# Photon and dilepton emission anisotropy for a magnetized quark-gluon plasma

Xinyang Wang<EMAIL_ADDRESS>Center for Fundamental Physics, School of
Mechanics and Physics, Anhui University of Science and Technology, Huainan,
Anhui 232001, People’s Republic of China  Department of Physics, Jiangsu
University, Zhenjiang 212013, People’s Republic of China  School of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, People’s Republic of China  Igor A. Shovkovy
<EMAIL_ADDRESS>School of Applied Sciences and Arts, Arizona State
University, Mesa, Arizona 85212, USA  Department of Physics, Arizona State
University, Tempe, Arizona 85287, USA

(March 8, 2024)

###### Abstract

We study the higher-order anisotropy coefficients $v_{4}$ and $v_{6}$ in the
photon and dilepton emission from a hot magnetized quark-gluon plasma.
Together with the earlier predictions for $v_{2}$, these results show a
distinctive pattern of the anisotropy coefficients in several kinematic
regimes. In the case of photon emission, nonzero coefficients $v_{n}$ (with
even $n$) have opposite signs at small and large values of the transverse
momentum (i.e., $k_{T}\lesssim\sqrt{|eB|}$ and $k_{T}\gtrsim\sqrt{|eB|}$,
respectively). Additionally, the $v_{n}$ signs alternate with increasing $n$,
and their approximate values decrease as $1/n^{2}$ in magnitude. The
anisotropy of dilepton emission is well pronounced only at large transverse
momenta and small invariant masses (i.e., when $k_{T}\gtrsim\sqrt{|eB|}$ and
$M\lesssim\sqrt{|eB|}$). The corresponding $v_{4}$ and $v_{6}$ coefficients
are of the same magnitude and show a similar alternating sign pattern with
increasing $n$ as in the photon emission.

##  I Introduction

Quark-gluon plasma (QGP) is a state of extremely hot matter made of deconfined
quarks and gluons that carry non-Abelian color charges [1, 2, 3]. The
existence of such a plasma state stems from the asymptotic freedom in quantum
chromodynamics (QCD) [4, 5]. QGP was present naturally in the early Universe
about a microsecond after the Big Bang. It can also be produced in heavy-ion
collisions at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven and the
Large Hadron Collider (LHC) at CERN. The corresponding “Little Bang”
experiments allow one to study the fundamental properties of QGP [6].

Despite small sizes and short interaction times in relativistic collisions,
experimental data provide a strong evidence that the QGP forms a strongly
interacting viscous liquid [7, 8, 9]. The flow measurements, quantified by the
anisotropy coefficients $v_{n}$, support the scenario of QGP evolving
hydrodynamically for a considerable fraction of its lifetime [10]. Theoretical
models also indicate that the plasma has low viscosity [11], consistent with a
strongly interacting regime.

The dynamics responsible for the QGP production in heavy-ion collisions are
complicated and only partially understood. One of the aspects in dire need of
better understanding is the possible generation and evolution of background
magnetic fields in noncentral collisions. Theoretical studies suggest that the
initial magnetic field $B$ could be of the order of $m_{\pi}^{2}/e\approx
3\times 10^{18}~{}G$ [12, 13, 14, 15, 16, 17]. Such an incredibly strong field
could modify the thermodynamic and transport properties of QGP, trigger chiral
anomalous effects (CME) [18, 19, 20], and ultimately affect numerous
observables. For reviews, see Refs. [21, 22, 23, 24].

To verify whether the QGP in noncentral collisions is magnetized and to
estimate the strength of the magnetic field, one can try scrutinizing the most
promising electromagnetic observables. It is reasonable to start by analyzing
the photon [25, 26, 27] and dilepton emission rates [28]. First, the magnetic
field affects the corresponding rates already at leading order in coupling.
Second, the photons and dileptons are clean probes of the QGP at early times.
Indeed, owing to their long mean-free path, they do not suffer much from
rescattering in a small volume of the plasma.

The heavy-ion experiments reveal that direct photons have a sizable elliptic
flow, quantified by a large ellipticity coefficient $v_{2}$ [29, 30, 31].
Their flow appears to be comparable to that of hadrons, which is truly
surprising. Unlike hadrons, the direct photons are emitted at early times of
QGP when collective flow may not have had the chance to form yet. This is
known as the “direct photon” puzzle. Many theoretical studies tried to address
it [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53]. In our detailed studies of the differential rates in Refs.
[54, 55], in particular, we argued that a large positive $v_{2}$ of the direct
photons may be explained by the presence of a strong background magnetic field
in the QGP. It is fair to note that further phenomenological investigations
are needed to settle the issue. This study is one of the key steps in that
direction. It extends the knowledge of the differential emission rates from a
strongly magnetized plasma.

The dilepton emission is another complementary probe of the QGP. Since their
spectra are not affected by the blueshift of the expanding medium, dileptons
can serve as an excellent thermometer of the QGP [56]. On the other hand, the
dilepton rate should be affected by the magnetic field [57, 58, 59, 60, 61,
62, 63, 64, 65]. Moreover, as we demonstrated in the earlier study [66],
dilepton emission is characterized by a sizable ellipticity at small values of
the invariant mass ($M\lesssim\sqrt{|eB|}$). In the same kinematic region, the
rate is also strongly enhanced. It is fair to mention that the corresponding
theoretical claims may be hard to verify systematically in current
experiments.

Here, we extend the previous studies by showing that the presence of a strong
magnetic field in the QGP should be encoded not only in $v_{2}$, but also in
high-order anisotropy coefficients. By using the same theoretical framework as
in Refs. [54, 55, 66], here we obtain detailed theoretical predictions for the
higher-order anisotropy coefficients $v_{4}$ and $v_{6}$ for a magnetized
plasma at rest. Similarly to $v_{2}$, they show nontrivial dependence on the
kinematic parameters. We argue that future detailed measurements of the photon
and dilepton anisotropy coefficients could provide a distinctive fingerprint
for verifying the presence of the background magnetic field in the plasma
produced by noncentral heavy-ion collisions. Of course, to produce theoretical
predictions for the corresponding heavy-ion observables, one needs to
convolute the differential rates with the specific dynamical models of plasma.
The latter task is left for future studies.

This paper is organized as follows. In Sec. II, we introduce the key
definitions and model assumptions in our study of the photon and dilepton
emission from a hot magnetized QGP. The numerical results for higher-order
anisotropy coefficients $v_{4}$ and $v_{6}$ are obtained and discussed in Sec.
III. The summary of the main findings and conclusions are given in Sec. IV. In
the Appendix, we quote the expression for the imaginary part of the Lorentz-
contracted polarization tensor, which is needed for calculating the photon and
dilepton rates.

##  II Model

Here we make the same model assumptions about the QGP as in Refs. [54, 55,
66]. We consider a plasma made of the lightest up and down quarks. While the
quantitative results may change slightly with the inclusion of the strange
quarks, all qualitative results are to remain the same. For simplicity, we
also assume that the masses of the up and down quarks are equal, i.e.,
$m_{u}=m_{d}=m=5~{}\mbox{MeV}$. It is a good approximation for the QGP with a
temperature of several hundred megaelectronvolts.

In this study, we consider the QGP plasma in the rest frame. By assumption,
the magnetic field points along the $z$ axis. The two setups for photon and
dilepton emission are illustrated schematically in the two panels in Fig. 1.

In the case of photon emission, the corresponding four-momentum
$k^{\mu}=(\Omega,\mathbf{k})$ satisfies the on-shell condition $k^{2}\equiv
k_{\mu}k^{\mu}=0$. In the dilepton case, on the other hand, the photon
$\gamma^{*}$ is virtual. Its momentum describes a lepton pair and satisfies a
different on-shell condition, i.e., $k^{2}=M^{2}$, where $M$ is the dilepton
invariant mass. Note that, without loss of generality, we can set $k_{x}=0$ in
the rest frame. The nonzero transverse components of the momentum are

$k_{y}=k_{T}\cos(\phi),\qquad k_{z}=k_{T}\sin(\phi),$ (1)

where $k_{T}=\sqrt{k_{y}^{2}+k_{z}^{2}}$ is the magnitude of the transverse
momentum and $\phi$ is the azimuthal angle measured from the $y$ axis. (The
transverse component of the momentum $k_{T}$ should not be confused with
$k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$, which is the component perpendicular
to the magnetic field.)

###  II.1 Photon emission rate and its anisotropy

The thermal photon production rate from the QGP can be conveniently expressed
in terms of the imaginary part of the retarded polarization tensor as follows
[67]:

$k^{0}\frac{d^{3}R}{dk_{x}dk_{y}dk_{z}}=\frac{d^{3}R}{k_{T}dk_{T}d\phi
dy}=-\frac{n_{B}(\Omega)}{(2\pi)^{3}}\mbox{Im}\left[\Pi^{\mu}_{R,\mu}\left(\Omega,\mathbf{k}\right)\right],$
(2)

where $n_{B}(\Omega)=1/\left[\exp\left(\Omega/T\right)-1\right]$ is the Bose-
Einstein distribution function and $T$ is the temperature. The latter
expression has the same form in QGP with and without a background field.
However, a nonzero magnetic field can strongly affect the photon polarization
tensor and, in turn, modify the photon emission rate. Below, we will utilize
the leading-order one-loop expression for
$\mbox{Im}\left[\Pi^{\mu}_{R,\mu}\left(\Omega,\mathbf{k}\right)\right]$
derived in Refs. [54, 68, 66] by using the Landau-level representation for
quarks. For convenience, we also quote the corresponding result in the
Appendix.

When the differential rate (2) is known, the anisotropy coefficients $v_{n}$
can be evaluated as follows:

$v_{n}(k_{T})=\frac{1}{\mathcal{R}_{0}}\int_{0}^{2\pi}\frac{d^{3}R}{k_{T}dk_{T}d\phi
dy}\cos(n\phi)d\phi,$ (3)

where the normalization factor is determined by integrating the emission rate
over the azimuthal angle $\phi$, i.e.,

$\mathcal{R}_{0}=\frac{d^{2}R}{k_{T}dk_{T}dy}=\int_{0}^{2\pi}\frac{d^{3}R}{k_{T}dk_{T}dyd\phi}d\phi.$
(4)

We will use the definition in Eq. (3) to quantify the anisotropy of the photon
emission from a hot magnetized QGP in Sec. III.

It is appropriate to comment on the approximation used here. When utilizing
the one-loop polarization tensor in Eq. (2), one accounts for the following
three leading-order processes: (i) the quark splitting ($q\rightarrow
q+\gamma$), (ii) the antiquark splitting ($\bar{q}\rightarrow\bar{q}+\gamma$),
and (iii) the quark-antiquark annihilation ($q+\bar{q}\rightarrow\gamma$) [54,
68]. Their contributions to the rate are of the order of $\alpha$, where
$\alpha\equiv e^{2}/(4\pi)=1/137$ is the fine structure constant. Recall that
the same processes are forbidden by energy-momentum conservation in the
absence of the magnetic field. Instead, leading contributions at $B=0$ come
from the gluon-mediated $2\to 2$ processes $q+g\rightarrow q+\gamma$,
$\bar{q}+g\rightarrow\bar{q}+\gamma$, and $q+\bar{q}\rightarrow g+\gamma$,
where $g$ represents a gluon [69, 70, 71, 72, 73, 74, 75]. Formally, they are
suppressed by an extra power of $\alpha_{s}$, where $\alpha_{s}\equiv
g_{s}^{2}/(4\pi)$ is the QCD strong coupling constant.

Unfortunately, the gluon-mediated $2\to 2$ processes have not been analyzed in
a magnetic field. Thus, it is unclear how the relative contributions of the
leading and subleading diagrams vary when one goes continuously from the zero-
field to the strong-field limit. Here we will assume that the magnetic field
is sufficiently strong for the leading-order contributions $\sim\alpha$ (from
the $1\to 2$ and $2\to 1$ processes) to dominate the anisotropy coefficients.
It can be true even in some cases when the subleading contributions
$\sim\alpha\alpha_{s}$ (from the gluon-mediated $2\to 2$ processes) dominate
the rates. With the current knowledge, however, we cannot establish a rigorous
range of validity for the approximation used. It is an important issue and
should be addressed in detail in future studies.

Figure 1: Schematic illustrations of the photon (a) and dilepton (b) emission
from a magnetized plasma in the rest frame. The transverse momenta $k_{T}$ of
the on-shell (a) and virtual (b) photons lie in the $y$-$z$ plane. The
azimuthal angle $\phi$ is measured from the $y$ axis. The lepton momenta
outside the magnetized plasma are $q_{1}$ and $q_{2}$ (b).

###  II.2 Dilepton emission rate and its anisotropy

Similarly to the photon emission, the differential dilepton production rate
can be expressed in terms of the imaginary part of the photon polarization
tensor, i.e.,

$\frac{dR_{l\bar{l}}}{d^{4}k}=\frac{\alpha}{12\pi^{4}}\frac{n_{B}\left(\Omega\right)}{M^{2}}\mbox{Im}\left[\Pi^{\mu}_{R,\mu}\left(\Omega,\mathbf{k}\right)\right],$
(5)

where $n_{B}(\Omega)=(e^{\Omega/T}-1)^{-1}$ is the Bose-Einstein distribution
function. Here we neglected the nonzero lepton masses and took into account
that $k^{2}\equiv\Omega^{2}-k_{\perp}^{2}-k_{z}^{2}=M^{2}$. Note that
$\Omega=\sqrt{M^{2}+k_{\perp}^{2}+k_{z}^{2}}$ and
$d^{4}K=MdMk_{T}dk_{T}dyd\phi$.

To quantify the anisotropy of dilepton emission in Sec. III, we will use the
Fourier coefficients similar to those in Eq. (3), i.e.,

$v_{n}(M,k_{T})=\frac{\int_{0}^{2\pi}d\phi\cos(n\phi)\left(dR_{l\bar{l}}/d^{4}k\right)}{\int_{0}^{2\pi}d\phi\left(dR_{l\bar{l}}/d^{4}k\right)}.$
(6)

It is instructive to emphasize that the approximation for the dilepton rate in
Eq. (5), given in terms of the one-loop photon polarization tensor, is
comparable to the leading-order result in the case of the vanishing magnetic
field [76]. Moreover, as shown in Ref. [66], it reduces to the zero-field Born
rate when the magnetic field goes to zero. Therefore, unlike the photon
emission, the leading-order dilepton emission is under theoretical control in
the whole range from the vanishing to strong magnetic fields.

##  III Results

To extend our previous studies of the photon and dilepton emission rates in
Refs. [54, 66], here we analyze the emission anisotropies in more detail. In
particular, we study the higher-order coefficients $v_{n}$, as defined by Eqs.
(3) and (6). Note that all odd coefficients $v_{3}$, $v_{5}$, etc. are
vanishing in a magnetized plasma. It is the consequence of the rotation
symmetry about the direction of the magnetic field. Here we will investigate
the effect of the magnetic field on the high-order anisotropy coefficients
$v_{4}$ and $v_{6}$. Note that the leading coefficient $v_{2}$, which measures
the ellipticity of emission, was investigated in detail in Refs. [54, 66]. By
scrutinizing the angular dependence of the emission rates below, we will argue
that such higher correlations hold interesting features that may become
invaluable in quantifying the properties of the QGP produced in noncentral
heavy-ion collisions.

In numerical calculations, we express all mass and energy quantities in units
of the (neutral) pion mass, $m_{\pi}\approx 0.135~{}\mbox{GeV}$. When
presenting the results, however, we will display the transverse momenta and
the dilepton invariant masses in gigaelectronvolts. To cover a substantial
range of the parameter space without producing an overwhelming amount of data
for the anisotropy coefficients, we will concentrate on the two representative
choices of the magnetic field strength, $|eB|=m_{\pi}^{2}$ and
$|eB|=5m_{\pi}^{2}$, and two representative values of temperature,
$T=0.2~{}\mbox{GeV}$ and $T=0.35~{}\mbox{GeV}$, which are typical under the
conditions in high-energy heavy-ion collisions [77, 78, 79].

As explained in Refs. [54, 66], the problem possesses a mirror symmetry with
respect to the reflection in the reaction plane. Thus, the rates remain
invariant when $\phi\to-\phi$. Taking into account also the parity symmetry
($y\to-y$), we see that the rate for the whole range of azimuthal angles from
$\phi=-\pi$ and $\phi=\pi$ can be obtained from that in the range between
$\phi=0$ and $\phi=\frac{\pi}{2}$.

###  III.1 Photon emission

Our earlier study in Ref. [54] showed that the photon emission from a
magnetized hot QGP has a well-pronounced ellipticity characterized by a
nonzero $v_{2}$. Moreover, its sign changes as some intermediate values of the
transverse momentum. It is predominantly negative at small $k_{T}$ (i.e.,
$k_{T}\lesssim\sqrt{|eB|}$) and positive at large $k_{T}$ (i.e.,
$k_{T}\gtrsim\sqrt{|eB|}$). Here we extend the study to the higher-order
anisotropy coefficients $v_{4}$ and $v_{6}$. As we will see, they also deviate
substantially from zero and show characteristic patterns of dependence on the
transverse momentum.

The numerical results for $v_{4}$ in the photon emission are shown in Fig. 2.
The left and right panels display the results for two different magnetic
fields, $|eB|=m_{\pi}^{2}$ and $|eB|=5m_{\pi}^{2}$, respectively. In both
cases, the blue solid and the red dashed lines correspond to two fixed
temperatures, i.e., $T=0.2~{}\mbox{GeV}$ and $T=0.35~{}\mbox{GeV}$,
respectively.

Figure 2: Anisotropic coefficient $v_{4}$ for the photon emission as a
function of the transverse momentum $k_{T}$ for two different temperatures,
$T=0.2~{}\mbox{GeV}$ (blue line) and $T=0.35~{}\mbox{GeV}$ (red line), and two
different strengths of the magnetic field, $|eB|=m_{\pi}^{2}$ (a) and
$|eB|=5m_{\pi}^{2}$ (b).

We should note that the numerical data for $v_{4}$ (as well as other
anisotropy coefficients below) appear to be very noisy, especially at small
values of $k_{T}$. There are several reasons for such a behavior. In part, it
is caused by the highly spiky dependence of the rates on the angular
coordinate in the vicinity of the Landau-level thresholds, which are
particularly strong at small $k_{T}$ [54, 66]. The jagged behavior is further
exacerbated by a finite angular resolution of the numerical data. While some
points happen to lie accidentally at or near sharp peaks, others fall near
local minima.

The results reveal a clear qualitative pattern in the behavior of $v_{4}$ as a
function of the transverse momentum. At relatively small momenta,
$k_{T}\lesssim\sqrt{|eB|}$, $v_{4}$ tends to be positive. However, it becomes
negative for $k_{T}\gtrsim\sqrt{|eB|}$. Notably, its absolute values are of
the order of $0.05$. Such large $v_{4}$ values can be detectable in heavy-ion
collisions if the background contributions due to other effects are under
control.

The numerical results for $v_{6}$ are shown in Fig. 3. As before, the left and
right panels display the results for two different magnetic fields,
$|eB|=m_{\pi}^{2}$ and $|eB|=5m_{\pi}^{2}$, respectively. In both cases, the
blue solid and the red dashed lines correspond to two fixed temperatures,
i.e., $T=0.2~{}\mbox{GeV}$ and $T=0.35~{}\mbox{GeV}$, respectively. In all
cases, the dependence of $v_{6}$ on the transverse momentum reveals similar
qualitative features. It changes from a negative value at relatively small
momenta, $k_{T}\lesssim\sqrt{|eB|}$, to positive values at relatively large
momenta $k_{T}\gtrsim\sqrt{|eB|}$. The absolute values of $v_{6}$ are of the
order of $0.02$, which are quite sizable too.

Figure 3: Anisotropic coefficient $v_{6}$ for the photon emission as a
function of the transverse momentum $k_{T}$ for two different temperatures,
$T=0.2~{}\mbox{GeV}$ (blue line) and $T=0.35~{}\mbox{GeV}$ (red line), and two
different strengths of the magnetic field, $|eB|=m_{\pi}^{2}$ (a) and
$|eB|=5m_{\pi}^{2}$ (b).

The characteristic features of the photon anisotropy are summarized in Table
1. It is interesting to note the alternating signs of the anisotropy
coefficients $v_{n}$ with increasing $n$. (Recall that all coefficients with
odd $n$ vanish.) Another curious feature is the overall scaling of the
magnitude, which goes as $1/n^{2}$. The latter may be an approximate numerical
result that holds only for the lowest three nonzero coefficients. However,
tentatively it appears to remain true also for $v_{8}$, although the data
become less reliable with increasing $n$ when the threshold effects from
Landau levels produce many spikes in the angular dependence.

Table 1: Summary of nonvanishing photon and dilepton anisotropy coefficients $v_{n}$. |  $v_{n}$ (photon emission) |  $v_{n}$ (dilepton emission)
---|---|---
| $k_{T}\lesssim\sqrt{|eB|}$ | $k_{T}\gtrsim\sqrt{|eB|}$ |  $k_{T}\gtrsim\sqrt{|eB|}$ & $M\lesssim\sqrt{|eB|}$
$v_{2}$ |  $\simeq-0.2$111From Ref. [54]. |  $\simeq+0.2$111From Ref. [54]. | $\simeq+0.2$
$v_{4}$ | $\simeq+0.05$ | $\simeq-0.05$ | $\simeq-0.05$
$v_{6}$ | $\simeq-0.02$ | $\simeq+0.02$ | $\simeq+0.02$

###  III.2 Dilepton emission

As demonstrated in Ref. [66], dilepton emission from a magnetized hot QGP
shows a sizable ellipticity, described by a positive $v_{2}$ of the order of
$0.2$, in the kinematic regime of small invariant masses (i.e.,
$M\lesssim\sqrt{|eB|}$) and large transverse momenta (i.e.,
$k_{T}\gtrsim\sqrt{|eB|}$). Here, we analyze the higher-order anisotropy
coefficients $v_{4}$ and $v_{6}$. They also deviate noticeably from zero in
the same kinematic regime.

Let us start by first reinforcing the results for the ellipticity of dilepton
emission obtained in Ref. [66]. In particular, here we extend the previous
calculations of the ellipticity coefficient $v_{2}$ to larger transverse
momenta (up to $k_{T}=2~{}\mbox{GeV}$) and increase the resolution in the
invariant mass (i.e., from $\Delta M=0.1~{}\mbox{GeV}$ down to $\Delta
M=0.01~{}\mbox{GeV}$). The corresponding new results are shown in Fig. 4. The
four panels show the ellipticity coefficient $v_{2}$ as a function of the
invariant mass for two temperatures, i.e., $T=0.2~{}\mbox{GeV}$ (two left
panels) and $T=0.35~{}\mbox{GeV}$ (two right panels), and two magnetic fields
$|eB|=m_{\pi}^{2}$ (two top panels) and $|eB|=5m_{\pi}^{2}$ (two bottom
panels). For reference, we also included one of the older low-resolution
datasets for $k_{T}=0.5~{}\mbox{GeV}$ from Ref. [66].

By comparing the dependence of $v_{2}$ on the invariant mass $M$ with the
results in Ref. [66], we find that the earlier conclusions are not only valid,
but they also become more robust with the increasing of the transverse
momentum. Furthermore, the current high-resolution data reconfirm that the
ellipticity coefficient $v_{2}$ takes generically large positive values
($v_{2}\sim 0.2$) in the region of small invariant masses (i.e.,
$M\lesssim\sqrt{|eB|}$). Its magnitude is comparable to the photon $v_{2}$
calculated in Ref. [54]. By comparing the data for the two different
temperatures in Fig. 4, we also see that the temperature dependence of the
dilepton $v_{2}$ is nearly negligible. For the large transverse momenta
considered, of course, it should not be surprising. As we will see below, both
$v_{4}$ and $v_{6}$ reveal a similarly weak temperature dependence.

Figure 4: Anisotropic coefficient $v_{2}$ for the dilepton emission as a
function of the invariant mass $M$ for several fixed values of the transverse
momentum $k_{T}$. The top two panels correspond to $|eB|=m_{\pi}^{2}$ and the
bottom two panels to $|eB|=5m_{\pi}^{2}$. The panels on the left are for
$T=0.2~{}\mbox{GeV}$, and the ones on the right are for $T=0.35~{}\mbox{GeV}$.
The data for $k_{T}=0.5~{}\mbox{GeV}$ are taken from Ref. [66].

Now, let us turn to the higher-order anisotropy coefficients $v_{4}$ and
$v_{6}$. We will concentrate our attention on the same kinematic region of
small invariant masses and large transverse momenta, where the anisotropy is
pronounced the most. The numerical results for the dilepton $v_{4}$ as a
function of the invariant mass are shown in Fig. 5. As before, the four panels
present the results for two temperatures, i.e., $T=0.2~{}\mbox{GeV}$ (two left
panels) and $T=0.35~{}\mbox{GeV}$ (two right panels), and two magnetic fields
$|eB|=m_{\pi}^{2}$ (two top panels) and $|eB|=5m_{\pi}^{2}$ (two bottom
panels). As we see from Fig. 5, at small invariant masses, the coefficient
$v_{4}$ tends to be negative with the absolute values of about $0.05$. Note
that these are sizable by any reasonable standards. They are also comparable
to the $v_{4}$ values in the photon emission at large transverse momenta, see
Fig. 2.

Figure 5: Anisotropic coefficient $v_{4}$ for the dilepton emission as a
function of the invariant mass $M$ for several fixed values of the transverse
momentum $k_{T}$. The top two panels correspond to $|eB|=m_{\pi}^{2}$ and the
bottom two panels to $|eB|=5m_{\pi}^{2}$. The panels on the left are for
$T=0.2~{}\mbox{GeV}$, and the ones on the right are for $T=0.35~{}\mbox{GeV}$.

The dilepton results for $v_{6}$ are shown in Fig. 6. The four panels show the
results for two temperatures, i.e., $T=0.2~{}\mbox{GeV}$ (two left panels) and
$T=0.35~{}\mbox{GeV}$ (two right panels), and two magnetic fields
$|eB|=m_{\pi}^{2}$ (two top panels) and $|eB|=5m_{\pi}^{2}$ (two bottom
panels). As we see, the anisotropy coefficient $v_{6}$ tends to be positive at
small $M$, with the maximal values of the order of $0.02$. Such values are
comparable to those of photon $v_{6}$ in Fig. 3.

Figure 6: Anisotropic coefficient $v_{6}$ for the dilepton emission as a
function of the invariant mass $M$ for several fixed values of the transverse
momentum $k_{T}$. The top two panels correspond to $|eB|=m_{\pi}^{2}$, and the
bottom two panels to $|eB|=5m_{\pi}^{2}$. The panels on the left are for
$T=0.2~{}\mbox{GeV}$, and the ones on the right are for $T=0.35~{}\mbox{GeV}$.

It should be noted that nonvanishing $v_{4}$ and $v_{6}$ are barely resolved
for the intermediate transverse momentum $k_{T}=0.5~{}\mbox{GeV}$, especially
in the case of the stronger field $|eB|=5m_{\pi}^{2}$. It is not surprising as
the corresponding $k_{T}$ is comparable to $\sqrt{|eB|}$. Nevertheless, the
trend becomes unambiguous for the larger values of $k_{T}$. As anticipated,
both $v_{4}$ and $v_{6}$ have a weak temperature dependence at sufficiently
large transverse momenta. The key features of the dilepton anisotropy are
summarized in Table 1.

In addition to the alternating sign pattern and the hierarchy of coefficients
$v_{n}\propto 1/n^{2}$ in the region of small invariant masses, we can also
identify other interesting features in the high-resolution data obtained here.
For example, we see well-pronounced modulations in the $v_{n}$ dependence on
the invariant mass. Indeed, by comparing the results in Figs. 4 through 6, one
can easily identify correlated patterns of peaks in all anisotropy
coefficients $v_{n}$. They are visible even in the region of moderately large
invariant masses. As is easy to verify, they come from the Landau-level
quantization of quarks. In heavy-ion physics, such modulations could be hard,
if not impossible, to observe. Perhaps, they could have some phenomenological
implications in other contexts.

###  III.3 Application to heavy-ion collisions

The main goal of our study is to characterize spatial profiles of the photon
and dilepton emission in the rest frame of a strongly magnetized plasma. We
found that both emission types could be highly anisotropic. This finding
implies that a background magnetic field serves as an additional (“intrinsic”)
source of anisotropy unrelated to the hydrodynamic flow of the plasma.
Therefore, it is natural to suggest that the anisotropy coefficients observed
in heavy-ion collisions should contain the following two distinct
contributions:

$v^{\rm obs}_{n}=v^{\rm flow}_{n}\oplus v^{\rm mag}_{n},$ (7)

where $v^{\rm flow}_{n}$ is the usual hydrodynamic part while $v^{\rm
mag}_{n}$ is an intrinsic part due to a nonzero magnetic field. The proxy for
the latter is given by our analysis of the photon and dilepton emission in the
rest frame of a magnetized plasma above. It should be noted that different
types of anisotropy contributions are not necessarily independent or additive.

To perform a systematic study of quantifying and separating the two
contributions in Eq. (7) in the context of heavy-ion collisions, one would
require detailed numerical investigations. Possible phenomenological
approaches include hydrodynamic simulations or molecular dynamics models that
take into account the space-time evolution of the plasma. The corresponding
studies in the presence of a magnetic field background have not been done yet.
Such studies are also beyond the scope of this paper.

Without detailed simulations, here we can give only qualitative arguments,
supporting the general idea of an additional intrinsic source of anisotropy
due to the background magnetic field that have been ignored before. We can
also speculate that the corresponding anisotropy contribution could be
substantial if the magnetic field is as strong as suggested by some estimates
[12, 13, 14, 15, 16, 17]. It could be also important that the magnetic field
is particularly strong during the early stages of the plasma evolution, when
hydrodynamic flow did not develop fully.

##  IV Summary and Conclusions

In this paper, we investigated the higher-order anisotropy coefficients
$v_{4}$ and $v_{6}$ for photon and dilepton emission from a magnetized hot QGP
in the rest frame. For both processes, we revealed several characteristic
features in the dependence of the anisotropy coefficients on the kinematic
parameters. The summary of the overall magnitudes and signs of the anisotropy
coefficients is given in Table 1.

In the case of photon emission, we find qualitatively different anisotropy
patterns at small and large transverse momenta. At small momenta (i.e.,
$k_{T}\lesssim\sqrt{|eB|}$), the signs and absolute values of the anisotropy
coefficients are as follows: $v_{4}\simeq+0.05$ and $v_{6}\simeq-0.02$. At
large momenta (i.e., $k_{T}\gtrsim\sqrt{|eB|}$), the signs of $v_{n}$ reverse,
but the absolute values remain about the same, i.e., $v_{4}\simeq-0.05$ and
$v_{6}\simeq+0.02$. Combining these findings with the $v_{2}$ results in Ref.
[54], we see that the signs of even coefficients $v_{n}$ alternate. The
absolute values gradually decrease with increasing $n$ in each kinematic
region. Quantitatively, the scaling appears to go as $v_{n}\propto 1/n^{2}$

We find that the dilepton emission also has a noticeable anisotropy. However,
it is well pronounced only in the kinematic regime with large transverse
momenta (i.e., $k_{T}\gtrsim\sqrt{|eB|}$) and small invariant masses (i.e.,
$M\lesssim\sqrt{|eB|}$). The signs and absolute values of the anisotropy
coefficients are as follows: $v_{4}\simeq-0.05$ and $v_{6}\simeq+0.02$.
Supplementing these findings with the results for $v_{2}$ obtained in Ref.
[66], we see that the signs of even coefficients $v_{n}$ alternate, and their
absolute values decrease with increasing $n$. The quantitative scaling is
similar to that in the photon emission.

In application to QGP produced by noncentral heavy-ion collisions, one may
argue that the magnetic field could be too weak, e.g., well below the scale
set by the pion mass, to have observable effects. It is possible and, perhaps,
even likely that the field is weak indeed. Nevertheless, we argue that even
weak fields can affect the anisotropy of both photon and dilepton emissions in
certain kinematic regions. Indeed, as we see from our calculations, the
anisotropy is sizable even for the transverse momenta that are much larger
than the magnetic field scale. This is analogous to the anisotropy of the
classical synchrotron radiation. Admittedly, the effects on the photon
emission may be diluted by the subleading gluon-mediated processes. Hopefully,
the anisotropy does not vanish completely and could remain observable. The
situation with dileptons might be better, however. Indeed, the same leading-
order diagrams contribute in the case with and without the background magnetic
field.

It is tempting to suggest that a set of the first few nonzero anisotropy
coefficients $v_{n}$, extracted from the photon and dilepton data, can provide
a distinctive fingerprint of the background magnetic field in a hot QGP
produced by heavy-ion collisions. The current data with overwhelming
background effects may not allow one to test this idea easily in experiment.
Additionally, the task is complicated by the convolution with other sources of
anisotropy such as hydrodynamics flow and initial state fluctuations.
Nevertheless, we find it valuable to have concrete theoretical predictions for
the intrinsic $v_{n}^{\rm mag}$ produced by the background magnetic field. The
advances in experimental techniques, collision simulations, and data analysis
could make the current hopeless task possible in the future.

To give reliable theoretical predictions for the observable signatures in
heavy-ion experiments, one needs to combine the results of this study with
realistic space-time models of QGP with expansion and nonuniform profiles. The
latter requires the use of phenomenological models, for example, such as $3+1$
viscous hydrodynamic simulations in Refs. [41, 80]. The corresponding task is
beyond the scope of this paper. It has to be undertaken, however, before one
can reach the final conclusions about the emission anisotropy as a likely
signature of a background magnetic field in heavy-ion collisions.

While the motivation of this study was triggered by potential applications in
heavy-ion physics, it is instructive to mention that our main results may also
find applications in astrophysics, where relativistic QED plasmas are common.
With a suitable rescaling of the model parameters, our analysis can be easily
generalized to QED plasmas under conditions in magnetars [81], supernovae
[82], and gamma-ray bursts [83]. It is reasonable to assume that the
anisotropy profiles of the photon and dilepton emission will be similar.

###### Acknowledgements.

The work of X. W. was supported by National Science Foundation of China under
Grant No. 12147103. The work of I. A. S. was supported by the U.S. National
Science Foundation under Grant No. PHY-2209470.

##  Appendix A Imaginary part of the Lorentz-contracted polarization tensor

For convenience, here we quote the expression for the imaginary part of the
Lorentz-contracted polarization tensor that appears in the photon and dilepton
rates; see Eqs. (2) and (5), respectively. In the Landau-level representation,
the corresponding analytical expression takes the following form [54, 68, 66]:

$\displaystyle\mbox{Im}\left[\Pi^{\mu}_{R,\mu}\left(\Omega,\mathbf{k}\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{2\pi\ell_{f}^{4}}\sum_{n>n^{\prime}}^{\infty}\frac{g(n,n^{\prime})\left\\{\Theta\left[(k_{-}^{f})^{2}-k_{\parallel}^{2}\right]-\Theta\left[k_{\parallel}^{2}-(k_{+}^{f})^{2}\right]\right\\}}{\sqrt{\left[(k_{-}^{f})^{2}-k_{\parallel}^{2}\right]\left[(k_{+}^{f})^{2}-k_{\parallel}^{2}\right]}}\mathcal{F}_{n,n^{\prime}}^{f}(\xi_{f})$
(8) $\displaystyle-$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{4\pi\ell_{f}^{4}}\sum_{n=0}^{\infty}\frac{g_{0}(n)\Theta\left[k_{\parallel}^{2}-(k_{+}^{f})^{2}\right]}{\sqrt{k_{\parallel}^{2}\left[k_{\parallel}^{2}-(k_{+}^{f})^{2}\right]}}\mathcal{F}_{n,n}^{f}(\xi_{f}),$

where $\Theta\left(x\right)$ is the Heaviside step function,
$\alpha_{f}=e_{f}^{2}/(4\pi)$, $e_{f}$ is the flavor-dependent electric charge
of the quark, $k_{\parallel}^{2}\equiv\Omega^{2}-k_{z}^{2}$,
$\xi_{f}=k_{\perp}^{2}\ell_{f}^{2}/2$ and $\ell_{f}=1/\sqrt{|e_{f}B|}$ is a
flavor-dependent magnetic length. The Landau-level thresholds are determined
by the following two transverse momenta:

$k_{\pm}^{f}=\left|\sqrt{m^{2}+2n|e_{f}B|}\pm\sqrt{m^{2}+2n^{\prime}|e_{f}B|}\right|.$
(9)

Functions $g(n,n^{\prime})$ and $g_{0}(n)$ are determined by the quark
distribution functions. In thermal equilibrium, they are given by

$\displaystyle g(n,n^{\prime})$ $\displaystyle=$ $\displaystyle
2-\sum_{s_{1},s_{2}=\pm}n_{F}\left(\frac{\Omega}{2}+s_{1}\frac{\Omega(n-n^{\prime})|e_{f}B|}{k_{\parallel}^{2}}+s_{2}\frac{|k_{z}|}{2k_{\parallel}^{2}}\sqrt{\left(k_{\parallel}^{2}-(k_{-}^{f})^{2}\right)\left(k_{\parallel}^{2}-(k_{+}^{f})^{2}\right)}\right),$
(10) $\displaystyle g_{0}(n)$ $\displaystyle=$ $\displaystyle
g(n,n)=2-2\sum_{s=\pm}n_{F}\left(\frac{\Omega}{2}+s\frac{|k_{z}|}{2|k_{\parallel}|}\sqrt{k_{\parallel}^{2}-4(m^{2}+2n|e_{f}B|)}\right),$
(11)

where $n_{F}(E)=1/\left[\exp\left(E/T\right)+1\right]$ is the Fermi-Dirac
distribution function. Finally, $\mathcal{F}_{n,n^{\prime}}^{f}(\xi)$ is the
following flavor-dependent function of the transverse momentum:

$\mathcal{F}^{f}_{n,n^{\prime}}(\xi)=8\pi\left(n+n^{\prime}+m^{2}\ell_{f}^{2}\right)\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right]+8\pi\left(\frac{k_{\parallel}^{2}-k_{\perp}^{2}}{2}\ell_{f}^{2}-(n+n^{\prime})\right)\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right],$
(12)

and function $\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ is defined in terms of the
Laguerre polynomials, i.e.,

$\mathcal{I}_{0}^{n,n^{\prime}}(\xi)=\frac{(n^{\prime})!}{n!}e^{-\xi}\xi^{n-n^{\prime}}\left(L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right)\right)^{2}=\frac{n!}{(n^{\prime})!}e^{-\xi}\xi^{n^{\prime}-n}\left(L_{n}^{n^{\prime}-n}\left(\xi\right)\right)^{2}.$
(13)

Note that the two different representations for
$\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ are equivalent. Note that, by
definition, Laguerre polynomials with negative lower indices vanish.

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Parallel Tree Kernel Computation

Souad Taoutisup1, Hadda Cherrounsup1 and  Djelloul Ziadisup2
sup1LIM, Université UATL Laghouat, Algeria
sup2Groupe de Recherche Rouennais en Informatique Fondamentale, Université de Rouen Normandie, France
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Tree kernels are fundamental tools that have been leveraged in many applications, particularly those based on machine learning for Natural Language Processing tasks.
In this paper, we devise a parallel implementation of the sequential algorithm for the computation of some tree kernels of two finite sets of trees [14]. Our comparison is narrowed on a  sequential implementation of SubTree kernel computation.  This latter  is mainly reduced to an intersection of  weighted tree automata.  Our  approach relies on the nature of the data parallelism source inherent in this computation  by deploying  the MapReduce paradigm.

One of the key benefits of our approach is its versatility in being adaptable to a wide range of substructure tree kernel-based learning methods.  To evaluate the efficacy of our parallel approach, we conducted a series of experiments that compared it against the sequential version using a diverse set of synthetic tree language datasets that were manually crafted for our analysis. The reached results  clearly demonstrate  that the proposed parallel algorithm outperforms the sequential one in terms of latency.

§ INTRODUCTION

Trees are basic data structures that are naturally used in real world applications to represent a wide range of objects in structured form, such as XML documents [11], molecular structures in chemistry [8] and parse trees in natural language processing [15].

In [9], Haussler provides a framework based on convolution kernels, which find the similarity between two structures by summing the similarity of their substructures. Many convolution kernels for trees are presented based on this principle and have been effectively used to a wide range of data types and applications.

Tree kernels, which were initially presented in [3, 4], as specific convolution kernels, have been shown to be interesting approaches for the modeling of many real world  applications. Mainly those related to  Natural Language Processing tasks, e.g. named entity recognition and relation extraction [13], text syntactic-semantic similarity [1], detection of text plagiarism [16], topic-to-question generation [2], source code plagiarism detection [7], linguistic pattern-aware dependency which  captures chemical–protein interaction patterns within biomedical literature [18].

Subtree (ST) kernel [17] and the subset tree (SST) [3] were the initially kernels introduced in the context of trees. The compared segments in ST Kernel, are subtrees, a node and its entire descendancy. While in the SST Kernel, the considered segments are subset-trees, a node and its partial descendancy.

The principle of tree kernels, as initially presented, is to compute the number of shared substructures (subtrees and subset trees) between two trees $t_1$ and $t_2$ with $m$ and $n$ nodes, respectively. It may be computed recursively as follows:

\begin{equation}
K(t_1, t_2)= \displaystyle\sum_{(n_1, n_2)  \in N_{t_1} \times N_{t_2}} \Delta (n_1, n_2)
\end{equation}

where $N_{t_1}$ and $N_{t_2}$ are the number of nodes in $t_1$ and $t_2$ respectively and $\Delta (n_1, n_2)= \sum^{|S|} _{i=1} I_i(n_1).I_i(n_2)$ for some finite set of
subtrees $S = \{s_1, s_2, \dots \}$, and $I_i(n)$ is an indicator function
which is equal to 1 if the subtree is rooted at node $n$ and to
$0$ otherwise.

An approach for computing the tree kernels of two finite sets of trees was proposed in [12]. That makes use of Rooted Weighted Tree Automata (RWTA) that are a class of weighted tree automata. Subtree, Subset tree, and Rooted tree kernels can be computed using a general intersection of RWTAs associated with the  two finite sets of trees and then the computation of weights on the resulting automaton.

In this paper, we narrow  our study to the parallel implementation using MapReduce paradigm of the construction of the RWTA from a finite set of trees (as this step is a common base for other trees kernels) and to the comparison with the sequential linear algorithm proposed in [10] for SubTree kernel computation. The main motivation behind this proposal is that despite the  linear complexity of the sequential version, it remains complex when we consider  Machine Learning computational cost requirements.

We begin by defining RWTA. Next, we present the sequential proposed algorithms. Then we provide our parallel implementation based on MapReduce programming model.

The rest of the paper is organized as follows: Section <ref> introduces Tree Kernels and Automata while presenting the sequential proposed algorithms. Section<ref> presents the details of the parallel implementation of the tree kernel computation based on MapReduce. Some experimental results and evaluations are shown in Section <ref>. Finally, the conclusion and perspectives are presented in Section <ref>

§ TREE KERNELS AND AUTOMATA

Let $\Sigma$ be a graded alphabet, $t$ is a tree over $\Sigma$, defined initially as $t = f(t_1, \dots , t_k)$ where $k$ can be any integer, $f$ any symbol from $\Sigma_k$ and $t_1, \dots , t_k$ are any $k$ trees over $\Sigma$. The set of trees over $\Sigma$ is referred as $T_\Sigma$. A tree language over $\Sigma$ is a subset of $T_\Sigma$.

Let $\mathbb{M}= (M, +)$ be a monoid with identity is $0$, a formal tree series $\mathbb{P}$ [4] [6] over a set $S$ represents a mapping from $T_\Sigma$ to $S$, where its support is the set $\text{Support}(\mathbb{P}) = \{t \in T_\Sigma | (\mathbb{P}, t) \neq 0 \}$. Any formal tree series is consistent to a formal sum $\mathbb{P} = \sum_{t \in T_{\sum}} (\mathbb{P}, t)t$, which is in this case both associative and commutative.

Weighted tree automata can realize formal tree series. In this paper, we employ specific automata, and the weights just indicate the finality of states. As a result, the automata we use are a particular subclasses of weighted tree automata.

§.§ Root Weighted Tree Automata

Let $\mathbb{M}= (M, +)$  be a commutative monoid. An $\mathbb{M}$-Root Weighted Tree Automaton ($\mathbb{M}$-RWTA) is a 4-tuple $(\Sigma, Q, \mu, \delta)$ with the following properties:

* $\Sigma = \bigcup_{k \in \mathbb{N}} \Sigma_k$: a graded alphabet,

* $Q$: a finite set of states,

* $\mu$: the root weight function, is a function from $Q$ to $M$,

* $\delta$: the transition set, is a subset of $Q \times \Sigma_k \times Q^k$.

A $\mathbb{M}$-RWTA is referred  as RWTA, if there is no ambiguity.

The root weight function $\mu$ is extended to $2^Q \to M$ for each subset $S$ of $Q$ by $\mu(S) = \Sigma_{s \in S} \mu(s)$. The function $\mu$ is equivalent to the finite subset of $Q \times M$ defined for any couple $(q, m)$ in $Q \times M$ by $(q, m) \in \mu \iff \mu(q) = m.$

For any subset $S$ of $Q$, the root weight function $\mu$ is expanded to $2^Q \to M$ by $\mu(S) = \Sigma_{s \in S} \mu(s)$. The function $\mu$

The transition set $\delta$ corresponds to the function in $\Sigma_{k} \times Q^{k} \rightarrow 2^{Q}$ defined for any symbol $f$ in $\Sigma_k$ and for any $k$-tuple $(q_1, \dots , q_k)$ in $Q^k$ by

$q \in \delta(f, q_1, . . . , q_k) \iff (q, f, q_1,.\dots, q_k) \in \delta$.

The function $\delta$ is extended to $\Sigma_{k} \times (2^Q)^k \rightarrow 2^Q$ as follows: for any symbol $f$ in $\Sigma_{k}$, for any $k$-tuple $(Q_1, \dots , Q_k)$ of subsets of $Q$,

$\delta(f, Q_1, \dots , Q_k) = \bigcup_{(q_1,\dots,q_k)\in Q_1 \times \dots \times Q_k} \delta(f, q_1, \dots , q_k)$.

Finally, the function $\Delta$ is the function from $T_\Sigma$ to $2^Q$ defined for any tree $t = f(t_1, \dots , t_k)$ in $T_{\Sigma}$ by

$\Delta(t) = \delta(f, \Delta(t_1), \dots , \Delta(t_k))$.

A weight of a tree  $t$ in a $\mathbb{M}$-RWTA $A$ is $\mu(\Delta(t))$. The formal tree series realized by $A$ is the formal tree series over $M$ denoted by $\mathbb{P}_A$ and defined by $\mathbb{P}_A=\sum_{t\in T_\Sigma}\mu(\Delta(t))$, with $\mu(\emptyset) = 0$ where $0$ is the identity of $\mathbb{M}$.

Let us consider the graded alphabet $\Sigma$ defined by $\Sigma_0 = \{a, b\}$, $\Sigma_1 = \{g, h\}$ and $\Sigma_2 = \{f\}$. Let $\mathbb{M} = (\mathbb{N}, +)$. The RWTA $A = (\Sigma, Q, \mu, \delta)$ defined by

* $Q = \{1, 2, 3, 4, 5, 6\},$

* $\mu = \{(1, 1), (2, 4), (3, 3), (4, 1), (5, 2), (6, 3)\},$

* $\delta$ = {$(1, a), (3, b), (2, h, 1), (4, g, 3), (5, f, 2, 3),$
$ (6, f, 4, 5)$},

This RWTA  is represented in Figure <ref>. It realizes the following tree series: $\mathbb{P}_A = 1a + 3b + 4h(a) + 1g(b) + 2g(h(a)) + 3f(g(h(a)), g(b))$

[node distance=2.5cm,bend angle=30,transform shape,scale=1]

[state] (q6) \( 6 \);
[state, below left of=q6] (q5) \( 5 \);
[state, below right of=q6] (q4) \( 4 \);
[state, below left of=q5] (q2) \( 2 \);
[state, below of=q2] (q1) \( 1 \);
[state, below of=q4] (q3) \( 3 \);

(q6) ++(-1cm,0cm) edge[above,<-] node \( 3 \) (q6);
(q5) ++(-1cm,0cm) edge[above,<-] node \( 2 \) (q5);
(q2) ++(-1cm,0cm) edge[above,<-] node \( 4 \) (q2);
(q4) ++(1cm,0cm) edge[above,<-] node \( 1 \) (q4);
(q3) ++(1cm,0cm) edge[above,<-] node \( 3 \) (q3);
(q1) ++(-1cm,0cm) edge[above,<-] node \( 1 \) (q1);

(q1) ++(0cm,-1cm) node \( a \)  edge[->] (q1);
(q3) ++(0cm,-1cm) node \( b \)  edge[->] (q3);

(q1) edge[->, above left] node \( h \) (q2)
(q3) edge[->, above right] node \( g \) (q4)
(q2) edge[->, above left] node \( g \) (q5);

(q4) ++(-1cm,1.0cm)  edge[->] node[above right,pos=0] \( f \) (q6) edge[shorten >=0pt,] (q4) edge[shorten >=0pt,] (q5);

The RWTA \( A \)

A RWTA appears to be a prefix tree constructed in the context of words. A finite set of trees could be represented by this compact structure.

In addition, many substructures' tree can be computed through this compact structure. In what follow, we introduce  Subtree,  Rooted tree and SubSet tree and their related automata [14].  However, in  this paper, we narrow our Kernel computation proposal to  the SubTree case.

§.§ Subtree

Let $\Sigma$ be a graded alphabet and $t = f(t_1, \dots , t_k)$ a tree in $T_\Sigma$. The set $\ST{t}$ is the set defined inductively by:

$\ST{t} = \{t\} \cup \bigcup_{1 \leq j \leq k} \ST{t_j}.$

Let $L$ be a tree language over $\Sigma$. The set $\STS{L}$ is the set defined by:

$\STS{L} = \bigcup_{t \in L} \ST{t}.$

The formal tree series $\STZ{t}$ is the tree series over $\mathbb{N}$ inductively defined by:

$\STZ{t} = t + \sum_{1 \leq j \leq k} \STZ{t_j}.$

If $L$ is finite, the rational series $\STZ{L}$ is the tree series over $\mathbb{N}$ defined by:

$\STZ{L} = \sum_{t \in L} \STZ{t}.$

Let $\Sigma$ be the graded alphabet defined by $\Sigma_{0} = \{a, b\}$, $\Sigma_{1} = \{g, h\}$ and $\Sigma_{2} = \{f\}$.

Let  $t$ be the tree  $f(h(a), g(b))$, we have:
$\ST{t} = \{a, b, h(a), g(b), f(h(a), g(b))\}$ $\STS{t} = t + h(a) + g(b) + a + b$

Let $\Sigma$ be a graded alphabet. Let $t$ be a tree in $T_\Sigma$. The Subtree automaton associated with $t$ is the RWTA $A_t = (\Sigma, Q, \mu, \delta)$ defined by:

* $Q =\STS{t}$,

* $\forall s \in Q,\quad \mu(s) = (\STZ{t}, s)$,

* $\forall f \in \Sigma, \forall s_1, \dots , s_{k+1} \in Q, s_{k+1} \in \delta(f, s_1, \dots, s_k) \iff  s_{k+1} = f(s_1, \dots , s_k).$

This  Weighted Tree Automaton (RWTA) requires less storage space because its states are exactly  its subsets.

Given the tree defined by $t = f(f(h(a),b),g(b))$. The RWTA $A_t$  associated with the tree $t$ is represented in Figure <ref>.

[node distance=2.5cm,bend angle=30,transform shape,scale=1]

[state, rounded rectangle] (q6) \( t \);
[state, below left of=q6, rounded rectangle] (q5) \( f(h(a), b) \);
[state, below right of=q6, rounded rectangle] (q4) \( g(b) \);
[state, below left of=q5, rounded rectangle] (q2) \( h(a) \);
[state, below of=q2, rounded rectangle] (q1) \( a \);
[state, below of=q4, rounded rectangle] (q3) \( b \);

(q6) ++(-1cm,0cm) edge[above,<-] node \( 1 \) (q6);
(q5) ++(-1.5cm,0cm) edge[above,<-] node \( 1 \) (q5);
(q2) ++(-1cm,0cm) edge[above,<-] node \( 1 \) (q2);
(q4) ++(1cm,0cm) edge[above,<-] node \( 1 \) (q4);
(q3) ++(1cm,0cm) edge[above,<-] node \( 2 \) (q3);
(q1) ++(-1cm,0cm) edge[above,<-] node \( 1 \) (q1);

(q1) ++(0cm,-1cm) node \( a \)  edge[->] (q1);
(q3) ++(0cm,-1cm) node \( b \)  edge[->] (q3);

(q3) edge[->, above right] node \( g \) (q4)
(q1) edge[->, above left] node \( h \) (q2);

(q2) ++(1cm,1.0cm)  edge[->] node[above left,pos=0] \( f \) (q5) edge[shorten >=0pt,] (q2) edge[shorten >=0pt,] (q3);
(q4) ++(-1cm,1.0cm)  edge[->] node[above right,pos=0] \( f \) (q6) edge[shorten >=0pt,] (q4) edge[shorten >=0pt,] (q5);

The RWTA \( A_{t} \) associated with the tree  \( t=f(f(h(a),b),g(b)) \).

§.§ Rooted Tree

Let $\Sigma$ be a graded alphabet and $t = f(t_1, \dots , t_k)$ a tree in $T_\Sigma$. We indicate with $\Sigma^\prime$, the set $\Sigma \cup \{\perp\}$, where $\perp \in \Sigma ^\prime_0$ and $\perp \notin \Sigma$.
$\PS{t}$ is the set of trees on $T_\Sigma^\prime$ inductively defined by:

\PS{t} =     & \{t\} \cup f(\perp,\ldots, \perp) \\
&\cup \ f(\PS{t_1},\ldots, \PS{t_k})
\end{array}$

It should be noted that $\perp$ is not a prefix of $t$.

Let $L$ be a tree language over $\Sigma$. The set
PrefixSet($L$) is the set defined by:

$\PS{L} = \bigcup_{t \in L} \PS{t}.$

The formal tree series $\PSZ{t}$ is the tree series over $\mathbb{N}$ inductively defined by:

$\PSZ{t} = \sum_{t^\prime \in \PS{t}}  t^\prime$

If $L$ is finite, the series $\PSZ{L}$ is the tree series over $\mathbb{N}$ defined by:

$$\PSZ{L} = \sum_{t^\prime \in L}  \PSZ{t^\prime}$$

If $L$ is not finite, as $\Sigma$ is a finite set of symbols, there exists a symbol $f$ in $\Sigma^k$ such that $f(\perp, ..., \perp)$ occurs as a prefix infinite number of times in $L$. Therefore, $\PSZ{L}$ is a series of trees over $\mathbb{N} \cup \{+\infty\}$.

Let $\Sigma$ be a graded alphabet. Let $t$ be a tree in $T_\Sigma$. The automaton of prefixes associated with $t$ is the RWTA $A_t = (\Sigma^\prime, Q, \mu, \delta)$ defined by:

* $Q =\STS{t} \cup \{\perp\}$,

* $\forall t^\prime \in Q,\quad \mu(t^\prime) =$
$ \begin{cases}
1, & \text{if}\ t^\prime=t, \\
0, & \text{else},
\end{cases}$

* $\forall t^\prime = f(t_1,\dots, t_k) \in Q, \delta (f, t_1,\dots, t_k) = {t^\prime}$

* $\forall f \in \Sigma^k, \delta(f, \perp, \dots, \perp) = \{f(t_1,\dots, t_k)\in Q\}.$

§.§ SubSet Tree (SST)

Let $\Sigma$ be a graded alphabet and $t = f(t_1, \dots , t_k)$ a tree in $T_\Sigma$. We indicate with $\Sigma^\prime$, the set $\Sigma \cup \{\perp\}$, where $\perp \in \Sigma ^\prime_0$ and $\perp \notin \Sigma$.
The set SSTSet($t$) is the set of trees on $T_{\Sigma^\prime}$ defined by:
$SSTSet(t) = PrefixSet(SubtreeSet(t))$

Let $L$ be a tree language over $\Sigma$. The set
SSTSeries($L$) is the set defined by:

$SSTSeries(L) = \sum_{t^\prime \in L} SSTSeries(t) t^\prime.$

The formal tree series SSTSeries$(t)$ is the tree series over $\mathbb{N}$ inductively defined by:

$SSTSeries(t) = SubtreeSeries(prefixSet(t))$

Let $\Sigma$ be a graded alphabet. Let $t$ be a tree in $T_\Sigma$. The SST automaton associated with $t$ is the RWTA $A_t = (\Sigma^\prime, Q, \mu, \delta)$ defined by:

* $Q =\STS{t} \cup \{\perp\}$,

* $\forall t^\prime \in Q,\quad \mu(t^\prime) =$
1, & \text{if}\ t^\prime=t, \\
0, & \text{else},
\end{cases}$

* $\forall t^\prime = f(t_1,\dots, t_k) \in Q, \delta (f, t_1,\dots, t_k) = {t^\prime}$

* $\forall f \in \Sigma^k, \delta(f, \perp, \dots, \perp) = \{f_j(t_1,\dots, t_k)\in Q | h(f_j)=f\}.$

§.§ Sequential Kernel Computation

In order to compute the kernel of two finite tree languages $X$ and $Y$, we act in three steps:

* First, we construct both RWTAs $A_{X}$ and $A_{Y}$.

* Then we compute the intersection of $A_{X}$ and $A_{Y}$;

* Finally, the kernel is simply computed through a  sum of  all of the root weights of this RWTA.

One can easily observe that  the set of states, denoted by  $Q$,  is equal to SubTreeSet$(t)$ in ST, plus $\{\perp\}$ for both Rooted Tree and SubSet Tree. However, their  root weight function ($\mu$ ) are different.
In addition, their  $\delta$,  denoting the transition functions,  are defined from the ST transition table. Consequently, the RWTA construction step remains  the same for the three tree substructures while the RWTA intersection step is distinct for each tree substructure.

Let us recall that in this paper we narrow the RWTA intersection and kernel computation to the ST kernel case.

Initially, we introduce the sequential step-by-step procedure that enable us to efficiently calculate tree kernels by utilizing the intersection of tree automata.

§.§.§ RWTA Construction

In this section, we describe through an algorithm the construction of an RWTA from a finite set of trees. Consider the finite set of trees $X$, first,  we extract all the prefixes of each tree in $X$ and sum their number of occurrences of each tree in $X$, which is equivalent to the sum of the subtrees series.

$\STZ{X} = \sum_{ t \in X} \STZ{t}$

Algorithm <ref> constructs an RWTA from a finite set of trees.

$X$: Set of trees
RWTA $A_X$  = $(\Sigma, Q, \mu, \delta)$
$Q_{X} = \emptyset$
$t  \in X$

$t \not\in Q_{X}$

Add($Q_X$, $t$);

$\mu_X(t) \gets 1$;

$\mu_X(s) \gets \mu_X(s) + 1$;

Computation of Automaton $A_X$ from $X$

§.§.§ RWTA Intersection

Let $\sum$ be an alphabet. Let $X$ and $Y$ be two finite tree languages over $\Sigma$. The Tree Series of  ($X$, $Y$) is defined by:

$\STZ{(X, Y)}=$
$\sum_{t \in T_{\sum}}(\STZ{X}, t) \times (\STZ{Y}, t).$

Let $\Sigma$ be the graded alphabet defined by $\Sigma_0 = \{a, b\}$, $\Sigma_1 = \{g, h\}$ and $\Sigma_2 = \{f\}$. Let us consider the three  trees $t_1 = f(f(h(a), b), g(b))$,  $t_2 = f(h(a), g(b))$ and $t_3 = f(f(h(a), b), f(h(a), g(b)))$.

We have:

$\STZ{t{_1}} = t_1 + f(h(a), b) + h(a) + g(b) + a + 2b$

$\STZ{t{_2}} = t_2 + h(a) + g(b) + a + b$

$\STZ{t{_3}} = t_3 + f(h(a), b) + t_2 + 2h(a) + g(b) + 2a + 2b$

$\STZ{\{t_1,t_2\}} = t_1 + t_2 + 2f(h(a), b) + 2h(a) + 2g(b) + 2a + 3b$

$\STZ{(\{t_1,t_2\}, \{t_3\})} = t_2 + f(h(a), b) + 4h(a) + 4g(b) + 4a + 3b$

By definition, any $t$ in $T_\Sigma$, is in $Q$ if and only if $t \in \STS{X} \cap \STS{Y}$. Moreover, by definition of $\mu$, for any tree $t$, $\mu(t) = \mu_X(t) \times \mu_Y(t)$, as for any tree $t$, if $t \notin \STS{X}$ (resp. $t \notin  \STS{Y})$, then $\mu_X(t) = 0$ (resp. $\mu_Y(t) = 0$).

In order to compute the kernel from two RWTA, Algorithm <ref> below, loops through the states of $A_X$, if any state $q$ is present in $A_Y$ then it is added to the automaton $A_{(X,Y)}$ with $\mu_{(X,Y)}(q) = \mu_X(q) \times \mu_Y(q)$, respectively to $A_Y$.

$2$ RWTA  $A_X$ and $A_Y$
an RWTA $A_{(X,Y)}=A_X \times A_Y$

$A_{(X, Y)} \gets  A_Y$;

$s \in Q_X$

$s \in  Q_Y$
$\mu_{(X,Y)}(s) \gets \mu_X(s) \times \mu_Y(s)$;

$\mu(s) \gets 0$;

Computation of Automaton $ A_{(X,Y)} = A_X \times A_Y$

§.§.§ Subtree Kernel Computation

After the construction of the RWTA for both sets of tree and then their intersection is built, in this last step we can effectively perform the  computation of tree kernel in subtree case. Let $X$ and $Y$ be two finite tree languages. Let $Z$ be the accessible part of their intersection tree automaton $A_{(X,Y)}$. Then, the kernel is simply computed through the sum of the weights:

$$\TK{X}{Y} = \sum_{q\in Z} \mu(q)$$

§ PARALLEL TREE KERNEL COMPUTATION

According to the inherent data parallelism in the Tree Kernel Computation, we design and implement a parallel version of the previous sequential SubTree Kernel Computation  based on MapReduce paradigm. The framework provides parallel implementation in a distributed environment. It has some advanced features of distributed computing without programming to coordinate the tasks, as the large task is partitioned into small chunks that can be executed concurrently, which provides ease of handling job scheduling, fault tolerance, distributed aggregation, and other management tasks.

Kernel Computation are very  close to the Big Data paradigm which processing is very  challenging   compared with conventional data processing techniques. Numerous solutions have been properly proposed to address the computational and storage issues of big data. The  MapReduce framework is one of the prominent methods.

Before dealing with our parallel version, we explain the principle of MapReduce programming model in what follow.

§.§ MapReduce Framework

Map-Reduce was created by Google as a parallel distributed programming approach that works on a cluster of computers due to large-scale data [5]. It is termed parallel because tasks are executed by dedicating multiple processing units in a parallel environment, and distributed over distinct storage. Hadoop is among most popular  open-source MapReduce implementation created primarily by the Apache Software Foundation.

MapReduce became the most popular framework for proposing programs without worrying about infrastructure complexity and difficulties since it provided such a stable, easy-to-use, abstract, and scalable environment. its  associated  programming paradigm is outlined through the basic Map-Reduce jobs.

§.§.§ Map Function

The first step of a Map-Reduce task is to map the input data to one or more specific reducers. Each mapper reads a set of files and generates a list of $<key, value>$ pairs for each unit of data based on a mapping schema. Key pairs from the same list would be collected in the same reducer. The most important element of a Map-Reduce method, which affects precision, time complexity, and space complexity, is the mapping schema.

§.§.§ Reduce Function

The output of mappers acts as input for the reducer function. Each reducer consequently receives a key associated with a list of records, denoted as $<key_{i}, record_{i_{1}}, record_{i_{2}},..., record_{i_{r}}>>$. Reducer output is also pairs of $<key, value>$, which can produce many values with the same keys. Within each reducer, the reducer function is applied in parallel to the input list of data and its output will be written to the Distributed File System.

For our parallel   Subtrees Kernel Computation of two finite sets of trees $X$ and $Y$,  we act as in the sequential version in a  three-step process,  where two steps are performed  in parallel:

* Parallel RWTA Construction,

* Parallel RWTA Intersection,

* Kernel Computation which is deduced   by means of the same manner as in the sequential version.

§.§ Parallel RWTA Construction

Let $X$ be a finite tree language. To construct the RWTA $A_X$ from $X$ based on MapReduce paradigm, we have to  determine the Map and Reduce jobs.  The prefixes of $X$ are listed in a file which is  used as an input.

First, the Map function (Algorithm <ref>) splits the prefixes list into subtrees. Next, the Map function distributes the key-value pairs as follows: $<key, 1>$, where the key represents the subtree, and the value is $1$ indicating the number of occurrence. At this step it represents its presence.

Then, the Map function sends the key-value pairs to  reducers by key, so if a subtree appears multiple times, it will be sent to the same reducer multiple times. This principle is similar to the popular word count program where words are subtree.

($X$: Set of trees) file

$file \neq empty$

Split$(\text{line}, S)$;

$s \in S$

Emit$(s, 1)$;

Map Function for the construction of $A_X$ from $X$

Next, the reducer (Algorithm <ref>) sums the value which is the number of occurrences and is equivalent to the weight of the subree. Finally, the subtrees are merged in tree series.

mapped $\textlangle s, 1\textrangle$

RWTA $A_X$

Weight$_s \gets 0$;

all mapped $s$

$\text{Weight}_s \gets \text{Weight}_s + 1$;

Add$(A_X, (s,\text{Weight}_s))$;

Reduce Function for the construction of $A_X$ from $X$

§.§ Parallel RWTA Intersection

Let $X$ and $Y$ be two finite tree languages, $A_X$ and $A_Y$ are their respective RWTA. In order to compute the intersection of automata $A_X$ and $A_Y$ using MapReduce programming model, we have to identify the Map and Reduce jobs. Let us mention that we have both RWTA details saved, from the last Construction step,  as an input file, each of them in a separate line.

First, the Map function (Algorithm <ref>) splits the tree series for $X$ and $Y$ resp. into subtrees with their weights in list. Then, the Map function distributes the key-value pairs as follows: $\textlangle \text{subtree}, (1, weight)\textrangle $, where the key represents the subtree, and the value is composed of the weight of the subtree and the number $1$ that acts as a Boolean indicating whether or not the subtree is present in the RWTA.

Next, the Map function distributes its output to the reducers by key i.e if any subtree is present in different RWTA, it will be sent to the same reducer.

(RWTA $A_X$, $A_Y$) file

$file \neq empty$

Split$(\text{line}, Q)$;

$s \in Q$

Emit$(s, (1, \mu(s)))$;

Map Function for the computation of the automaton $A_X \times A_Y$

After that, every reducer (Algorithm <ref>) sums the first part of the value, which indicates the presence of the subtree in the RWTA to check if it is present in both RWTAs. Then, if the presence is equal to 2 the reducer multiplies the weights of the received subtree from $A_X$ and $A_Y$.

$\textlangle s, (1, \mu(s)) \textrangle $
RWTA $A_X \times A_Y$

Presence$_s \gets 0$;

Weight$_s \gets 1$;

mapped $s$

$\text{Presence}_s \gets \text{Presence}_s + 1$;

$\text{Weight}_s \gets \text{Weight}_s \times \mu(s)$;

Presence$_s = 2$

Add$(A_X \times A_Y, (s, \text{Weight}_s))$;

Reduce Function for the computation of the intersection  automaton $A_X \times A_Y$

§ EXPERIMENTS AND RESULTS

To analyse our parallel RWTA-based SubTree Kernel computation we have performed a batch of  comparative experiments in order to demonstrate the difference in terms of latency   between our parallel  algorithm and the sequential  one using MapReduce paradigm.

Additionally,  we use the absolute acceleration metric defined by $A^{abs} = T_{seq}/T_{par}$ where $T_{seq}$ and $T_{par}$ are the running times of the sequential  and the parallel algorithms respectively.

Prior to presenting our findings, let us initially provide a description of the benchmark we constructed and outline the implementation details.

§.§ Dataset

In order to perform the  comparative study  of both variants of algorithms, we need  a testbed of  multiple datasets that cover the  variety wide of tree characteristics in order to have a deep algorithm analysis, which is not the case in the real world  datasets that are standard benchmarks for learning on relatively small trees. For that purpose, in our experiments, we are brought in   generating  synthetic datasets.

For this building  dataset task,  we have considered into account mainly three criteria:
i) the  alphabet size (varying between $2$ and $12$),
ii) the range of the  maximal alphabet arity (between $1$ and $5$), and
iii) the tree depth ($TD$) that we have varied within the range of  $10$ to $50$.

The constructed tree datasets are  divided  into two batches   according to the alphabet size in [$2$,$12$]. The first batch (respectively the second batch) gathered three  datasets      $D1$, $D2$, $D3$ and $D4$ (respectively $D1^s$, $D2^s$, $D3^s$ and $D4^s$ ). Each of them contains  two  tree sets generated using this above principle.  Into each batch, each dataset has different size that we have classified  according to their average tree size in three classes (small: less or equal to  $1.5$GB, medium: less than  $2.5$GB, and large: more than  $4.4$GB and $7$GB).
Table <ref> illustrates more details on the generated datasets.

# Trees      $\Sigma$     Arity      $TD$      size Gb
$D1$       500      [2, 12]        [1, 5]     [10,20]       $ 1.5$
$D2$       800      [2, 12]       [1, 5]      [10, 50]     $ 2.5$
$D3$       3000      [2, 12]       [2, 5]      [10, 50]       $ 4.4$

$D4$       4500      [2, 12]       [2, 5]      [10, 50]       $ 7$
$D1^s$       500      [2, 6]       [1, 5]     [10,20]       $ 1.5$
$D2^s$       800      [2, 6]      [1, 5]      [10, 50]     $ 2.5$
$D3^s$       3000      [2, 6]       [2, 5]      [10, 50]       $ 4.4$

$D4^s$       4500      [2, 6]       [2, 5]      [10, 50]       $ 7$

Details on the generated datasets.

Both sequential and MapReduce-based algorithms are implemented in Java 11. All experiments were performed on a server equipped with an Intel(R) Xeon(R) Silver $4216$ CPU (2.10GHz) processor with  $32$ cores and $128$GB of RAM running Linux.

Sequential and parallel   codes  in addition to the  generated datasets are available on Github [ <https://github.com/souadtaouti/parallel-Subtree-kernel-computation-using-RWTA/>].

In this study, we have established a fixed cluster architecture, utilizing Docker containers for conducting all tests. Our cluster comprises one container designated as the Master, and five containers serving as Slaves. It is important to note that this choice of cluster configuration was made arbitrarily for the purpose of this research.
The Hadoop [<https://hadoop.apache.org/>] V 3.3.0 is installed as MapReduce implementation platform on our cluster.

§.§ Results

y axis line style =  opacity = 0 ,
axis x line       = none,
tickwidth         = 0 pt,
enlarge y limits  = 0.2,
enlarge x limits  = 0.02,
symbolic y coords = $D4$,$D3$,$D2$,$D1$,
nodes near coords,

coordinates  (97.2,$D1$)
coordinates (0.87,$D1$) (1.96,$D2$)

Performances of Parallel algorithm  vs the  Sequential one in terms of running time (minutes).

Figure <ref> reports the performances of  our parallel SubTree Kernel computation versus the sequential algorithm in terms of running time on the generated datasets.
Let us mention that the sequential time is obtained on one node (container) of  our  cluster.

It is obvious to observe that  our parallel computation  is significantly faster compared to the sequential version across all dataset instances. Furthermore, our analysis reveals that the average absolute acceleration achieved is
$50,89$ times,  as demonstrated  in Table <ref>.  This substantial acceleration is notable, which is a good acceleration, taking into account the utilization of six nodes in our cluster, and reflects the effectiveness of our parallel computation approach.

$D1$          $D2$      $D3$      $D4$        Average
$A^{abs}$        $57,23$        $62.96$     $42.61$     $40.75$       $50,89$

Acceleration  of Parallel SubTree Kernel computation on different  datasets.

y axis line style =  opacity = 0 ,
axis x line       = none,
tickwidth         = 0pt,
enlarge y limits  = 0.2,
enlarge x limits  = 0.02,
symbolic y coords = Small,Medium,Large1,Large2,
nodes near coords,
coordinates  (0.87,Small) (1.96,Medium)  (3.75,Large1)  (8.67,Large2);
coordinates   (3.7,Large1) (1.95,Medium)  (1.65,Small)
$|\Sigma|$=[2; 6], $|\Sigma|$=[2; 12]

Performances of our  Parallel algorithm in terms of running time (minutes)  when varying  the alphabet size.

Considering the influence of the  alphabet size variation on the performances of our algorithm,  we have measured  the performances reached by both datasets batches  ({$D1$, $D2$,  $D3$ and and $D4$}  and { $D1^s$, $D2^s$, $D3^s$  and $D4^s$}.

In Figure <ref>, we present the performance results of our parallel SubTree Kernel computation, showcasing the impact of the Alphabet size on the running time for the generated datasets.

The results demonstrate that our parallel algorithm performs better when the alphabet size is larger  at least for our datasets. This outcome is in line with our prediction, as a smaller alphabet size implies a higher frequency of common prefixes. A more detailed analysis of the number of common prefixes further supports this finding.

§ CONCLUSION

The prefix tree automaton constitutes a common base for the computation of different tree kernels: SubTree, RootedTree, and SubSequenceTree kernels [14]. In this paper, we have shown a parallel Algorithm that efficiently compute this common structure (RWTA automaton) and we have used it for the computation of the SubTree Kernel using MapReduce paradigm.

Our parallel implementation of the SubTree kernel computation  has been tested on  synthetic datasets with different parameters. The results showed that our parallel computation is by far more speed than the sequential version for all instances of datasets.

Despite that this work has shown the efficiency of the parallel implementation compared to the sequential algorithms, three main future works  are envisaged.
Firstly, we have to devise some algorithms that generalise the computation of others kernels such  RootedTree, and SubSequenceTree ….
Some of them  will deploy tree automata intersection  in addition to the associated weights computation. In fact, while the subtree kernel is a simple summation of weights, the SubSequenceTree needs more investigation on the weight computations using  the resulted RWTAs intersection.

Secondly, more large datasets have to be generated and tested to confirm the output-sensitive results of our solutions.
Finally, one can investigate different cluster architectures in order to give more insights and recommendations on the cluster' parameters tuning.

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