.001 | 0.001 | 0.021 | 0.018 | 0.695 | 0.234 | 0.001 | 7.106 | 0.024 | 0.016 | 0.715 | 0.211
4 | 0.5429 | 0.57032 | 0.001 | 0.001 | 0.03 | 0.038 | 0.709 | 0.204 | 0.001 | 11.321 | 0.031 | 0.051 | 0.72 | 0.215
5 | 0.28952 | 0.44493 | 0.002 | 0.0 | 0.046 | 0.04 | 0.662 | 0.205 | 0.002 | 0.0 | 0.031 | 0.04 | 0.66 | 0.239
10 | 0.10269 | 0.10299 | -0.004 | 0.004 | 0.325 | 0.1 | 0.787 | 0.183 | -0.003 | 0.126 | 0.357 | 0.11 | 0.752 | 0.151
15 | 0.07469 | 0.07482 | 0.04 | 0.079 | 0.439 | 0.076 | 0.855 | 0.125 | 0.04 | 0.069 | 0.445 | 0.085 | 0.785 | 0.129
20 | 0.06125 | 0.06157 | 0.2 | 0.146 | 0.501 | 0.058 | 0.828 | 0.123 | 0.109 | 0.155 | 0.516 | 0.104 | 0.909 | 0.098
25 | 0.05385 | 0.05438 | -0.03 | 0.309 | 0.548 | 0.101 | 0.922 | 0.08 | -0.166 | 0.211 | 0.513 | 0.12 | 0.979 | 0.142
Mean: | 0.027 |  | 0.24 |  | 0.76 |  | -0.002 |  | 0.241 |  | 0.773 |

Table 46: Clustering details with EEG Eye State (normalized)

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 30 | 14979 | 326 | 1.0 | 4.4E+07 | 14979 | 379 | 1.0 | 4.6E+07 | 14979 | 1469 | 1.0 | 2.2E+08 | 14979 | 1926 | 1.0 | 2.3E+08 | 14979 | 2078 | 0.233 | 0.767 | 2.2E+08
3 | 30 | 14979 | 211 | 1.0 | 5.3E+07 | 14979 | 260 | 1.0 | 4.1E+07 | 14979 | 1462 | 1.0 | 2.4E+08 | 14979 | 1465 | 1.0 | 2.4E+08 | 14979 | 1376 | 0.533 | 0.467 | 2.3E+08
4 | 30 | 14979 | 228 | 1.0 | 5.1E+07 | 14979 | 214 | 1.0 | 5.5E+07 | 14979 | 1203 | 1.0 | 2.6E+08 | 14979 | 1218 | 1.0 | 2.6E+08 | 14979 | 1332 | 0.933 | 0.067 | 2.5E+08
5 | 30 | 14979 | 160 | 1.0 | 6.3E+07 | 14979 | 202 | 1.0 | 7.3E+07 | 14979 | 816 | 1.0 | 2.8E+08 | 14979 | 912 | 1.0 | 2.8E+08 | 14979 | 716 | 0.733 | 0.267 | 2.6E+08
10 | 30 | 14979 | 81 | 1.0 | 7.4E+07 | 14979 | 214 | 1.0 | 1.0E+08 | 14979 | 347 | 1.0 | 2.9E+08 | 14979 | 443 | 1.0 | 2.9E+08 | 14979 | 372 | 0.5 | 0.5 | 2.5E+08
15 | 30 | 14979 | 102 | 1.0 | 7.9E+07 | 14979 | 190 | 1.0 | 1.5E+08 | 14979 | 240 | 1.0 | 2.9E+08 | 14979 | 379 | 1.0 | 3.2E+08 | 14979 | 186 | 0.333 | 0.667 | 2.7E+08
20 | 30 | 14979 | 66 | 1.0 | 8.2E+07 | 14979 | 181 | 1.0 | 2.0E+08 | 14979 | 229 | 1.0 | 3.3E+08 | 14979 | 248 | 1.0 | 3.2E+08 | 14979 | 202 | 0.733 | 0.267 | 2.7E+08
25 | 30 | 14979 | 41 | 1.0 | 7.6E+07 | 14979 | 153 | 1.0 | 2.4E+08 | 14979 | 202 | 1.0 | 3.2E+08 | 14979 | 198 | 1.0 | 3.2E+08 | 14979 | 220 | 0.933 | 0.067 | 3.1E+08

###  A.22 Pla85900

Dimensions: $m$ = 85900, $n$ = 2.

Description: a data set contains cities coordinates for traveling salesman
problem.

Table 47: Summary of the results with Pla85900 ($\times 10^{15}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 3.74908 | 0.60567 | 0.044 | 0.717 | 0.025 | 0.122 | 0.758 | 0.357 | 1.44 | 0.716 | 0.024 | 0.024 | 0.626 | 0.431 | 0.014 | 0.025 | 0.063 | 0.07 | 0.884 | 0.37
3 | 2.28057 | 0.36429 | 0.029 | 0.028 | 0.422 | 0.43 | 0.869 | 0.453 | 0.028 | 0.036 | 0.351 | 0.532 | 0.817 | 0.488 | 0.029 | 0.027 | 0.488 | 0.231 | 0.802 | 0.39
5 | 1.33972 | 0.21485 | 0.068 | 0.553 | 0.477 | 0.44 | 0.955 | 0.463 | 0.056 | 0.546 | 0.198 | 0.298 | 0.757 | 0.414 | 0.037 | 0.022 | 0.359 | 0.19 | 0.992 | 0.386
10 | 0.68294 | 0.10939 | 0.536 | 0.308 | 0.587 | 0.384 | 0.712 | 0.414 | 0.542 | 0.292 | 0.434 | 0.438 | 0.848 | 0.41 | 0.122 | 0.156 | 0.538 | 0.309 | 0.99 | 0.306
15 | 0.46029 | 0.07374 | 0.363 | 0.591 | 0.555 | 0.376 | 0.819 | 0.395 | 0.326 | 0.534 | 0.558 | 0.342 | 0.793 | 0.441 | 0.217 | 0.127 | 0.556 | 0.236 | 1.148 | 0.328
20 | 0.34988 | 0.056 | 0.374 | 0.504 | 0.797 | 0.405 | 1.087 | 0.392 | 0.419 | 0.418 | 0.633 | 0.455 | 0.92 | 0.412 | 0.321 | 0.225 | 0.611 | 0.33 | 1.099 | 0.381
25 | 0.28259 | 0.04538 | 0.843 | 0.39 | 0.534 | 0.42 | 0.866 | 0.4 | 0.778 | 0.313 | 0.346 | 0.38 | 0.801 | 0.467 | 0.563 | 0.326 | 0.568 | 0.223 | 1.105 | 0.306
Mean: | 0.322 |  | 0.485 |  | 0.866 |  | 0.513 |  | 0.363 |  | 0.794 |  | 0.186 |  | 0.455 |  | 1.003 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 3.74908 | 0.60567 | 0.012 | 0.012 | 0.054 | 0.053 | 0.81 | 0.376 | 0.015 | 0.014 | 0.093 | 0.072 | 0.986 | 0.371
3 | 2.28057 | 0.36429 | 0.024 | 0.026 | 0.467 | 0.248 | 0.831 | 0.375 | 0.023 | 0.029 | 0.505 | 0.232 | 0.981 | 0.373
5 | 1.33972 | 0.21485 | 0.045 | 0.277 | 0.292 | 0.099 | 0.844 | 0.372 | 0.033 | 0.023 | 0.397 | 0.217 | 1.167 | 0.294
10 | 0.68294 | 0.10939 | 0.316 | 0.298 | 0.527 | 0.275 | 1.049 | 0.31 | 0.117 | 0.13 | 0.619 | 0.311 | 1.202 | 0.295
15 | 0.46029 | 0.07374 | 0.241 | 0.172 | 0.467 | 0.167 | 0.923 | 0.316 | 0.258 | 0.111 | 0.583 | 0.202 | 0.922 | 0.301
20 | 0.34988 | 0.056 | 0.371 | 0.27 | 0.486 | 0.231 | 0.889 | 0.374 | 0.315 | 0.23 | 0.659 | 0.309 | 1.183 | 0.338
25 | 0.28259 | 0.04538 | 0.682 | 0.339 | 0.451 | 0.142 | 1.033 | 0.365 | 0.577 | 0.288 | 0.568 | 0.209 | 1.103 | 0.328
Mean: | 0.242 |  | 0.392 |  | 0.911 |  | 0.191 |  | 0.489 |  | 1.078 |

Table 48: Clustering details with Pla85900

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 40 | 14000 | 267 | 1.5 | 5.1E+07 | 14000 | 205 | 1.5 | 4.4E+07 | 14000 | 1844 | 1.5 | 3.0E+08 | 14000 | 1482 | 1.5 | 3.1E+08 | 14000 | 2054 | 1.0 | 0.5 | 3.0E+08
3 | 40 | 14000 | 306 | 1.5 | 9.0E+07 | 14000 | 258 | 1.5 | 8.3E+07 | 14000 | 1319 | 1.5 | 5.5E+08 | 14000 | 1303 | 1.5 | 5.3E+08 | 14000 | 1667 | 1.4 | 0.1 | 5.2E+08
5 | 40 | 14000 | 276 | 1.5 | 1.4E+08 | 14000 | 196 | 1.5 | 1.2E+08 | 14000 | 1413 | 1.5 | 8.0E+08 | 14000 | 1188 | 1.5 | 7.9E+08 | 14000 | 1744 | 1.05 | 0.45 | 7.7E+08
10 | 40 | 14000 | 108 | 1.5 | 2.6E+08 | 14000 | 144 | 1.5 | 2.7E+08 | 14000 | 853 | 1.5 | 1.5E+09 | 14000 | 882 | 1.5 | 1.5E+09 | 14000 | 1014 | 1.2 | 0.3 | 1.5E+09
15 | 40 | 14000 | 94 | 1.5 | 3.7E+08 | 14000 | 130 | 1.5 | 5.4E+08 | 14000 | 620 | 1.5 | 2.0E+09 | 14000 | 470 | 1.5 | 1.9E+09 | 14000 | 466 | 0.6 | 0.9 | 1.9E+09
20 | 40 | 14000 | 84 | 1.5 | 4.3E+08 | 14000 | 112 | 1.5 | 6.7E+08 | 14000 | 408 | 1.5 | 2.2E+09 | 14000 | 322 | 1.5 | 2.3E+09 | 14000 | 450 | 1.4 | 0.1 | 2.3E+09
25 | 40 | 14000 | 48 | 1.5 | 4.6E+08 | 14000 | 110 | 1.5 | 9.3E+08 | 14000 | 292 | 1.5 | 2.5E+09 | 14000 | 297 | 1.5 | 2.5E+09 | 14000 | 270 | 1.15 | 0.35 | 2.4E+09

###  A.23 D15112

Dimensions: $m$ = 15112, $n$ = 2.

Description: a data set with German cities coordinates for travelling salesman
problem.

Table 49: Summary of the results with D15112 ($\times 10^{11}$)

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-sequential | Big-means-inner | Big-means-competitive
---|---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std | med | std
2 | 3.68403 | 1.91281 | 0.01 | 0.014 | 0.746 | 0.405 | 0.725 | 0.387 | 0.018 | 0.017 | 0.639 | 0.39 | 0.59 | 0.432 | 0.015 | 0.009 | 0.456 | 0.249 | 0.569 | 0.359
3 | 2.5324 | 1.30806 | 0.027 | 0.024 | 0.623 | 0.383 | 0.626 | 0.392 | 0.039 | 0.026 | 1.078 | 0.369 | 0.969 | 0.363 | 0.037 | 0.021 | 0.504 | 0.208 | 0.959 | 0.193
5 | 1.32707 | 0.68685 | 0.039 | 0.015 | 0.325 | 0.447 | 0.355 | 0.506 | 0.039 | 5.665 | 0.708 | 0.361 | 0.931 | 0.389 | 0.036 | 0.027 | 0.417 | 0.214 | 0.789 | 0.4
10 | 0.64491 | 0.33634 | 0.749 | 1.286 | 0.18 | 0.348 | 0.604 | 0.386 | 0.158 | 1.277 | 0.099 | 0.27 | 0.692 | 0.406 | 0.097 | 0.172 | 0.289 | 0.088 | 1.091 | 0.412
15 | 0.43136 | 0.22585 | 0.711 | 0.886 | 0.194 | 0.39 | 0.527 | 0.425 | 1.26 | 1.078 | 0.032 | 0.356 | 0.826 | 0.455 | 0.169 | 0.091 | 0.153 | 0.075 | 1.102 | 0.44
20 | 0.32177 | 0.16883 | 0.994 | 0.651 | 0.163 | 0.421 | 0.813 | 0.434 | 1.39 | 0.841 | 0.36 | 0.335 | 0.729 | 0.466 | 0.455 | 0.394 | 0.293 | 0.084 | 1.007 | 0.366
25 | 0.25308 | 0.1323 | 1.078 | 1.176 | 0.493 | 0.542 | 0.978 | 0.354 | 0.933 | 0.265 | 0.262 | 0.37 | 0.916 | 0.407 | 0.296 | 0.333 | 0.488 | 0.201 | 0.895 | 0.374
Mean: | 0.515 |  | 0.389 |  | 0.661 |  | 0.548 |  | 0.454 |  | 0.808 |  | 0.158 |  | 0.371 |  | 0.916 |

$k$ | $f^{*}$ | $\overline{f}$ | Big-means-collective | Big-means-hybrid
---|---|---|---|---
$\varepsilon$ | $\overline{t}$ | $t$ | $\varepsilon$ | $\overline{t}$ | $t$
med | std | med | std | med | std | med | std | med | std | med | std
2 | 3.68403 | 1.91281 | 0.033 | 0.019 | 0.486 | 0.282 | 0.808 | 0.317 | 0.024 | 0.014 | 0.495 | 0.217 | 0.849 | 0.342
3 | 2.5324 | 1.30806 | 0.026 | 0.017 | 0.403 | 0.196 | 0.46 | 0.339 | 0.024 | 0.039 | 0.53 | 0.224 | 0.997 | 0.35
5 | 1.32707 | 0.68685 | 0.04 | 0.017 | 0.405 | 0.267 | 1.088 | 0.361 | 0.05 | 0.031 | 0.514 | 0.182 | 1.005 | 0.332
10 | 0.64491 | 0.33634 | 0.132 | 0.389 | 0.202 | 0.092 | 1.047 | 0.269 | 0.117 | 0.148 | 0.244 | 0.162 | 0.929 | 0.287
15 | 0.43136 | 0.22585 | 0.21 | 0.323 | 0.136 | 0.092 | 0.912 | 0.308 | 0.188 | 0.099 | 0.173 | 0.077 | 0.962 | 0.389
20 | 0.32177 | 0.16883 | 1.015 | 0.602 | 0.33 | 0.339 | 0.952 | 0.274 | 0.431 | 0.28 | 0.336 | 0.127 | 0.847 | 0.375
25 | 0.25308 | 0.1323 | 0.698 | 0.397 | 0.397 | 0.324 | 0.973 | 0.333 | 0.427 | 0.357 | 0.369 | 0.287 | 1.16 | 0.397
Mean: | 0.308 |  | 0.337 |  | 0.891 |  | 0.18 |  | 0.38 |  | 0.964 |

Table 50: Clustering details with D15112

$k$ | $n_{exec}$ | Big-means-sequential | Big-means-inner | Big-means-competitive | Big-means-collective | Big-means-hybrid
---|---|---|---|---|---|---
$s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}$ | $n_{d}$ | $s$ | $n_{s}$ | $t_{max}^{1}$ | $t_{max}^{2}$ | $n_{d}$
2 | 15 | 8000 | 1231 | 1.5 | 1.1E+08 | 8000 | 701 | 1.5 | 8.3E+07 | 8000 | 3677 | 1.5 | 4.6E+08 | 8000 | 6205 | 1.5 | 4.7E+08 | 8000 | 6527 | 0.95 | 0.55 | 6.3E+08
3 | 15 | 8000 | 815 | 1.5 | 1.7E+08 | 8000 | 991 | 1.5 | 1.3E+08 | 8000 | 6558 | 1.5 | 8.4E+08 | 8000 | 1688 | 1.5 | 8.2E+08 | 8000 | 6256 | 0.7 | 0.8 | 9.4E+08
5 | 15 | 8000 | 398 | 1.5 | 2.2E+08 | 8000 | 648 | 1.5 | 1.5E+08 | 8000 | 3757 | 1.5 | 1.2E+09 | 8000 | 5951 | 1.5 | 1.2E+09 | 8000 | 5367 | 0.7 | 0.8 | 1.3E+09
10 | 15 | 8000 | 373 | 1.5 | 3.9E+08 | 8000 | 546 | 1.5 | 5.3E+08 | 8000 | 2715 | 1.5 | 1.9E+09 | 8000 | 2842 | 1.5 | 2.0E+09 | 8000 | 2390 | 0.9 | 0.6 | 2.0E+09
15 | 15 | 8000 | 164 | 1.5 | 4.3E+08 | 8000 | 568 | 1.5 | 7.8E+08 | 8000 | 1938 | 1.5 | 2.4E+09 | 8000 | 1352 | 1.5 | 2.3E+09 | 8000 | 1613 | 1.35 | 0.15 | 2.2E+09
20 | 15 | 8000 | 192 | 1.5 | 5.1E+08 | 8000 | 358 | 1.5 | 9.8E+08 | 8000 | 1176 | 1.5 | 2.6E+09 | 8000 | 1056 | 1.5 | 2.7E+09 | 8000 | 834 | 1.35 | 0.15 | 2.6E+09
25 | 15 | 8000 | 175 | 1.5 | 5.2E+08 | 8000 | 386 | 1.5 | 1.2E+09 | 8000 | 681 | 1.5 | 2.8E+09 | 8000 | 884 | 1.5 | 2.8E+09 | 8000 | 1033 | 0.85 | 0.65 | 2.7E+09

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$\cdot 10^{8}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (a) CORD-19 Embeddings

$0$$5$$10$$15$$20$$25$$0$$0.5$$1$$1.5$$2$$2.5$$\cdot 10^{9}$No of clustersNo
of dist. func. eval.Big-means-sequentialBig-means-innerBig-means-
competitiveBig-means-collectiveBig-means-hybrid (b) HEPMASS

$0$$5$$10$$15$$20$$25$$0$$0.5$$1$$1.5$$2$$\cdot 10^{8}$No of clustersNo of
dist. func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-
means-collectiveBig-means-hybrid (c) US Census Data 1990

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$4$$5$$\cdot 10^{7}$No of clustersNo of
dist. func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-
means-collectiveBig-means-hybrid (d) Gisette

$0$$5$$10$$15$$20$$25$$0.2$$0.4$$0.6$$0.8$$1$$\cdot 10^{8}$No of clustersNo of
dist. func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-
means-collectiveBig-means-hybrid (e) Music Analysis

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$4$$5$$\cdot 10^{8}$No of clustersNo of
dist. func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-
means-collectiveBig-means-hybrid (f) Protein Homology

$0$$5$$10$$15$$20$$25$$0$$0.2$$0.4$$0.6$$0.8$$1$$\cdot 10^{9}$No of clustersNo
of dist. func. eval.Big-means-sequentialBig-means-innerBig-means-
competitiveBig-means-collectiveBig-means-hybrid (g) MiniBooNE Particle
Identification

$0$$5$$10$$15$$20$$25$$0$$0.2$$0.4$$0.6$$0.8$$1$$\cdot 10^{8}$No of clustersNo
of dist. func. eval.Big-means-sequentialBig-means-innerBig-means-
competitiveBig-means-collectiveBig-means-hybrid (h) MiniBooNE Particle
Identification (normalized)

$0$$5$$10$$15$$20$$25$$2$$4$$6$$8$$\cdot 10^{7}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (i) MFCCs for Speech Emotion Recognition

Figure 5: Distance function evaluations. Set 1

$0$$5$$10$$15$$20$$25$$1$$2$$3$$4$$5$$\cdot 10^{7}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (a) ISOLET

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$4$$\cdot 10^{8}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (b) Sensorless Drive Diagnosis

$0$$5$$10$$15$$20$$25$$0$$0.5$$1$$1.5$$2$$2.5$$\cdot 10^{7}$No of clustersNo
of dist. func. eval.Big-means-sequentialBig-means-innerBig-means-
competitiveBig-means-collectiveBig-means-hybrid (c) Sensorless Drive Diagnosis
(normalized)

$0$$5$$10$$15$$20$$25$$1$$2$$3$$4$$5$$\cdot 10^{7}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (d) Online News Popularity

$0$$5$$10$$15$$20$$25$$2$$4$$6$$8$$\cdot 10^{7}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (e) Gas Sensor Array Drift

$0$$5$$10$$15$$20$$25$$0$$0.5$$1$$\cdot 10^{9}$No of clustersNo of dist. func.
eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (f) 3D Road Network

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$4$$\cdot 10^{7}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (g) Skin Segmentation

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$\cdot 10^{8}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (h) KEGG Metabolic Relation Network (Directed)

$0$$5$$10$$15$$20$$25$$2$$4$$6$$8$$\cdot 10^{8}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (i) Shuttle Control

Figure 6: Distance function evaluations. Set 2

$0$$5$$10$$15$$20$$25$$0$$0.2$$0.4$$0.6$$0.8$$1$$\cdot 10^{8}$No of clustersNo
of dist. func. eval.Big-means-sequentialBig-means-innerBig-means-
competitiveBig-means-collectiveBig-means-hybrid (a) Shuttle Control
(normalized)

$0$$5$$10$$15$$20$$25$$0$$2$$4$$6$$\cdot 10^{8}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (b) EEG Eye State

$0$$5$$10$$15$$20$$25$$1$$2$$3$$\cdot 10^{8}$No of clustersNo of dist. func.
eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (c) EEG Eye State (normalized)

$0$$5$$10$$15$$20$$25$$0$$0.5$$1$$1.5$$2$$2.5$$\cdot 10^{9}$No of clustersNo
of dist. func. eval.Big-means-sequentialBig-means-innerBig-means-
competitiveBig-means-collectiveBig-means-hybrid (d) Pla85900

$0$$5$$10$$15$$20$$25$$0$$1$$2$$3$$\cdot 10^{9}$No of clustersNo of dist.
func. eval.Big-means-sequentialBig-means-innerBig-means-competitiveBig-means-
collectiveBig-means-hybrid (e) D15112

Figure 7: Distance function evaluations. Set 3