\section{Introduction}

Among the various approaches used in meson physics, the formalism of
Bethe-Salpeter and Dyson-Schwinger equations (DSEs) plays a traditional and
indispensable role. The Bethe-Salpeter equation (BSE) provides a
field-theoretical starting point to describe hadrons as relativistic bound
states of quarks and/or antiquarks. For instance, the DSE and BSE framework has
been widely used in order to obtain nonperturbative information about the
spectra and decays of the whole lightest pseudoscalar nonet, with an
emphasis on the QCD pseudo-Goldstone boson --- the pion \cite{PSEUDO}.
Moreover, the formalism satisfactorily provides a window to the 'next-scale'
meson sector, too, including vector, scalar \cite{SCALARY} and excited mesons.
Finally, electromagnetic form factors of mesons have been calculated with this
approach for space-like momenta \cite{FORMFAKTORY}.

When dealing with bound states composed of light quarks, then it is unavoidable
to use the full covariant BSE framework. Nonperturbative knowledge of the
Green's function, which makes part of the BSE kernel, is  required. Very often,
the problem is solved in Euclidean space, where it is more tractable, as
there are no Green's function singularities there. The  physical amplitudes
can be then obtained by continuation to Minkowski space. Note that the
extraction of mass spectra is already a complicated task \cite{BHKRW2007}, not
to speak of an analytic continuation of Euclidean-space form factors.

When dealing with heavy quarkonia or mixed heavy mesons like $B_c$
(found at Fermilab by the CDF Collaboration \cite{BCmesons}),
 some simplifying  approximations are possible. Different
approaches have been developed to reduce the computational complexity of the
full four-dimensional (4D) BSE. The so-called instantaneous \cite{INSTA} and
quasi-potential approximations \cite{QUASI}
can reduce the 4D BSE to a 3D equation in a Lorentz-covariant manner. In
practice, such 3D equations are much more tractable, since their resolution is
less involved, especially if one exploits the considerable freedom in
performing the 3D reduction. Also note that, contrary to the BSE in the ladder
approximation, these equations reduce to the Schr\"{o}dinger equation of
nonrelativistic Heavy-Meson Effective Theory and nonrelativistic QCD
\cite{HEAVY}. However, the interaction kernels of the reduced equations
often correspond to input based on economical phenomenological models, and the
connection to the underlying theory (QCD) is less clear (if not abandoned from
the onset).

In the present paper, we extend the method of solving the full 4D BSE,
originally developed for pure scalar theories \cite{NAKAN,KUSIWI,SAUADA2003},
to theories with nontrivial spin degrees of freedom. Under a certain
assumption on the functional form of Green's functions, we develop a method
of solving the BSE directly in Minkowski space, in its original manifestly
Lorentz-covariant 4D form. In order to make our paper as self-contained as
possible, we shall next supply some basic facts about the BSE approach to
relativistic mesonic bound states.

The crucial step to derive the homogeneous BSE for bound states is the
assumption that the bound state reflects itself in a pole of the four-point
Green's function for on-shell total momentum $P$, with $P^2=M_j^2$, viz.\
\begin{equation}
G^{(4)}(p,p',P)=\sum_j\frac{-i}{(2\pi)^4}\frac{\psi_j(p,P_{os})
\bar{\psi_j}(p',P_{os})}{2E_{p_j}(P^0-E_{p_j}+i\epsilon)}+\mbox{regular terms}\;,
\end{equation}
where $E_{p_j}=\sqrt{\vec{p}\,{}^2+M_j^2}$ and $M_j$ is the (positive) mass of
the bound state characterized by the BS wave function $\psi_j$ carrying the
set of quantum numbers $j$.
Then the BSE can be conventionally written in momentum space like
\begin{eqnarray}
S_1^{-1}(p_+,P)\psi(p,P)S_2^{-1}(p_-,P)&&=-i\int\frac{d^4k}{(2\pi)^4}
V(p,k,P)\psi(p,P)\, ,
 \\
p_+&&=p+\alpha P \, ,
\nonumber \\
p_-&&=p-(1-\alpha)P \, ,
\nonumber
\end{eqnarray}
or, equivalently, in terms of BS vertex function $\Gamma$ as
\begin{eqnarray}  \label{wakantanka}
\Gamma(p,P)&=&-i\int\frac{d^4k}{(2\pi)^4}V(p,k,P)S_1(k_+,P)
\Gamma(p,P)S_2(k_-,P) \, ,
\end{eqnarray}
where we suppress all Dirac, flavor and Lorentz indices, and $\alpha\in(0,1)$.
The function $V $ represents  the two-body-irreducible interaction kernel, and
 $S_i$ ($i=1,2$) are the dressed propagators of the constituents. The free
propagators read
\begin{equation}
S_i^0(p)=\frac{\not p+m_i}{p^2-m^2_i+i\epsilon}.
\end{equation}

Concerning solutions to the BSE (\ref{wakantanka}) for pseudoscalar mesons,
they have the generic form \cite{LEW}
\begin{equation} \label{gen.form}
\Gamma(q,P)=\gamma_5[\Gamma_A+\Gamma_Bq.P\not\!q+\Gamma_C\not\!P+
\Gamma_D\not\!q\not\!P+  \Gamma_E\not\!P\not\!q] ,
\end{equation}
where
the $\Gamma_i$, with $i=A,B,C,D,E$, are scalar functions of their arguments
$ P,q$. If the bound state has a well-defined charge parity, say
${\cal{C}}=1$, then these functions are even in $q.P$, and furthermore
$\Gamma_D=-\Gamma_E$.

As was already discussed in Ref.~\cite{MUNCZEK}, the dominant contribution to
the BSE vertex function for pseudoscalar mesons comes from the first term in
Eq.~(\ref{gen.form}). This is already true, at a 15\% accuracy level, for the
light pseudoscalars $\pi,K,\eta$, while in the case of ground-state heavy
pseudoscalars, like the $\eta_c$ and $\eta_b$, the contributions from the other
tensor components in Eq.~(\ref{gen.form}) are even more negligible.
Hence, at this stage of our Minkowski calculation, we also approximate our
solution by taking  $\Gamma=\gamma_5\Gamma_A$.

The interaction kernel is approximated by the dressed gluon propagator,
with the interaction gluon-quark-antiquark vertices taken in their bare forms.
Thus, we may write
\begin{equation} \label{landau}
V(p,q,P)=g^2(\kappa) D_{\mu\nu}(p-q,\kappa)\gamma^{\nu}\otimes\gamma^{\mu} \, ,
\end{equation}
where the full gluon propagator is renormalized at a scale $\kappa$. The
effective running strong coupling $\alpha_s$ is then related to $g$ through the
equations
\begin{eqnarray}  \label{gluon}
g^2(\kappa)D_{\mu\nu}(l,\kappa)&&=
\alpha_s(l,\kappa)\frac{ P^T_{\mu\nu}(l)}{l^2+i\epsilon}-\xi g^2(\kappa)
\frac{l_{\mu}l_{\nu}}{l^4+i\epsilon}\, ,
\\
\alpha_s(q,\kappa)&&=\frac{g^2(\kappa)}{1-\Pi(q^2,\kappa)}\, ,
\nonumber \\
P^T_{\mu\nu}(l)&&=-g_{\mu\nu}+\frac{l_{\mu}l_{\nu}}{l^2}\, .
\nonumber
\end{eqnarray}

From the class of $\xi$-linear covariant gauges, the Landau gauge $\xi=0$ will
be employed throughout the present paper.

In the next section, we shall derive the solution for the dressed-ladder
approximation to the BSE, i.e., all propagators are considered dressed ones,
and no crossed diagrams are taken into account. The BSE for quark-antiquark
states has many times been treated in Euclidean space, even beyond the ladder
approximation. Most notably, the importance of dressing the proper vertices in
the light-quark sector was already stressed in Ref.~\cite{ACHJO}, so our
approximations are certainly expected to have a limited validity.
Going beyond the rainbow ($\gamma_{\mu}$) approximation is straightforward
but rather involved. (For comparison, see the Minkowski study of
Schwinger-Dyson equations published in Refs.~\cite{SAULIJHEP,SAULI2}),
the latter paper including the minimal-gauge covariant vertex instead of the
bare one). In the present paper, we prefer to describe the computational
method rather than carrying out a BSE study  with the most sophisticated
kernel known in the literature.

The set-up of this paper is as follows. In Sec.~2 we describe the method of
solving the BSE. As a demonstration, numerical results are presented in
Sec.~3. Conclusions are drawn in Sec.~4. The detailed derivations of the integral
equation, that we actually solved numerically, are presented in the Appendices.

\section{Integral representation and solution of the BSE}

In this section we describe our method of solving the BSE in Minkowski space.
It basically assumes that the various Green's functions appearing in the
interaction kernel can be written as weighted integrals over the various
spectral functions (i.e., the real distribution) $\rho$.

More explicitly stated, the full quark and gluon propagators, the latter ones
in the Landau gauge, are assumed to satisfy the standard Lehmann
representation, which reads
\begin{equation} \label{srforquark}
S(l)=\int_{0}^{\infty}d\omega\frac{\rho_v(\omega)\not l+
\rho_s(\omega)}{l^2-\omega+i\epsilon}\,,
\end{equation}
\begin{equation} \label{srforgluon}
G_{\mu\nu}(l)=\int_{0}^{\infty}d\omega\frac{\rho_g(\omega)}{l^2-\omega+i\epsilon}
P^T_{\mu\nu}(l) \, ,
\end{equation}
where $\rho$ is a real istribution.
 Until now, with certain limitations, the
integral representations ~(\ref{srforquark}) and (\ref{srforgluon}) have been used for the nonperturbative evaluation
of Green's functions in various models \cite{SAFBP}. However, we should note
here that the true analytic structure of QCD Green's functions is not reliably
known (also see Refs.~\cite{ALKSME,FISHER1,SABIA}), which studies suggests the tructure given by ~(\ref{srforquark}) and (\ref{srforgluon}) is not sufficient if not excluded. In this case, the lehmann representation or perhaps the ussage of real $\rho$ in the integral representation ~(\ref{srforquark}) and (\ref{srforgluon}) can be regarded as an analyticized approximation of the true
quark propagator. The complexification of $rho$ within the complex integration path is one of the straightforward  and  questionable generalization \cite{ARRBRO}. The general question of the existence of Lehamnn represintation in QCD is beyond the scope of presented paper and we do not discussed the problem furthermore.

Furthermore, we generalize here the idea of the Perturbation Theory Integral
Representation (PTIR) \cite{NAKAN}, specifically for our case. The PTIR
represents a unique integral representation (IR) for an $n$-point Green's function
defined by an $n$-leg Feynman integral.
The generalized PTIR formula for the $n$-point function in a theory involving
fields with arbitrary spin is exactly the same as in the original scalar theory
considered in Ref.~\cite{NAKAN}, but the spectral function now acquires a
nontrivial tensor structure. Let us denote such a generalized weight function by
$\rho(\alpha,x_i)$.  Then, it can be clearly decomposed into the sum
\begin{equation}
\rho(\alpha,x_i)_{\mbox{\scriptsize scalar theory}}\rightarrow \sum_j
\rho_j(\alpha,x_i){\cal{P}}_j  ,
\end{equation}
where $\alpha,x_i$ represent the set of spectral variables, and $j$ runs over
all possible independent combinations of Lorentz tensors and Dirac matrices
$P_j$. The function $\rho_j(\alpha,x_i)$ just represents the PTIR weight
function of the $j$-th form factor (the scalar function by definition), since
it can obviously be written as a suitable scalar Feynman integral. Leaving
aside the question of (renormalization) scheme dependence, we refer the reader
to the textbook by Nakanishi \cite{NAKAN} for a detailed derivation of the
PTIR. The simplest examples of such "generalized" integral representations corresponds with Lehmann
representations for spin half ~(\ref{srforquark}) and spin one propagators  (\ref{srforgluon}).

Let us now apply our idea to the pseudoscalar bound-state vertex function
keeping in mind that the singularity structure (given by the denominators)
of the r.h.s.\ of the  BSE is the same as in the scalar models studied in
Refs.~\cite{KUSIWI,SAUADA2003}, the appropriate IR for the pseudoscalar
bound- state vertex function $\Gamma_A(q,P)$ should read
\begin{equation}  \label{repr}
\Gamma_A(q,P)=\int_{0}^{\infty} d\omega \int_{-1}^{1}dz
\frac{\rho_A^{[N]}(\omega,z)}
{\left[F(\omega,z;P,q)\right]^N}\, ,
\end{equation}
where we have introduced a useful abbreviation for the denominator of the
IR~(\ref{repr}), viz.\
\begin{equation} \label{efko}
F(\omega,z;P,q)=\omega-(q^2+q.Pz+P^2/4)-i\epsilon \, ,
\end{equation}
with $N$ a free integer parameter.

Substituting the IRs~(\ref{repr}), (\ref{srforgluon}), (\ref{srforquark}) into
the r.h.s.\ of the BSE~(\ref{wakantanka}), one can analytically integrate over
the loop momenta. Assuming the uniqueness theorem \cite{NAKAN}, we should
arrive at the same IR~(\ref{repr}), because  of the r.h.s.\ of the
BSE~(\ref{wakantanka}). The derivation is given in Appendix A for the cases
$N=1,2$.

In other words, we have converted the momentum BSE (with a singular kernel)
into a homogeneous two-dimensional integral equation for the real weight
function $\rho_A^{[N]}(\omega,z)$, i.e.,
\begin{equation} \label{madrid}
\rho^{[N]}_A(\tilde{\omega},\tilde{z})=
\int_{0}^{\infty} d\omega \int_{-1}^{1}dz
V^{[N]}(\tilde{\omega}, \tilde{z};\omega,z)\rho^{[N]}_A(\omega,z) ,
\end{equation}
where the kernel $V^{[N]}(\tilde{\omega}, \tilde{z};\omega,z)$ is a regular
multivariable function.

The kernel $V^{[N]}$ also automatically supports the domain $\Omega$ where
the function $\rho^{[N]}_A(\omega,z)$ is nontrivial. This domain is always
smaller then the infinite strip $[0,\infty)\times[-1,1]$, as is explicitly
assumed by the boundaries of the integrals over $\omega$ and $z$.
For instance, with the simplest kernel parametrized by a free gluon
propagator and constituent quarks of mass $m$, we get for the flavor-singlet
meson $\rho^{[N]}_A(\omega,z)\neq 0$ only if $\omega>m^2$.

In our approach, to solve the momentum BSE in Minkowski space is equivalent
to finding a real solution to the real integral equation~(\ref{madrid}). No
special choice of frame is required. If one needs the resulting vertex
function, can be obtained by numerical integration over $\rho_N$ in an
arbitrary reference frame.

\section{ Numerical Results}

In this section we discuss the numerical solution of the BSE with various
interaction kernels. For that purpose, we shall vary the coupling strength
as well as the effective gluon mass $m_g$. We are mainly concerned with the
range of binding energies that coincide with those of heavy quarkonia, which
systems we shall study in future work. Moreover, we take a discrete set of
values for the mass $m_g$, such that it runs from zero to the value of the
constituent quark mass. These values are expected to be relevant for the case
of a true gluon propagator (when $m_g$ is replaced by the continuous spectral
variable $\omega$ (\ref{srforgluon})). Thus, in each case, the corresponding
gluon density is  $\rho_g(c)=N_g\delta(c-m^2_g)$, which specifies the kernel
of the BSE to be (in the Landau gauge)
\begin{equation}  \label{gluon2}
V(q-p)=g^2
\frac{-g_{\mu\nu}+\frac{(q-p)_{\mu}(q-p)_{\nu}}{(q-p)^2}}
{(q-p)^2-m_g^2+i\epsilon}
\gamma^{\nu}\otimes\gamma^{\mu}
\end{equation}
where the prefactor (including the trace of the color matrices) is simply
absorbed in the coupling constant. For our actual calculation, we use the bare
constituent propagator $S_i(p_i)$ with heavy quark mass $M\equiv m$ (see Appendix A
for this approximation).

Firstly, we follow the standard procedure: after fixing the bound-state mass
($\sqrt{P^2}$), we look for a solution by iterating the BSE for a spectral
function with fixed coupling constant $\alpha=g^2/(4\pi)$.
Very similarly to the scalar case \cite{SAUADA2003}, the choice $N=2$ for the
power of $F$ in the IR of the bound-state vertex function is the preferred one.
This choice is a reasonable compromise between on the one hand limiting
numerical errors and on the other hand avoiding the computational obstacles
for high $N$. Here we note that using $N=1$ is rather unsatisfactory
(comparing with the massive Wick-Cutkosky model), since then we do not find any
stable solution for a wide class of input parameters $g$, $m_g$. In contrast,
using the value $N=2$ we obtain stable results for all possible interaction
kernels considered here. This includes the cases with vanishing $m_g$, which
means that the numerical problems originally present in the  scalar models
\cite{SAUADA2003} are fully overcome here. The details of our numerical
treatment are given in Appendix B.

As is more usual in the nonrelativistic case, we fix the coupling constant
$\alpha=g^2/(4\pi)$ and then look for the bound-state mass spectrum. We find
the same results in either case, whether $P$ or $\alpha$ is fixed first,
noting however that in the latter case the whole integration in the kernel
$K$ needs to be carried out in each iteration step, which makes the problem
more computer-time consuming.

\begin{figure}
\centerline{  \mbox{\psfig{figure=soub3.ps,height=14.0truecm,
width=14.0truecm,angle=0}} }
\caption[99]{\label{figBSE} The rescaled weight function
$\tau=\frac{\rho^{[2]}(\omega,z)}{\omega^2}$ for the following model parameters:
$\eta=0.95$, $m_g=0.001M$, $\alpha_s=0.666$; the small mass $m_g$
 approximates the one-gluon-exchange interaction kernel. }
\end{figure}

The obtained solutions for varying $\alpha$ and mass $m_g$, with
a fixed fractional binding $\eta=\sqrt{P^2}/(2M)=0.95$, are given in Table~1.
If we fix the gluon mass at $m_g=0.5$ and vary the fractional binding $\eta$,
we obtain the spectrum of Table~2.

\begin{center}
\small{\begin{tabular}{|c|c|c|c|c|}
\hline \hline
$m_g/m_q $ &   $10^{-3}$  & 0.01  & 0.1 & 0.5   \\
\hline
 $\alpha$ & 0.666 & 0.669  & 0.745 & 1.029  \\
\hline \hline
\end{tabular}}
\end{center}

\begin{center}
TABLE 1. Coupling constant $\alpha_s=g^2/(4\pi )$  for
several choices of $ m_g/M$,
with given binding fraction $\eta=\sqrt{P^2}/(2M)=0.95$.
 \end{center}

\begin{center}
\small{\begin{tabular}{|c|c|c|c|c|}
\hline \hline
$\eta: $ &0.8 &  0.9  & 0.95 & 0.99   \\
\hline
 $\alpha$ &1.20 & 1.12 & 1.03  & 0.816  \\
\hline \hline
\end{tabular}}
\end{center}

\begin{center}
TABLE 2. Coupling $\alpha_s=g^2/(4\pi )$ as a
function of binding fraction $\eta=\sqrt{P^2}/(2M)$, for
exchanged massive gluon with $m_g=0.5M$.
\end{center}

For illustration, the  weight function $\tilde{\rho}^{[2]}$  is
displayed in Fig.~\ref{figBSE}.

\section{ Summary and  Conclusions}

The main result of the present paper is the development of a technical
framework  to solve the bound-state BSE in Minkowski
space. In order to obtain the spectrum, no preferred reference frame is
needed, and the wave function can be obtained in an arbitrary frame
--- without numerical boosting --- by a simple integration of the weight
function.

The treatment is based on the usage of an IR for the
Green's functions of a given theory, including the bound-state vertices
themselves. The method has been explained and checked numerically on the
samples of pseudoscalar fermion-antifermion bound states. It was shown
that the momentum-space BSE can be converted into a real equation for a
real weight functions $\rho$, which is easily solved numerically.
The main motivation of the author was to develop a practical tool respecting
selfconsistency of DSEs and BSEs. Generalizing this study to other mesons,
such as vectors and scalars, and considering more general flavor or isospin
structures, with the simultaneous improvement of the approximations
(correctly dressed gluon propagator, dressed vertices, etc.), will be an
essential step towards a fully Lorentz-covariant description of a plethora of
transitions and form factors in the time-like four-momentum region.

\

\

{\Large{ \bf Acknowledgments}}

\

I would like to thank George Rupp for his careful reading of the manuscript.


\section{Introduction}
Wireless sensor networks (WSNs) have attracted attention from wireless network research communities for more than a decade \cite{Akyildiz:2002:WSN:Survey}. Their abilities to collect environmental data and, to effectively and wirelessly send those data back to central, processing nodes, have been identified by research work such as \cite{senvm}. Recently, with the emerging Internet of Things (IoT) \cite{stankovic_iot}, researchers foresee its potentials to bring a new generation of the Internet where things are connected to engender unprecedented intelligence that facilitates people's daily lives, enterprise's tasks, and city-wide missions. WSNs are an integral part of the IoT because they are the sources of sensed data to be processed and analyzed by the computing clouds. Therefore, WSNs in the IoT are expected to handle operations with more density, diversity, and tighter integration.

WSNs often demand that sensed data be incorporated with timestamps so that the systems can fuse, distinguish and sequence the data consistently. Therefore, several protocols proposing to synchronize a global time in WSNs include \cite{Maroti:2004:FTS:1031495.1031501,5211944,Apicharttrisorn:2010:EGT:1900724.1901046}. However, some systems do not require a global notion of time; they demand sensor nodes to sample data at the same time to take sequential snapshots of an environment. Such systems include \cite{Werner-Allen:2005:FSN:1098918.1098934}, which is the first protocol to achieve this task of ``synchronicity''.  Desynchronization is an \emph{inverse} of synchronicity because it requires nodes \emph{not} to work at same time; hence, desynchronization can provide nodes with collision-free and even equitable access to a shared resource. A concrete example is a system using a Time Division Multiple Access or TDMA protocol, in which nodes utilize a shared medium at different time slots to avoid collision. In addition, desynchronization can schedule duty cycles of sensor nodes; in other words, nodes covering the same sensing area take turns waking up to sense the environment while others are scheduled to sleep to save the limited energy. Other potential applications of desynchronization include techniques to increase a sampling rate in multiple analog-to-digital converters, to schedule resources in multi-core processors, and to control traffic at intersections \cite{4274893}.

In this paper, we propose a stable desynchronization algorithm for multi-hop wireless sensor networks called Multi-hop Desynchronization With an ARtificial Force field or M-DWARF. We use TDMA to validate our algorithm and evaluate its performance. M-DWARF uses basic concepts of artificial force fields in DWARF \cite{Choochaisri:2012:DAF:2185376.2185378}. However, to support multi-hop networks and to avoid their hidden terminal problems, M-DWARF adds two mechanisms called \textit{Relative Time Relaying} and \textit{Force Absorption}. With these features added, M-DWARF is able to desynchronize multi-hop networks without collision from hidden terminals while maintaining maximum channel utilization. We evaluate M-DWARF's functionality on TelosB motes \cite{telosb} and its performance on TOSSIM \cite{Levis:2003:TAS:958491.958506}, a TinyOS simulator. We compare M-DWARF with Extended Desynchronization for Multi-hop Networks or EXTENDED-DESYNC (also referred as EXT-DESYNC) \cite{MK09DESYNC} and Lightweight coloring and desynchronization for networks or LIGHTWEIGHT \cite{5062165} on several topologies of multi-hop wireless networks. According to the simulation results, M-DWARF is the only desynchronization that has all the three properties - fast convergence, high stability, and maintained fairness. Moreover, in \cite{Choochaisri:2012:DAF:2185376.2185378}, we prove that desynchronization using artificial force fields is a convex function; that is, it converges to a global minimum without local ones. In addition, our stability analysis (in a supplementary document) proves that once DWARF or M-DWARF systems reach a state of desynchrony, they become stable. In other words, DWARF or M-DWARF provides a static equilibrium of desynchronized force fields at steady states. Once nodes in the systems deviate from a state of desynchrony, they will attempt to return back to the balance of the forces immediately. Our stability analysis not only suggests the stability of our desynchronization our algorithms but also proves that the systems will eventually converge to an equilibrium.

In the next section, we briefly explain how our proposed desynchronization, M-DWARF, works and describe its contributions. In Section \ref{sec:related_work}, we survey related literature of desynchronization in temporal and spatial domains, as well as TDMA scheduling algorithms. Section \ref{sec:desync_algo} recalls the basic concepts of DWARF and then explains the M-DWARF algorithm in detail. Finally, Appendix \ref{sec:psuedocode_m_dwarf} shows the psuedocodes of M-DWARF.

\section{Contributions}
\label{sec:contribution}
In order to understand the contributions of the M-DWARF algorithm, it is important to know basically how it works.
In addition to DWARF, M-DWARF has two mechanisms, which are named relative time relaying and force absorption, to support multi-hop topologies. The algorithmic steps of M-DWARF can be enumerated as follows.

\begin{enumerate}
\item Nodes, which are not initially desynchronized, set a timer to fire in $T$ time unit and listen to all one-hop neighbors.
\item Upon receiving a firing message from its one-hop neighbor, the node marks the current time to be the relative phase reference. Then, it reads relative phases of its two-hop neighbors which are included within the firing message. After that, it calculates their two-hop neighbors' phases by using the relative phase reference as the offset. The details are explained in Section \ref{sec:relative}.
\item When the timer, $T$, expires, a node broadcasts a firing message containing relative phases of its one-hop neighbors. Then, it calculates a new time phase to move on the phase circle, which is based on the summation of artificial forces from all phase neighbors within two hops where some forces are absorbed as explained in Section \ref{sec:absorption}. Then, the node sets a new timer according to the new calculated phase.
\item Similar to DWARF, a node adjusts its phase by using Eq. \ref{eq:newphase} with  $K$ calculated from Eq. \ref{eq:arbitrary_T}.
\end{enumerate}

All nodes in the artificial force field (in the period circle) iteratively run the same algorithm until the force is balanced.
The pseudo-code of this algorithm is shown in Appendix \ref{sec:psuedocode_m_dwarf}. The contributions of our proposed desynchronization algorithms are listed as follows.

\paragraph{Autonomous operations} M-DWARF works without any master or root nodes, so it does not need to elect or select such nodes and also does not have a single point of failure in this sense. Moreover, nodes do not need knowledge about network topology; they only have to know their one-hop and two-hop neighbors in order for the desynchronization to function correctly. In addition, M-DWARF is easy to be deployed because it works independently without any deployment setup. Finally, M-DWARF adapts itself very well to dynamics such as leaving or joining nodes.

\paragraph{Determinism} M-DWARF does not use any random operation, so its protocol behavior is deterministic and predictable.

\paragraph{Throughput} Thanks to its high stability, at a state of desynchrony, a node's time slot firmly stays at the same position in the next iteration, causing no or minimal interference with adjacent time slots. As a result, M-DWARF requires less \emph{guard time} between slots, allowing nodes to fully utilize the resource or medium because of its stable desynchronization. Without stability, the beginning and the end of the frames are likely to collide or interfere with the adjacent ones because time slots of the next iteration do not strictly stay at the same position. Moreover, M-DWARF requires low overhead. In each time period \textit{T}, each node only broadcasts a desynchronization message containing one-hop and two-hop neighbor information; in other words, it does not need any two-way or three-way handshakes like many other protocols.

\paragraph{Delay} A node that starts to join the desynchronization network needs to broadcast a desynchronization message to declare itself to the network and occupy its own time slot. Therefore, it has to wait one time period to send the first data frame. By determining the slot size and transmission bitrate, the node can predict how many iterations it needs to finish transmitting one packet. In consequence, all the nodes in the network route can share information regarding the end-to-end delay of a packet traversal. Therefore, upper and lower limits of such a delay can be determined.

\paragraph{Interoperability} Because M-DWARF does not assume any radio hardware specifications or MAC-layer techniques, it is highly interoperable with open standards. It assumes only that all nodes can transmit and receive data at the same spectrum. Although WSNs are a target platform of this paper, we argue that our desynchronization and TDMA algorithms can be applied to generic wireless multi-hop networks \cite{Sgora:2015:TDMA} or wireless mesh networks \cite{Vijayalayan:2013:Scheduling}.

\paragraph{Complexity} M-DWARF has low complexity for the following three reasons. First, it does not require time synchronization between nodes. Second, there are no handshakes; only one-way broadcasting is necessary. Third, its computational complexity depends only on the number of one-hop and two-hop neighbors, instead of the entire network size.

\paragraph{Energy Efficiency} Because of M-DWARF's high stability, data frames of adjacent time slots are less likely to collide or interfere with each other. Therefore, it has lower probability for packet retransmission, which is a factor for energy inefficiency in networks.

\paragraph{Channel Utilization Fairness} M-DWARF can provide equal access for all the neighbor nodes. The artificial forces are balanced between neighbor nodes, so all the nodes converge to occupy equal slot sizes.

\section{Related Work}
\label{sec:related_work}
Our related work can be mainly divided into three categories - 1) desynchronization on a temporal domain in wireless networks 2) desynchronization on a spatial domain in robotics 3) TDMA scheduling algorithms. Then, we summarize the properties of related work of the first category in Table \ref{tab:compared}.

\subsection{Desynchronization on a Temporal Domain in Wireless Networks}
\label{sec:timedesync}
To the best of our knowledge, Self-organizing desynchronization or DESYNC \cite{4379660}  is the first to introduce the desynchronization problem. In DESYNC, a node simply attempts to stay in the middle phase position between its previous and next phase neighbors. By repeating this simple algorithm, all nodes will eventually and evenly be spread out on a temporal ring. However, the error from one phase neighbor is also propagated to the other phase neighbors and is indefinitely circulated inside the network. As a consequence, DESYNC's error is quite high even after convergence. C. M. Lien et al. propose Anchored desynchronization or ANCHORED \cite{anchored} that uses the same method as DESYNC but requires one anchored node to fix the phase of its oscillator. However, because ANCHORED uses only the phase information of the phase neighbors, it shall suffer the similar desynchronization error as DESYNC. In contrast, our work relies on all received neighbors' information and is therefore more robust to the error from one phase neighbor. In  \cite{4663417}, the authors describe how DESYNC works on multi-hop networks and explain an extension for DESYNC by exchanging two-hop neighbor information.

Inversed Mirollo-Strogatz or INVERSE-MS \cite{4274893}, designed to converge faster than DESYNC, is an inverse algorithm of the synchronicity work by \cite{MS1990}.
At a steady state, INVERSE-MS maintains a dynamic equilibrium (\textit{i.e.}, nodes keep changing time positions while maintaining desynchronization). However, in INVERSE-MS, the time period is distorted whereas our algorithm does not distort the time period.
In Extended Desynchronization or EXT-DESYNC \cite{MK09DESYNC}, the authors propose a desynchronization algorithm that is similar to the extension proposed in \cite{4663417}. Each node sends its one-hop neighbors' relative time information to all of its one-hop neighbors.
Then, the one-hop neighbors relay such information to two-hop neighbors so that each node knows two-hop relative time information.
Consequently, each node can presume that there are two-hop neighbors appearing on the time circle.
Therefore, each node uses time information of both one-hop and two-hop neighbors to desynchronize with the same algorithm as in DESYNC. One mechanism in our multi-hop algorithm proposed in this paper is partly based on this notion.

M-DESYNC \cite{5062256} is a localized multi-hop desynchronization algorithm that works on a granularity of time slots. This protocol uses a graph coloring model for solving desynchronization. It starts by estimating the required number of time slots with a two-hop maximum degree or the maximum number of colors. This estimation allows nodes in M-DESYNC to immediately choose each predefined slot or color and helps M-DESYNC converge very fast. However, M-DESYNC requires that all nodes have a global notion of time in order to share the common perception of time slots. Furthermore, M-DESYNC claims that it works only on acyclic networks. On the contrary, our algorithm does not require a global notion of time and can work on both acyclic and cyclic networks.

A. Motskin et al. \cite{5062165} propose a simple, lightweight desynchronization algorithm, namely LIGHTWEIGHT, that is also based on a graph coloring model. Unlike M-DESYNC, the algorithm works on general graph networks and does not need the global time. To ensure that the selected time slot does not overlap with others', a node needs to listen to the shared medium for a full time period before claiming the slot. The listening mechanism can only avoid collision with one-hop neighbors but cannot avoid collision with two-hop neighbors (\textit{i.e.}, the hidden terminal problem). On the contrary, our algorithm works well on multi-hop networks; each node can effectively avoid collision with two-hop neighbors.
Furthermore, without a common notion of time, the starting time of each slot is quite random; as a result, several time gaps are too small to be used as time slots. This external fragmentation problem poorly reduces resource utilization of the system. Finally, to converge faster, their algorithm overestimates the number of required time slots. Hence, several large time gaps are also left unused and the resource utilization is undoubtedly and significantly lowered. In our work, nodes gradually adapt their phases to be separated from each other as far as possible. Therefore, the external fragmentation problem is reduced and the resource utilization is maximized.

T. Pongpakdi et al. propose Orthodontics-inspired Desynchronization or DESYNC-ORT \cite{desyncort}. In their work, they use information from all one-hop neighbors and attempt to find nodes that are already in corrected time positions and tie them up together. This method is similar to the Orthodontics method that attempts to tie teeth which are already in corrected positions together.
The desynchronization errors of DESYNC-ORT are lower than those of DESYNC.
However, to calculate the correct positions, each node is required to know the total number of nodes in the system in advance. Additionally, the algorithm does not propose to solve the problem in multi-hop networks because nodes in two-hop neighbors cannot be tied together with one-hop neighbors. In contrast, our algorithm does not require nodes to have knowledge about the total number of nodes in advance but gradually and automatically adapts itself based on the current number of neighbors. Finally, our algorithm works on multi-hop networks.

Vehicular-network Desynchronization, abbreviated as V-DESYNC \cite{v-desync}, is proposed to desynchronize nodes in vehicular ad-hoc networks. Their work has a different objective; that is, the algorithm does not focus on fairness (\textit{i.e.}, nodes are not necessary to be equitably separated) because vehicular networks are highly dynamic. In our work, we focus on wireless sensor networks with static sensor nodes and we attempt to render resource utilization fairness among sensor nodes.

Table \ref{tab:compared} summarizes the features of works in this category. Note that the overhead of the proposed algorithm depends on whether the algorithm works on the single-hop or multi-hop mode.

\begin{table*}[t]
	\centering
	\renewcommand{\arraystretch}{1.2}
\caption{Comparison of Desynchronization Protocols}
	{
		\begin{tabular}{|m{2cm}|m{1.5cm}|m{1.5cm}|m{1.2cm}|m{1.2cm}|m{1.5cm}|m{1.2cm}|m{1.5cm}|m{1cm}|}
			\hline
			\multirow{2}{*}{Algorithms} & \multicolumn{8}{c|}{Properties}\\
			\cline{2-9}
			& Period & Time sync & Fair Space & Multi- hop & Conver- gence & Error & Scalable & Over- head\\
			\hline
			\hline
			DESYNC & Fixed & No & Yes & No & Moderate & High & Poor & Zero\\
			\hline
			ANCHORED & Fixed & No & Yes & No & Moderate & High & Poor & Zero\\
			\hline
			INVERSE-MS & Distorted & No & Yes & No & Fast & Low & Good & Zero\\
			\hline
			EXT-DESYNC & Fixed & No & Yes & Yes & Moderate & High & Poor & High\\
			\hline
			M-DESYNC & Fixed & Required & No & Yes & Fast & High & Good & Low\\
			\hline
			LIGHT- WEIGHT & Fixed & No & No & Yes & Fast & High & Good & Zero\\
			\hline
			DESYNC-ORT & Fixed & No & Yes & No & Moderate & Low & Good & Zero\\
			\hline
			V-DESYNC & Fixed & No & No & No & No & High & Good & Zero\\
			\hline
			DWARF & Fixed & No & Yes & No & Moderate & Low & Good & Very Low\\
			\hline
			M-DWARF (Proposed) & Fixed & No & Yes & Yes & Moderate & Low & Good & Low\\
			\hline
		\end{tabular}
	}

	\label{tab:compared}
\end{table*}

\subsection{Desynchronization on a Spatial Domain in Robotics}
\label{sec:spacedesync}

\begin{figure}
	\centering
	\subfloat[Robotic Close Ring]{
		\label{fig:subfig:robotic_closed_ring}
		\includegraphics[width=1.2in]{figure/robotic-closed-ring}
	}
	\hspace{1.5cm}
	\subfloat[Robotic Perfect Close Ring]{
		\label{fig:subfig:robotic_closed_ring_perfect}
		\includegraphics[width=1.2in]{figure/robotic-closed-ring-perfect}
	}
	\caption{Robotic pattern formation on a closed ring. (\ref{fig:subfig:robotic_closed_ring}) Robots are randomly placed on a closed ring. (\ref{fig:subfig:robotic_closed_ring_perfect}) In the perfect configuration, robots are equitably separated from each other.}
	\label{fig:robotic_ring}
\end{figure}

In robotic pattern formation, multiple robots distributedly group and align themselves in geometric patterns such as circles, rectangles, and triangles. Without an explicit argument, robotic pattern formation can be abstracted as desynchronization on a spatial domain. Robots attempt to separate away from each other as far as possible to form such patterns; in other words, robots desynchronize themselves spatially to avoid the collision with each other in the spatial domain.

These two pieces of work  \cite{suzuki-96,suzuki-99} investigate several robotic pattern formations. However, the pattern formation that is similar to desynchronization on the temporal domain is the formation on a closed ring.
Figure \ref{fig:robotic_ring} illustrates a robotic formation on a closed ring. In Figure \ref{fig:subfig:robotic_closed_ring}, initially, robots are randomly placed on any position on the closed ring. The perfect configuration of the formation is illustrated in Figure \ref{fig:subfig:robotic_closed_ring_perfect}; that is, robots are equivalently separated away on the ring.

Other papers such as \cite{defago04,cohen-08,flocchini-08} propose the algorithms that are similar to each other for robotic formations on a closed ring, assuming that robots have limited visibility range. Each robot attempts to adjust its position to the middle between two nearest robots on its left side and right side (Figure \ref{fig:closedring-desync}). In their papers, they prove that this simple algorithm eventually converges the robotic formation to the perfect configuration (Figure \ref{fig:closedring-desync-perfect}).

\begin{figure}
	\centering
	\subfloat[A Robotic Move]{
		\label{fig:closedring-desync}
		\includegraphics[width=1.2in]{figure/robotic-closed-ring-desync}
	}
	\hspace{1.5cm}
	\subfloat[Convergence to Perfect Configuration]{
		\label{fig:closedring-desync-perfect}
		\includegraphics[width=1.2in]{figure/robotic-closed-ring-desync-perfect}
	}
	\caption{Moving to the midpoint algorithm. (a) Each robot moves to the midpoint between two nearest visible neighbors. (b) The algorithm converges to the perfect configuration..}
	\label{fig:robotic-closed-ring-desync}
\end{figure}

In \cite{4141997}, heterogeneous robots are grouped in a distributed manner into teams that are equally spread out to cover the monitored area. Each robot has no global knowledge of others' absolute positions but can detect relative positions of the others with respect to itself as well as the type of the others.
To form a circle, an artificial force is used as an abstraction for velocity adaptation of a robot.
Robots of different types have attractive forces to each other, while robots of the same type have repulsive forces. As a result, a circle of heterogeneous robots is formed and robots are spaced on the circle  (see Figure \ref{fig:circular_formation}). This work inspires our desynchronization algorithm.

\begin{figure}
	\centering
	\includegraphics[width=3.0in]{figure/robo_new}
	\caption{Results of Robotic Circular Formation. Robots with two different types form the circle.}
	\label{fig:circular_formation}
\end{figure}

\subsection{TDMA Scheduling Algorithms}
\label{sec:tdma}

Other works that are related to desynchronization protocols are distributed Time Division Multiple Access (TDMA) protocols. Distributed TDMA protocols are similar to M-DESYNC \cite{5062256}; their protocols work on a granularity of time slots. Similar to M-DESYNC, many of the distributed TDMA protocols such as TRAMA \cite{trama}, Parthasarathy \cite{Parthasarathy}, ED-TDMA \cite{edtdma}, and Herman \cite{herman} assume time is already slotted or all nodes are synchronized to achieve the same global clock.
In our work, we do not require time synchronization and do not assume already slotted time.

S. C. Ergen et al. \cite{Ergen:2010:TDMAAlgo} propose node-based and level-based TDMA scheduling algorithms for WSNs. Their techniques mainly derive from graph coloring algorithms in which the \textit{colors} must have been predefined. In contrast, our desynchronization algorithms never predefine slots but rather allow nodes to adjust their slots with those in the neighborhood on-the-fly.
K. S. Vijayalayan et al. \cite{Vijayalayan:2013:Scheduling} survey distributed scheduling techniques for wireless mesh networks and A. Sgora et al. \cite{Sgora:2015:TDMA} provide an extensive survey of recent advances in TDMA algorithms in wireless multi-hop networks.

Both the survey papers interestingly give a comprehensive overview of the TDMA scheduling algorithms and techniques that are still being investigated and further developed by wireless network researchers worldwide.

\section{DWARF and M-DWARF Desynchronization Algorithms}
\label{sec:desync_algo}
In this section, we briefly introduce the concept of \textit{artificial force field} and our previous work of desynchronization for single-hop networks \cite{Choochaisri:2012:DAF:2185376.2185378} because it is necessary to understand the basic concepts before the proposed algorithm for multi-hop networks can be understood.

\subsection{Desynchronization Framework, Artificial Force Field and DWARF Algorithms}
\label{AFF_DWARF}
\subsubsection{Desynchronization Framework}
The desynchronization framework is depicted as a time circle in Figure \ref{fig:time_circle}.
The perimeter of a time circle represents a configurable time period $T$ of nodes' oscillators.
The time position or \textit{phase} of each node represents its turn to perform a task (\textit{e.g.}, accessing a shared resource, sampling data, or firing a message).
The system is desynchronized when all nodes are separated in the time circle. We define the terms used in the desynchronization context as follows.

\begin{figure}
	\centering
	\includegraphics[width=3.2in]{figure/time_circle_new}
	\caption{Desynchronization framework: $\phi_1$ and $\phi_2$ are phases of node 1 and 2 respectively. While all the four nodes are phase neighbors to each other, node 2 and 4 are the previous and next phase neighbor of node 1 respectively. The left figure shows a desynchrony state that will converge to the perfect desynchrony state as in the right figure.}
	\label{fig:time_circle}
\end{figure}

\begin{definition}[Phase]
	A phase $\phi_i$ of node $i$ is the time position on the circle of a time period $T$, where $0 \leq \phi_i <  T$ and $T \in \mathbb{R}^+$.
\end{definition}

\begin{definition}[Phase Neighbor]
	Node $j$ is a phase neighbor of node $i$ if node $i$ perceives the existence of node $j$ through reception of $j$'s messages at the phase $\phi_i + \phi_{i,j}$, where $\phi_{i,j}$ is the phase difference between node $j$ and node $i$,
	\begin{equation}
	\phi_{i,j} = \left\{
	\begin{array}{l l}
	\phi_j - \phi_i & \quad \mbox{if $\phi_j \geq \phi_i$,}\\
	T - (\phi_i - \phi_j) & \quad \mbox{if $\phi_j < \phi_i$.}\\ \end{array} \right.
	\end{equation}
\end{definition}

\begin{definition}[Next Phase Neighbor]
	Node $j$ is the next phase neighbor of node $i$ if $\phi_{i,j} = \underset{k \in S}{\min}\{\phi_{i,k}\}$, where $S$ is a set of node $i$'s neighbors.

\end{definition}
\begin{definition}[Previous Phase Neighbor]
	Node $j$ is the previous phase neighbor of node $i$ if \\$\phi_{i,j} = \underset{k \in S}{\max}\{\phi_{i,k}\}$, where $S$ is a set of node $i$'s neighbors.

\end{definition}
\begin{definition}[Desynchrony State]
	The system is in a desynchrony state if $\phi_i \neq \phi_j$ for all $i, j \in V$ and  $i \neq j$, where $V$ is a set of nodes in a network that cannot share the same phase.
\end{definition}
\begin{definition}[Perfect Desynchrony State]
	The system is in the perfect desynchrony state if it is in the desynchrony state and $\phi_{i,j} = T/N$ for all $i \in V$, $j$ is $i$'s previous phase neighbor, and $N$ is the number of nodes in a network that cannot share the same phase.
\end{definition}

We note that two nodes can share the same phase if they are not within the two-hop communication range of each other.

\subsubsection{Artificial Force Field}
\label{AFF}
An artificial force field is an analogy to the circle of a time period where nodes have repulsive forces to each other.
Nodes are in the same force field if they can communicate with each other or share the same medium.
If node $i$ and node $j$ are on the same force field, they have repulsive forces to push one another away.
A closer pair of nodes has a higher magnitude of force than a farther pair does.
The time interval between two nodes is derived from the phase difference between them.
If two nodes have a small phase difference, they have a high magnitude of force and vice versa.
In other words, a repulsive force is an inverse of a phase difference between two nodes:

\begin{equation}
f_{i,j} = - \frac{1}{\Delta \phi_{i,j} / T}, \quad \Delta \phi_{i,j} \in (-\frac{T}{2}, \frac{T}{2}),
\label{eq:force}
\end{equation}
where $f_{i,j}$ is the repulsive force from node $j$ to node $i$ on a time period $T$ and $\Delta \phi_{i,j}$ is the phase difference between node $i$ and $j$.
We note that $\Delta \phi_{i,j}$ is not equal to 0 because if two nodes fire at the same time, their firings collide and two nodes do not record other's firing. Additionally, at $T/2$ or $-T/2$, a node does not repel an opposite node because the forces are balanced.

A repulsive force can be positive (clockwise repulsion) or negative (counterclockwise repulsion).
A positive force is created by a node on the left half of the circle relative to the node being considered whereas a negative force is created by a node on the right half.
Figure \ref{fig:force_field} represents a field of repulsive forces on node 1.

\begin{figure}
	\centering
	\includegraphics[width=1.5in]{figure/forcefield}
	\caption{Artificial Force Field. Arrow lines represent repulsive forces from node 2, 3, and 4 to node 1. A shorter and thicker line is a stronger force. A force from node 4 is a positive force and two forces from node 2 and 3 are negative forces.}
	\label{fig:force_field}
\end{figure}
Each node in the force field moves to a new time position or phase proportional to the total received force.
Given $n$ nodes in a force field, the total force on a node $i$ is the following:
\begin{equation}
\mathcal{F}_i = \sum_{\substack{j=1\\ j \neq i}}^{n}{f_{i,j}}.
\label{eq:fsum}
\end{equation}

Eventually, nodes reach an equilibrium where the total force of the system is close to zero and each pair of phase neighbor nodes has the same time interval.
This equilibrium state also indicates the perfect desynchrony state because all nodes are equally spaced on the time circle.
\subsubsection{DWARF, the Single-Hop Desynchronization Algorithm}
\label{DWARF}
We assume that, initially, nodes are not desynchronized and each node sets a timer to fire in $T$ time unit.
After setting the timer, each node listens to all its neighbors until the timer expires.
When receiving a firing message from its neighbor, the (positive or negative) repulsive force from that neighbor is calculated based on the phase difference.
When the timer expires, a node broadcasts a firing message to neighbors.
Then, the node calculates a new time phase to move on the circle based on the summation of forces from all neighbors and sets a new timer according to the new time phase.

It is reasonable now to question how far a node should move or adjust its phase.
In our work, given the total received force $\mathcal{F}_i$, the node $i$ adjusts to a new time phase $\phi_i^{'}$,
\begin{equation}
\phi_i^{'} = (\phi_i + K\mathcal{F}_i) \mod T,
\label{eq:newphase}
\end{equation}
where $\phi_i$ is the current phase of the node $i$.

Undoubtedly, the proper value of the coefficient $K$ leads to the proper new phase.
The value of $K$ is similar to a step size which is used in artificial intelligence techniques.
Therefore, if the value of $K$ is too small, the system takes much time to converge.
On the other hand, if the value of $K$ is too large, the system may overshoot the optimal value and fail to converge.
We observe that, given the same time period, fewer nodes in the system result in bigger phase difference between two phase neighbors. To be desynchronized, nodes in sparse networks must make a larger adjustment to their time phases than nodes in dense networks.
Therefore, the same total received force should have a greater impact on a node in sparse networks than on a node in dense networks.
To reflect this observation, the coefficient $K$ is inversely proportional to a power of the number of nodes $n$,
\begin{equation}
K = c_1 \times n^{-c_2}, \text{ where } c_1, c_2 \geq 0.
\end{equation}

Therefore, we have conducted an experiment to find the proper value of $c_1$ and $c_2$.

We set a time period $T$ to 1000 and vary the number of nodes.
In the specific number of nodes, we first simulate to see the trend of the value $K$ that leads to small errors.
Then, we select a range of good $K$ values and simulate 100 times to obtain the average desynchronization error for each $K$ value.
In each simulation, we randomly set an initial phase of each node between 0 and $T$ (period value).
Finally, we select the $K$ value that results in the lowest error.
After getting the proper $K$ value for each number of nodes, we plot the relation between $K$ and the number of nodes (Figure \ref{fig:relation_k_n}) and use a mathematical tool to calculate the power regression. The obtained relation function between $K$ and $n$ (the trendline in Figure \ref{fig:relation_k_n}) consists of $c_1$ and $c_2$  values as follows:
\begin{equation}
K = 38.597 \times n^{-1.874}.  \nonumber
\end{equation}
\begin{figure}
	\centering
	\includegraphics[width=2.2in]{figure/k-trendline}
	\caption{Relation of the coefficient $K$ with a number of nodes $n$}
	\label{fig:relation_k_n}
\end{figure}
However, this $K$ value is derived by setting $T$ equal to 1000.
Therefore, for an arbitrary value of $T$,
\begin{equation}
\label{eq:arbitrary_T}
K = 38.597 \times n^{-1.874} \times \frac{T}{1000}.
\end{equation}
The proof of Eq. \ref{eq:arbitrary_T} can be found in \cite{Choochaisri:2012:DAF:2185376.2185378}. Moreover, in \cite{Choochaisri:2012:DAF:2185376.2185378}, we also prove that the force function of DWARF has the convexity property; that is, it has one global minima and no local minima. Additionally, in this paper, we provide stability analysis of DWARF and M-DWARF in the Supplementary Material.

\subsection{M-DWARF, the Multi-hop Desynchronization Algorithm (Proposed)}
\label{sec:m_dwarf}
In this section, we extend the artificial force field concept to desynchronization in multi-hop networks. We begin with applying DWARF directly to a simple multi-hop topology and find out how the algorithm fails on such a topology in Section \ref{sec:hidden-terminal}. Then, we propose two simple yet effective resolutions, relative time relaying and force absorption, to extend DWARF for multi-hop networks in Section \ref{sec:relative} and \ref{sec:absorption} respectively. Additionally, we provide pseudo-code of M-DWARF in Appendix \ref{sec:psuedocode_m_dwarf}.

\subsubsection{The Hidden Terminal Problem}
\label{sec:hidden-terminal}
To see how DWARF works on a multi-hop network, we set a simple 3-node chain topology as illustrated in Figure \ref{fig:3nodes-chain-hidden}. In this topology, node 1 can receive firing messages from node 2 but cannot receive from node 3; on the other hand, node 3 can receive firing messages from node 2 but cannot receive from node 1. However, node 2 can receive firing messages from both node 1 and 3 which are not aware of each other's transmission. This simple scenario causes messages to collide at node 2.

\begin{figure}
	\centering
	\includegraphics[width=2.5in]{figure/3nodes-chain-hidden}
	\caption{The hidden terminal problem.}
	\label{fig:3nodes-chain-hidden}
\end{figure}

\begin{figure}
	\centering
	\subfloat[]{
		\includegraphics[width=2in]{figure/3nodes-chain-dwarf}
		\label{fig:3nodes-chain-dwarf}
	}
	\hspace{1.0cm}
	\subfloat[]{
		\includegraphics[width=2in]{figure/3nodes-chain-expected}
		\label{fig:3nodes-chain-expected}
	}
	\caption{(a) shows noisy phases due to message collision at node 2. (b) shows the expected result of the perfect desynchrony state.}
\end{figure}

We simulate DWARF by setting the time period to 1000 milliseconds with nodes starting up randomly.
The simulation result is shown in Figure \ref{fig:3nodes-chain-dwarf}. Node 2's and node 3's phases are plotted relatively to the node 1's phase which is relatively plotted to itself at 0. The noisy vertical line is the wrapping-around phase of node 3. It shows that node 1 and node 3 fire messages approximately at the same phase, causing message collision at node 2.
However, the expected result (\textit{i.e.} perfect desynchrony state) should be that three nodes are separated equivalently because all nodes will interfere each other if they fire messages at the same phase. The expected result is shown in Figure \ref{fig:3nodes-chain-expected} where each node is equivalently separated from each other approximately by 1000/3 milliseconds.

The problematic result is caused by the hidden terminal problem as demonstrated in Figure \ref{fig:3nodes-chain-hidden}; node 1 and node 3 are hidden to each other in this multi-hop topology. While node 3 is firing a message, node 1 senses a wireless channel and does not detect any signal from node 3 because the signal from node 3 is not strong enough within the signal range of node 1 and vice versa. Therefore, node 1 and node 3 notice only that there are two nodes, which are itself and node 2, in their perceived networks. Therefore, node 1 and node 3 simultaneously attempt to adjust their phases to the opposite side of node 2 in their time circles which are the same phase. As a result, messages of node 1 and 3 recursively collide at node 2.

The hidden terminal problem does not only affect the performance of DWARF but also affect that of DESYNC.
This is due to the fact that, in DESYNC, a node adjusts its phase based on firing messages from its perceived phase neighbors. Therefore, if it fails to know the phase or presence of its two-hop neighbors, none of their phases is perceived. In \cite{MK09DESYNC}, EXT-DESYNC, an extension of DESYNC, is proposed to solve the hidden terminal problem based on a relative time relaying mechanism. Based on the similar idea, we extend DWARF to support multi-hop topologies.
However, only relative time relaying mechanism does not lead DWARF to an optimal solution in some cases. Therefore, this paper also proposes a \textit{force absorption} mechanism for extending DWARF to support multi-hop networks more efficiently.

\subsubsection{Relative Time Relaying}
\label{sec:relative}
The first idea to solve the hidden terminal problem is straightforward. If a node does not know the firing times of its second-hop neighbors, its one-hop neighbors relay such information. Therefore, instead of firing only to notify others of its phase, each node includes their one-hop neighbors' firing time into a firing message.

However, due to our assumption that nodes' clocks are not synchronized, relying on second-hop neighbors' firing timestamps from its one-hop neighbors could lead to wrong phase adjustment. This problematic scenario is demonstrated in Figure \ref{fig:broadcast-problem}. Figure \ref{fig:broadcast-problem-topo} illustrates the firing message of node 2 that contains timestamps of its one-hop neighbors and Figure \ref{fig:broadcast-problem-ring} displays the problem. The inner circle represents the local time of node 1 and the outer circle represents the local time of node 2. The figure indicates that the local reference times (at 0 millisecond) of node 1 and node 2 are different. Therefore, if node 1 uses the node 3's firing time relayed by node 2, which is 125 milliseconds, node 1 will misunderstand the exact time phase of node 3. The misunderstood phase of node 3 is depicted as a dash circle.

\begin{figure}
	\centering
	\subfloat[]{
		\label{fig:broadcast-problem-topo}
		\includegraphics[width=1.8in]{figure/broadcast-problem-topo}
	}
	\hspace{2cm}
	\subfloat[]{
		\label{fig:broadcast-problem-ring}
		\includegraphics[width=1.8in]{figure/broadcast-problem-ring}
	}
	\caption{A problem of relying on second-hop neighbors' firing timestamps from its one-hop neighbors. (a) shows node 2 firing a message. (b) displays node 1's misperception of phases.}
	\label{fig:broadcast-problem}
\end{figure}

This problem can be simply solved by using relative phases instead of actual local timestamps. Each node fires a message that includes relative phases of its one-hop neighbors. A receiving node marks the firing phase of the firing node as a reference phase. Then, the receiving node perceives its second-hop neighbors' phases as relative phase offsets to the reference phase. Figure \ref{fig:broadcast-relative} shows how M-DWARF desynchronizes a three-node multi-hop chain network.

\begin{figure*}
	\centering
	\subfloat[]{
		\label{fig:broadcast-relative-topo}
		\includegraphics[width=1.8in]{figure/broadcast-relative-topo}
	}
	\hfill
	\subfloat[]{
		\label{fig:broadcast-relative-ring}
		\includegraphics[width=1.8in]{figure/broadcast-relative-ring_new}
	}
	\hfill
	\subfloat[]{
		\label{fig:broadcast-relative-perfect}
		\includegraphics[width=1.8in]{figure/broadcast-relative-perfect_new}
	}
	\caption{M-DWARF solves the problem by using one-hop neighbors' relative phases. (a) shows node 2's firing message.  (b) shows how node 1 marks the node 2's phase as a reference phase and uses it as an offset for calculating the node 3's phase. (c) Eventually, nodes are in the perfect desynchrony state.}
	\label{fig:broadcast-relative}
\end{figure*}

\subsubsection{Force Absorption}
\label{sec:absorption}
As we mentioned earlier, DWARF with the relative time relaying mechanism does not solve some cases.
These cases are when there are at least two second-hop neighbors that can share the same phase without interference. For example, in a 4-node chain network illustrated in Figure \ref{fig:4nodes-chain-topo}, node 2 and node 3 are physically far beyond two hops. Therefore, they can fire messages at the same time phase without causing message collisions, as shown in Figure \ref{fig:4nodes-chain-expected}.
However, in M-DWARF, node 0 perceives that node 2 and node 3 are at the same phase. Therefore, there are two forces from node 2 and node 3 to repel node 0 clockwise but there is only force from node 1 to repel node 0 counter-clockwise. Consequently, node 0 cannot stay at the middle between node 1 and the group of node 2 and 3 (see Figure \ref{fig:4nodes-chain-dwarf}).

\begin{figure*}
	\centerline{
		\subfloat[]{\includegraphics[scale=0.3]{figure/4nodes-chain-topo
			\label{fig:4nodes-chain-topo}}
		\hfill
		\subfloat[]{\includegraphics[scale=0.20]{figure/4nodes-chain-expected
			\label{fig:4nodes-chain-expected}}
		\hfill
		\subfloat[]{\includegraphics[scale=0.20]{figure/4nodes-chain-dwarf
			\label{fig:4nodes-chain-dwarf}}
	}
	\caption{The problem of the DWARF algorithm. (a) displays 4-node chain topology. (b) shows node 0's local view. (c) displays an imperfect desynchrony state.}
	\label{fig:4nodes-chain}
\end{figure*}

Therefore, we propose a novel force absorption mechanism for multi-hop desynchronization based on the artificial force field. The objective of this mechanism is to absorb the overwhelming force from at least two nodes that can fire at the same phase without interference.

The mechanism works as follows. A node receives a full repulsive force from the next/previous phase neighbor as in DWARF. However, a force from the second-next / second-previous phase neighbor is partially absorbed by the next / previous phase neighbor. The magnitude of the absorbed force depends on the phase interval between the next / previous and the second-next / second-previous phase neighbors.  The closer the second-next / second-previous phase neighbor moves to the next / previous phase neighbor, the lower the magnitude of the absorbed force becomes. Eventually, when the second-next / second-previous phase neighbor moves to the same phase as the next / previous phase neighbor, the additional force from the second-next / second-previous phase neighbor is fully absorbed. Consequently, the magnitude of two forces repelling the considered node is approximately equal to only the magnitude of one force. This principle is applied recursively; that is, the force from the third-next / third-previous phase neighbors is absorbed by the second-next / second-previous phase neighbor, and the force from the fourth-next / fourth-previous phase neighbor is absorbed by the third-next/third-previous phase neighbor, and so forth.
Figure \ref{fig:4nodes-chain-dwarf-absorb} illustrates this mechanism. In Figure \ref{fig:4nodes-chain-dwarf-absorb-split}, the force from node 2 to node 0 is absorbed by node 3 (the absorbed force is displayed in a blur line). Thus, from node 2, there is only small magnitude of force left to node 0. Eventually, in Figure \ref{fig:4nodes-chain-dwarf-absorb-perfect}, node 2 moves to the same phase as node 3 because they do not interfere each other and the force from node 2 is fully absorbed. Consequently, the network can be in the perfect desynchrony state.

\begin{figure*}
	\centerline{
		\subfloat[]{\includegraphics[scale=0.2]{figure/4nodes-chain-dwarf-absorb-split
			\label{fig:4nodes-chain-dwarf-absorb-split}}
		\hfil
		\subfloat[]{\includegraphics[scale=0.2]{figure/4nodes-chain-dwarf-absorb-perfect
			\label{fig:4nodes-chain-dwarf-absorb-perfect}}
	}
	\caption{M-DWARF solves the problem with force absorption; the blur line represented an absorbed force. (a) shows node 2's force is being absorbed by node 3. (b) displays the perfect desynchrony state.}
	\label{fig:4nodes-chain-dwarf-absorb}
\end{figure*}

Let $f_{i,j}$ be a full repulsive force from node $j$ to node $i$, $f_{i,j}^{'}$ be an absorbed force from node $j$ to node $i$, $T$ is the time period, and $\Delta \phi_{i,j}$ is the phase difference between node $i$ and $j$.
The force function for multi-hop networks is the following:
\begin{alignat}{2}
f_{i,j} &= \frac{1}{\Delta \phi_{i,j} / T}, \text{where }\Delta \phi_{i,j} \in (-\frac{T}{2}, \frac{T}{2}) \nonumber \\
f_{i,i + 1}^{'} &= f_{i,i + 1} \nonumber \\
f_{i, i - 1}^{'} &= f_{i,i - 1} \nonumber \\
f_{i,j}^{'} &= f_{i,x} - f_{i,j},
\label{eq:force-absorb}
\end{alignat}
where $j \notin \left\{i -1, i + 1\right\}$ and $x = (j - \frac{\Delta \phi_{i,j}}{|\Delta \phi_{i,j}|}) \mod n$.
For $f_{i,x}$, if node $j$ repels node $i$ forward, $x$ is $j + 1$. In contrast, if node $j$ repels node $i$ backward, $x$ is $j - 1$. At $T/2$ or $-T/2$, a node does not repel an opposite node because they are balanced.

For example, in Figure \ref{fig:4nodes-chain-dwarf-absorb}, node 0 calculates the force from node 2 as the following:
\begin{alignat}{2}
f_{0,2}^{'} &= f_{0,3} - f_{0,2} \nonumber \\
&= \frac{1}{\Delta \phi_{0,3} / T} - \frac{1}{\Delta \phi_{0,2} / T}. \nonumber
\end{alignat}

Noticeably, if node 2 moves close to node 3, the value of $\Delta \phi_{0,2}$ is close to the value of $\Delta \phi_{0,3}$. Then, the magnitude of force $f_{0,2}$ is reduced.
Finally, when $\Delta \phi_{0,2}$ is equal to $\Delta \phi_{0,3}$ as in Figure \ref{fig:4nodes-chain-dwarf-absorb-perfect}, the magnitude of force $f_{0,2}$ becomes 0; in other words, the force is fully absorbed.

\newpage


\section{Model}

We consider a collection of N three-level V-type atoms located at the same position. We label the ground state as $\ket{1}$ and the two excited states as $\ket{2}$ and $\ket{3}$, and the transition frequency from level j to i as $\omega_{ij}$. A weak drive field which is resonantly tuned to $\omega_{21}$ prepares the atomic system in a timed-Dicke state. As the drive field is turned off, we detect the photons emitted from the cloud in the forward direction. In the experiment, the atomic cloud has a finite size, but for theoretical simplicity we can assume it to be point-like ensemble interacting each other through the vacuum field modes. This is because we are measuring the forward scattering, where any phases of emitted photons due to the atomic position distribution is exactly compensated by the phases initially imprinted on the atoms by the drive field \cite{Scully_2006}. Additionally, the transitions $\ket{1}\leftrightarrow\ket{2}$ and $\ket{1}\leftrightarrow\ket{3}$  interact with the field effectively with the same  phase considering that the atomic cloud size is much smaller compared to $2\pi c/\omega_{23}$. We note that while the forward-scattered field is collectively enhanced, the decay rate of the atoms arising from interaction with the rest of the modes is not cooperative \cite{Bienaime_2011}.

The atomic Hamiltonian $H_A$ and the vacuum field Hamiltonian $H_F$ are
\eqn{\begin{split}
		H_A &= \sum_{m=1}^{N}\sum_{j=2,3} \hbar \omega_{j1} \hat{\sigma}_{m,j}^+ \hat{\sigma}_{m,j}^-,\\
		H_F &= \sum_{k} \hbar \omega_{k} \hat{a}_{k}^{\dagger} \hat{a}_{k},
		\label{eq:H-0}
\end{split}}
where $\hat{\sigma}_{m,j}^{\pm}$ is the raising/lowering operator acting on $m^\mr{th}$ atom and $j^\mr{th}$ level, $\hat{a}_k^{\dagger}$ and $\hat{a}_k$ are the field creation/annihilation operators of the corresponding frequency mode $\omega_{k}$, and $N$ refers to the effective number of atoms acting cooperatively in the forward direction.

First, we prepare the atomic system by a weak drive field. The atom-drive field interaction Hamiltonian is
\eqn{H_{\text{AD}}=-\sum_{m=1}^N\sum_{j=2,3} \hbar \Omega_j^m \bkt{ \hat{\sigma}_{m,j}^+ e^{-i \omega_D t} + \hat{\sigma}_{m,j}^- e^{i \omega_D t} }.\label{eq:H-AD}}
Here, $\omega_D$ is the drive frequency and $\Omega_j^m \equiv \vec{d}_{j1}^m\cdot\vec{\epsilon_{D}}\,E_D$ is the Rabi frequency of $j^\mr{th}$ level, where $\vec{d}_{j1}^{m}$ is the dipole moment of $\ket{j}\leftrightarrow\ket{1}$ transition of $m^\mr{th}$ atom, $\vec{\epsilon}_D$ is the polarization unit vector of the drive field, and $E_D$ is the electric field of the drive field. Given that the atomic ensemble is driven with the common field in our experiment, we will assume that the atomic dipoles are aligned with the drive and each other. We can thus omit the atomic labels to write $\Omega_j$.

The interaction Hamiltonian describing the atom-vacuum field interaction, under the rotating wave approximation, is given as
\eqn{
	H_{\text{AV}} = -\sum_{m=1}^N\sum_{j=2,3}\sum_{k} \hbar g_{m,j}(\omega_k) \bkt{ \hat{\sigma}_{m,j}^+\hat{a}_{k} + \hat{\sigma}_{m,j}^-\hat{a}_k^{\dagger}}.
}
Here, the atom-field coupling strength $g_{m,j}(\omega_k) \equiv \vec{d}_{j1}^{m} \cdot \vec{\epsilon}_k\sqrt{\frac{\omega_k}{2\hbar\varepsilon_0 V}}$,
where $\vec{\epsilon}_k$ is the polarization unit vector of the field mode, $\varepsilon_0$ is the vacuum permittivity,  and $V$ is the field mode volume. As justified previously, the atomic dipoles are aligned to each other and we write $g_{j}(\omega_k)$.
Also, note that the sum over k only refers to the forward-scattered modes. The spontaneous emission arising from the rest of the modes is to be considered separately later.

\section{Driven dynamics}

\begin{figure*}[t]
	\centering
	\includegraphics[width = 3.5 in]{fig_laser_extinction.eps}
	\caption{\textbf{(a)} The drive field intensity (red circles) at turn-off edge characterized as the truncated $\cos^4\bkt{\frac{\pi}{2}\frac{t-t_0}{\tau}}$ function (red solid line) bridging the on and off state of the intensity. Here, $t_0 = -4$ ns and the fall-time $\tau=3.5$ ns are assumed. While the intensity of the drive field turns off mostly within $\approx$ 3.5 ns, additional 0.5-ns waiting time is provided before the the data analysis of the collective emission begins at $t=0$ as shown in Fig.\,\ref{fig_decay}\,(b), to further remove the residual drive intensity and the transient effect from our measurement.}
	\label{fig_laser_extinction}
\end{figure*}

We consider here the driven dynamics of a atoms. Moving to the rotating frame with respect to the drive frequency,  and tracing out the vacuum field modes, we can write the following Born-Markov master equation for the atomic density matrix:
\eqn{
	\der{\hat{\rho}_A}{t} = -\frac{i}{\hbar} \sbkt{\widehat{H}_A + \widehat H_{AD}, \hat {\rho}_A} - \sum_{m,n = 1}^N\sum_{i,j = 2,3} \frac{\Gamma_{ij,mn}^{(D)}}{2} \sbkt{ \hat \rho_A \widehat{\sigma}_{m,i} ^+  \widehat{\sigma}_{n,j} ^- +  \widehat{\sigma}_{m,i} ^+  \widehat{\sigma}_{n,j} ^- \hat \rho_A - 2\widehat{\sigma}_{n,j} ^- \hat \rho_A  \widehat{\sigma}_{m,i} ^+  },
}
where $\widehat H_A = - \sum_{m=1}^{N}\sum_{j=2,3} \hbar \Delta_{j} \widehat{\sigma}_{m,j}^+ \widehat{\sigma}_{m,j}^-$ is the free atomic Hamiltonian and $\widehat H_{AD} =  -\sum_{m=1}^N\sum_{j=2,3} \hbar \Omega_j^m \bkt{ \widehat{\sigma}_{m,j}^+  + \widehat{\sigma}_{m,j}^-  }$ is the atom-drive interaction Hamiltonian  in the rotating frame, with   $ \Delta_j \equiv \omega_{j1} - \omega_D$. The driven damping rates  are defined as $  \Gamma_{ij,mn}^{(D) }  \equiv \frac{\vec{d}^m_{i1} \cdot \vec{d}^n_{j1}\omega_{D}^3}{3\pi \varepsilon_0 \hbar c^3}$, with the indices $i,j$ referring to the atomic levels, and $m,n$  to different atoms.

Using the above master equation, one can obtain the following optical Bloch equations for the case of a single atom:
\begin{subequations}
	\eqn{\label{eq:optical-Bloch-eqa}
		\partial_t \rho_{33} &= i\Omega_3(\rho_{13}-\rho_{31}) - \Gamma_{33}^{(D)}\rho_{33}  - \frac{\Gamma^{(D)}_{23}}{2} \rho_{23} - \frac{\Gamma^{(D)}_{23}}{2} \rho_{32} \\
		\partial_t \rho_{22} &= i\Omega_2(\rho_{12}-\rho_{21}) - \Gamma_{22}^{(D)}\rho_{22}- \frac{\Gamma^{(D)}_{23}}{2} \rho_{23} - \frac{\Gamma^{(D)}_{23}}{2} \rho_{32}\\
		\partial_t \rho_{11} &= -i\Omega_3(\rho_{13}-\rho_{31}) -i\Omega_2(\rho_{12}-\rho_{21}) + \Gamma^{(D)}_{33}\rho_{33}+ \Gamma^{(D)}_{22}\rho_{22} +  \Gamma^{(D)}_{23}\bkt{ \rho_{23} +  \rho_{32}} \\
		\partial_t \rho_{31} &= -i \Omega_2 \rho_{32} - i \Omega_3(\rho_{33}-\rho_{11})-\bkt{\frac{\Gamma^{(D)}_{33}}{2}-i\Delta_3}\rho_{31}- \frac{\Gamma^{(D)}_{23}}{2} \rho_{21}\\
		\partial_t \rho_{13} &= i \Omega_2 \rho_{23} + i \Omega_3(\rho_{33}-\rho_{11})-\bkt{\frac{\Gamma^{(D)}_{33}}{2}+i\Delta_3}\rho_{13}- \frac{\Gamma^{(D)}_{23}}{2} \rho_{12}\\
		\partial_t \rho_{21} &= -i \Omega_3 \rho_{23} - i \Omega_2(\rho_{22}-\rho_{11})-\bkt{\frac{\Gamma^{(D)}_{22}}{2}-i\Delta_2}\rho_{21} - \frac{\Gamma^{(D)}_{23}}{2} \rho_{31}\\
		\partial_t \rho_{12} &= i \Omega_3 \rho_{32} + i \Omega_2(\rho_{22}-\rho_{11})-\bkt{\frac{\Gamma^{(D)}_{22}}{2}+i\Delta_2}\rho_{12}- \frac{\Gamma^{(D)}_{23}}{2} \rho_{13}\\
		\partial_t \rho_{32} &= -i \Omega_2 \rho_{31} + i\Omega_3\rho_{12}-\bkt{\frac{\Gamma^{(D)}_{22}+\Gamma^{(D)}_{33}}{2} -i\omega_{23}}\rho_{32} - \frac{\Gamma^{(D)}_{23}}{2} \bkt{\rho_{22} + \rho_{33}}\\
		\partial_t \rho_{23} &= i \Omega_2 \rho_{13} - i\Omega_3\rho_{21}-\bkt{\frac{\Gamma^{(D)}_{22}+\Gamma^{(D)}_{33}}{2}+i\omega_{23}}\rho_{23}  - \frac{\Gamma^{(D)}_{23}}{2} \bkt{\rho_{22} + \rho_{33}},
		\label{eq:optical-Bloch-eqi}}
\end{subequations}
where we have defined the single atom driven damping rate as $\Gamma_{ij}^{(D) }\equiv \frac{\vec{d}_{i1} \cdot \vec{d}_{j1}\omega_{D}^3}{3\pi \varepsilon_0 \hbar c^3}$.

Numerically solving Eq.\,\eqref{eq:optical-Bloch-eqa}--Eq.\,\eqref{eq:optical-Bloch-eqi} along with the normalization condition $\rho_{33}+\rho_{22}+\rho_{11}=1$ gives us the steady state density matrix $\rho_S$ for the atom. Substituting our experimental parameters, we get the populations: $\rho_{S,33}\approx 0$, $\rho_{S,22}\approx 10^{-10}$, and $\rho_{S,11}\approx 1$. The absolute value of the coherences are: $|\rho_{S,23}|\approx 0$, $|\rho_{S,21}|\approx 10^{-5}$, and $|\rho_{S,31}|\approx 0$. These estimates are made for $N \approx1 - 10$, assuming the collective driven damping rate to be $ \Gamma_{ij}^{(D)} (N) \approx (1 + Nf) \Gamma_{ij}^{(D) }$ with phenomenological value f=1 and the collective Rabi frequency to be $ \Omega_{j} \approx \sqrt{N} \Omega_{j}$. Thus we can conclude that  the atomic ensemble is well within the  single excitation regime in $\ket{2}$.

The 3.5 ns time window of laser extinction has broad spectral component and may excite extra population to $\ket{2}$ and $\ket{3}$. We numerically simulate the optical Bloch equation for this time window to find the density matrix after the laser turn-off. We model the laser turn-off shape as $\cos^4$ (see Fig. \ref{fig_laser_extinction}) and vary the Rabi frequency accordingly. Note that this is a calculation for estimate purposes and may not convey the full dynamics in the laser extinction period. Within the numerical precision limit which is set by the evolution time step ($10^{-5}$ ns) multiplied by $\Gamma_{ij} \approx 0.01$ GHz, we obtain the following density matrix values after the turn-off: $\rho_{33}\approx 0$,  $\rho_{22}\approx 0$, $\rho_{11}\approx 1$,   $\rho_{23}\approx 0$, $\rho_{12}\approx 10^{-5}$, and $\rho_{13}\approx 10^{-7}-10^{-6}$. Thus the laser turn-off edge doesn't produce any significant excitation in $\ket{3}$.

\section{Quantum beat dynamics}

As the drive field is turned off, the system evolves with the atom-vacuum field interaction Hamiltonian. Moving to the interaction representation with respect to $H_A+H_F$, we get the interaction  Hamiltonian in the interaction picture:

\eqn{
	\tilde{H}_{\text{AV}} = -\sum_{m=1}^N\sum_{j=2,3}\sum_{k} \hbar g_{j}(\omega_k)
	\bkt{ \hat{\sigma}_{m,j}^+\hat{a}_{k}e^{i(\omega_{j1}-\omega_{k})t }
		+ \hat{\sigma}_{m,j}^-\hat{a}_k^{\dagger}e^{-i(\omega_{j1}-\omega_{k})t}},
	\label{eq:H-AV}
}

Initially the system shares one excitation in $\ket{2}$ symmetrically, and the EM field is in the vaccum state such that
\eqn{
	\ket{\Psi(0)} = \frac{1}{\sqrt{N}}\sum_{m=1}^{N}\hat{\sigma}_{m,2}^{+}\ket{11\cdots 1}\ket{\{0\}}.
	\label{eq:psi-initial}
}
As the system evolves due to the atom-vacuum field interaction, it remains in the single-excitation manifold of total atom + field Hilbert space, as one can see from the interaction Hamiltonian (Eq.\,\eqref{eq:H-AV}):
\eqn{
	\ket{\Psi(t)}=\bkt{\sum_{m=1}^{N}\sum_{j=2,3}c_{m,j}(t)\hat{\sigma}_{m,j}^{+}+\sum_{k}c_{k}(t)\hat{a}_k^{\dagger}}\ket{11\cdots 1}\ket{\{0\}}.
	\label{eq:psi-evolved}
}

Now we solve the Schr\"odinger equation to find the time evolution of the atom + field system under the atom-field interaction using Eqs.\eqref{eq:psi-evolved} and \eqref{eq:H-AV} to obtain
\begin{subequations}
	\begin{align}
	& \partial_t c_{m,j}(t) = i\sum_{k} g_j(\omega_k) e^{i(\omega_{j1}-\omega_{k})t} c_{\omega_k}(t), \\
	& \partial_t c_{\omega_k}(t) = i\sum_{m=1}^{N}\sum_{j=2,3}g_j(\omega_k)e^{-i(\omega_{j1}-\omega_{k})t}c_{m,j}(t).
	\end{align}
	\label{eq:de-1}
\end{subequations}
Formally integrating  Eq.\,\eqref{eq:de-1}(b) and plugging it in Eq.\,\eqref{eq:de-1}(a), we have
\eqn{
	\partial_t c_{m,j}(t) = -\sum_{k}g_j(\omega_k) e^{i(\omega_{j1}-\omega_{k})t}\int_0^t \mathrm{d}{\tau}\sum_{n=1}^{N}\sum_{l=2,3}g_l(\omega_k) e^{-i(\omega_{l1}-\omega_k)\tau} c_{n,l}(\tau).
}
We observe that $c_{m,2}(t)$'s ($c_{m,3}(t)$'s) have the same initial conditions and the same evolution equation, thus we can justifiably define $c_2(t) \equiv c_{m,2}(t)$ ($c_3(t)\equiv c_{m,3}(t)$).

Assuming a flat spectral density of the field and making the Born-Markov approximation we get
\begin{subequations}
	\begin{align}
	\partial_t c_2(t) &= -\frac{\Gamma_{22}^{(N)}}{2} c_2(t)-\frac{\Gamma_{23}^{(N)}}{2} e^{i\omega_{23}t}c_3(t),\\
	\partial_t c_3(t) &= -\frac{\Gamma_{33}^{(N)}}{2} c_3(t)-\frac{\Gamma_{32}^{(N)}}{2} e^{-i\omega_{23}t}c_2(t),
	\end{align}
	\label{eq:de-2}
\end{subequations}
where we have defined $\Gamma_{jl}^{(N)}\equiv \Gamma_{jl} + Nf\Gamma_{jl}$, with $\Gamma_{jl} = \frac{\vec{d}_{j1}\cdot \vec{d}_{l1}\omega_{l1}^3}{3\pi \varepsilon_0 \hbar c^3}$ as the generalized decay rate into the quasi-isotropic modes and $Nf\Gamma_{jl}$ as the collective decay rate in the forward direction \cite{Bienaime_2011, Araujo_2016}. The factor $f$ represents the geometrical factor coming from restricting the emission to the forward scattered modes. We emphasize here that the  emission into all the modes (not specifically the forward direction) denoted by $\Gamma_{jl}$ is added phenomenologically and is not collective. Considering that the atomic dipole moments induced by the drive field are oriented along the polarization of the driving field, we can obtain  $\Gamma_{23}=\sqrt{ \Gamma_{22}\Gamma_{33}}$, which can be extended to $\Gamma_{23}^{(N)}=\sqrt{ \Gamma_{22}^{(N)}\Gamma_{33}^{(N)}}$.

To solve the coupled differential equations, we take the Laplace transform of Eq.\,\eqref{eq:de-2}(a) and (b):
\begin{subequations}
	\begin{align}
	s\tilde{c}_2(s)&=c_2(0)-\frac{\Gamma_{22}^{(N)}}{2}\tilde{c}_2(s)-\frac{\Gamma_{23}^{(N)}}{2}\tilde{c}_3(s-i\omega_{23}),\\
	s\tilde{c}_3(s)&=c_3(0)-\frac{\Gamma_{33}^{(N)}}{2}\tilde{c}_3(s)-\frac{\Gamma_{32}^{(N)}}{2}\tilde{c}_2(s+i\omega_{23}),
	\end{align}
\end{subequations}
where we have defined  $\tilde{c}_j(s) \equiv \int_0^{\infty} c_j(t) e^{-st} \mathrm{d}(t)$ as the Laplace transform of $c_j \bkt{t}$. Substituting  the initial conditions, we obtain the Laplace coefficients as
\begin{subequations}\begin{align}
	\tilde{c}_2(s)&=\,\frac{1}{\sqrt{N}}\frac{s+\frac{\Gamma_{33}^{(N)}}{2}-i\omega_{23}}{s^2+(\Gamma_{\text{avg}}^{(N)}-i\omega_{23})s-i\omega_{23}\frac{\Gamma_{22}^{(N)}}{2}},\\
	\tilde{c}_3(s)&=-\frac{\Gamma^{(N)}_{32}}{2\sqrt{N}}\,\frac{1}{s^2+(\Gamma^{(N)}_{\text{avg}}+i\omega_{23})s+i\omega_{23}\frac{\Gamma^{(N)}_{33}}{2}}.
	\end{align}\end{subequations}
And the poles of the denominators are, respectively,
\begin{subequations}\begin{align}
	s_{\pm}^{(2)}=&-\frac{\Gamma^{(N)}_{\text{avg}}}{2} + \frac{i\omega_{23}}{2} \pm \frac{i\delta}{2}, \\
	s_{\pm}^{(3)}=&-\frac{\Gamma^{(N)}_{\text{avg}}}{2} - \frac{i\omega_{23}}{2} \pm \frac{i\delta}{2},
	\end{align}\end{subequations}
where we have defined  $\Gamma_{\text{avg}}^{(N)}=\frac{\Gamma_{33}^{(N)}+\Gamma_{22}^{(N)}}{2}$, $\Gamma_{\text{d}}=\frac{\Gamma_{33}^{(N)}-\Gamma_{22}^{(N)}}{2}$, and $\delta = \sqrt{\omega_{23}^2-\bkt{\Gamma^{(N)}_{\text{avg}}}^2+2i\omega_{23}\Gamma^{(N)}_{\text{d}}}$. The real part of the above roots corresponds to  the collective decay rate of each of the excited states, while the imaginary part corresponds to the frequencies. The fact that $\delta$ is generally a complex number unless $\Gamma_{22}\neq\Gamma_{33}$ means that we will have modification to both the decay rate and the frequency. To see this more clearly, we can expand $\delta$ up to second order in $\Gamma_{jl}^{(N)}/\omega_{23}$, considering we are working in a spectroscopically well-separated regime ($\Gamma_{jl}^{(N)}\ll\omega_{23}$);
\eqn{
	\delta \approx \omega_{23}\sbkt{1-\frac{1}{2}\bkt{\frac{\Gamma^{(N)}_{23}}{\omega_{23}}}^2}+i\Gamma^{(N)}_{d}\sbkt{1+\frac{1}{2}\bkt{\frac{\Gamma^{(N)}_{23}}{\omega_{23}}}^2},
}
the above poles become
\begin{subequations}\begin{align}
	s_{+}^{(2)}=&-\frac{\Gamma^{(N)}_{33}}{2}\bkt{1+\frac{\Gamma^{(N)}_{\text{d}}\Gamma_{22}^{(N)}}{2\omega_{23}^2}} + i\omega_{23}\sbkt{1-\bkt{\frac{\Gamma^{(N)}_{23}}{2\omega_{23}}}^2}, \\
	s_{-}^{(2)}=&-\frac{\Gamma^{(N)}_{22}}{2}\bkt{1-\frac{\Gamma^{(N)}_{\text{d}}\Gamma_{33}^{(N)}}{2{\omega_{23}^2}}} + i\omega_{23}\bkt{\frac{\Gamma^{(N)}_{23}}{2\omega_{23}}}^2, \\
	s_{+}^{(3)}=&-\frac{\Gamma^{(N)}_{33}}{2}\bkt{1+\frac{\Gamma^{(N)}_{\text{d}}\Gamma^{(N)}_{22}}{2\omega_{23}^2}} - i\omega_{23}\bkt{\frac{\Gamma^{(N)}_{23}}{2\omega_{23}}}^2 \\
	s_{-}^{(3)}=&-\frac{\Gamma^{(N)}_{22}}{2}\bkt{1-\frac{\Gamma^{(N)}_{\text{d}}\Gamma^{(N)}_{33}}{2\omega_{23}^2}} - i\omega_{23}\sbkt{1-\bkt{\frac{\Gamma^{(N)}_{23}}{2\omega_{23}}}^2}.
	\end{align}\end{subequations}

The atomic state coefficients in time domain are
\begin{subequations}
	\begin{align}
	c_2(t) &= \frac{1}{2\sqrt{N}\delta}  e^{- \Gamma^{(N)}_{\text{avg}}t/2}  e^{i\omega_{23}t/2} \sbkt{(-i\Gamma^{(N)}_{d}-\omega_{23}+\delta)e^{i\delta t/2} + (i\Gamma^{(N)}_{d}+\omega_{23}+\delta)e^{-i\delta t/2}},\\
	c_3(t) & =\frac{i\Gamma^{(N)}_{32}}{2\sqrt{N}\delta}  e^{-\Gamma^{(N)}_{\text{avg}}t/2} e^{-i\omega_{23}t/2} \sbkt{e^{i\delta t/2}-e^{-i\delta t/2}}.
	\end{align}
	\label{eq:atom-coefficients}
\end{subequations}

Again, expanding $\delta$ under the condition $\Gamma_{jl}^{(N)}\ll\omega_{23}$, we get
\begin{subequations}
	\begin{align}
	c_2(t) &= \frac{1}{\sqrt{N}}\sbkt{ e^{- \Gamma^{(N)}_{22}t/2} - \bkt{\frac{\Gamma^{(N)}_{23}}{2\omega_{23}}}^2\frac{\delta^*}{\delta\,}e^{-\Gamma^{(N)}_{33}t/2}e^{i\omega_{23}t}},\\
	c_3(t) & = -\frac{i\Gamma^{(N)}_{32}}{2\sqrt{N}\delta} \sbkt{e^{-\Gamma^{(N)}_{22}t/2}e^{-i\omega_{23}t}-e^{-\Gamma^{(N)}_{33}t/2}}.
	\end{align}
	\label{eq:atom-coefficients-2}
\end{subequations}
Note that the collection of $N$ atoms behaves like one ``super-atom'' which decays with a  rate that is $N$-times that of an individual atom in the forward direction. We note that the system is not only superradiant with respect to the transition involving the initially excited level, but  also with respect  to other transitions as well as a result of the vacuum-induced  coupling between the levels.  Most population in $\ket{2}$ decays with the decay rate $\Gamma_{22}^{(N)}$, and small amount of it decays with $\Gamma_{33}^{(N)}$ and has corresponding level shift $\omega_{23}$. In $\ket{3}$ are the equal amount of components decaying with $\Gamma^{(N)}_{22}$ (and level shifted $-\omega_{23}$) and $\Gamma^{(N)}_{33}$. The small but nonzero contribution of $\ket{3}$ makes beating of frequency about $\omega_{23}$.

\section{Field Intensity}

The light intensity at position $x$ and time $t$ (assuming the atom is at position $x=0$ and it starts to evolve at time $t=0$) is
\eqn{
	I(x,t) = \frac{\epsilon_0 c}{2}\bra{\Psi(t)}\hat{E}^{\dagger}(x,t) \hat{E}(x,t) \ket{\Psi(t)},
}
where the electric field operator is
\eqn{
	\hat{E}(x,t) = \int_{-\infty}^{\infty} \mathrm{d} k \, E_k \hat{a}_k e^{ikx}e^{-i\omega_k t}.
}
Plugging in the electric field operator and the single-excitation ansatz (Eq.\,\eqref{eq:psi-evolved}), we obtain the intensity up to a constant factor:

\eqn{
	I(x,t) \simeq N^2 \abs{e^{-i\omega_{23}\tau}c_2(\tau) + \frac{\Gamma_{23}}{\Gamma_{22}}c_3(\tau) }^2 \Theta(\tau),
	\label{eq:field-intensity}
}
where $\tau = t-\abs{x/v}$.

Substituting Eqs. \eqref{eq:atom-coefficients}(a) and (b) in the above and approximating $\delta$ in the regime $\Gamma_{jl}^{(N)}\ll\omega_{23}$, we get
\eqn{
	\frac{I(\tau)}{I_0} = e^{-\Gamma^{(N)}_{22}\tau}+\bkt{\frac{ \Gamma^{(N)}_{33}}{2\omega_{23}}}^2 e^{-\Gamma^{(N)}_{33}\tau} + \frac{\Gamma^{(N)}_{33}}{\omega_{23}} e^{-\Gamma^{(N)}_{\text{avg}}\tau} \sin(\omega_{23}\tau+\phi),
}
where $I_0$ is a normalization factor which increases as the number of atom increases. Neglecting the small second term in the right hand side, we get the relative beat intensity normalized to the main decay amplitude:
\eqn{
	\text{beat amp.} = \frac{\Gamma^{(N)}_{33}}{\omega_{23}},
}
and the beat phase $\phi$:
\eqn{
	\phi = \arctan\bkt{\frac{\Gamma^{(N)}_{22}}{\omega_{23}}}.
}
We see that even if there was no population in level 3 in the beginning, the vacuum field builds up a coherence between level 2 and level 3 to make a quantum beat. This is in line with the quantum trajectory calculation of single atom case \cite{Hegerfeldt_1994}, where the individual decay rates are replaced with collective decay rates. We can verify that the collective effect manifests in the beat size and the beat phase.

\section{Data analysis in Fig. \ref{fig_decay} (b)}

The modulated decay profiles of the flash after the peak are magnified in Fig.\,\ref{fig_decay} (b). The purpose of the figure is to visually compare the decay rate and the relative beat intensity $I_\mathrm{b}$, so we normalize each curve with the exponential decay amplitude such that the normalized intensity starts to decay from $\approx1$ at $t=0$. In practice, we fit the $I(t)$ shown in Fig.\,\ref{fig_decay} (a) after $t=0$ using Eq.\,\eqref{eq_intensity} to get $I_0$ for each curve, to get $I(t)/I_0$ curves as in Fig.\,\ref{fig_decay} (b). Note that, more precisely, it is the fitting curve that decays from $I(t)/I_0\approx1$, not the experimental data. In fact, the plotted data tend to be lower than the fitting curves near $t=0$, due to the effect of the transient behavior around the flash peak.

The inset displays the FFT of the beat signal shown in the main figure. We first subtract from $I(t)/I_0$ data the exponential decay profile the first term of the fitting function Eq.\,\eqref{eq_intensity} as well as the dc offset. The residual, which is a sinusoidal oscillation with an exponentially decaying envelop, is the beat signal represented by the second term of Eq.\,\eqref{eq_intensity}. The FFT of the beat signal has the lower background at $\omega = 0$ due to the pre-removal of the exponential decay and the offset. The linewidth of each spectrum is limited by the finite lifetime of the beat signal, which corresponds to $\Gamma^{(N)}_\mathrm{avg}$ as in Eq.\,\eqref{eq_intensity}.

\vspace{2cm}

\end{document}


\section{Introduction}
A huge amount of information flows through enterprise documents; thus, it is imperative to develop efficient information extraction techniques to extract and use this information productively. While documents comprise multiple components such as text, tables, figures etc.; tables are the most commonly used structural representation that organize the information into rows and columns. It captures structural and geometrical relationships between different elements and attributes in the data. Moreover, important facts/numbers are often presented in tables instead of verbose paragraphs. For instance, tables in financial domain are a good example where different financial metrics such as ``revenue", ``income" etc. are presented for different quarters/years. Extracting the content of a table into a structured format (csv or JSON) \cite{gao2019icdar}, \cite{gobel2013icdar}, \cite{jimeno2021icdar} is a key step in many information extraction pipelines.

Unlike traditional machine learning problems where the output is a class (classification) or number (regression), the outcome of a table parsing algorithm is always a structure. There needs to be a way to compare one structure against another structure and define some measure of ``similarity/distance" to evaluate different methods. A number of metrics quantifying this ``distance" have been proposed in literature and multiple competitions. Existing metrics evaluates the performance of table parsing algorithms using the structural and textual information. This paper presents the limitation of existing metrics based on their dependence on the textual information. We emphasize that textual information introduces additional dependency on the OCR (text detection/recognition), which is a separate area in itself and should not be included in evaluating how good is the detected table structure. This paper presents a ``true" metric which is agnostic to the textual details and accounts only for the layout of cells in terms of its row number/column number and bounding box.

\vspace{-6pt}
\section{Existing Metrics in Table Parsing}
Two of the existing metrics are adjacency relation set-based F1 scores with different definitions of the set. They break and linearize the table structure into two dimensions, one along the row and one along the column. Adjacency Relation (Text) \cite{gobel2013icdar} computes pair-wise relations between non-empty adjacent cells and the relation is considered correct only if the direction (horizontal/vertical) and text of both the participating cells match. It does not take into account empty cells and multi-hop cell alignment. Adjacency Relation (IOU) \cite{gao2019icdar} is a text-independent metric where original non-empty cells are mapped to predicted cells by leveraging (multiple) IOU thresholds and then adjacency relations are calculated. This metric takes a weighted average of the computed F1-scores at different IOU thresholds \{0.6, 0.7, 0.8, 0.9\}. Finally, the predicted relations are compared to the ground truth relations and precision/recall/F1 scores are computed.

The third metric considers the structure as a HTML encoding of the table. In this representation, the table is viewed as a tree with the rows being the children of the root $<table>$ node, and cells being the children (represented by $<td>[text]</td>$) of the individual rows. A Tree edit-distance (TEDS) metric \cite{zhong2020image} is proposed which compares  two trees and reports a single number summarizing the similarity.

While there are other metrics used in literature such as BLEU-4 \cite{li2019tablebank} (which is more language based), this paper only considers the above three most widely used metrics for evaluating the performance of table structure recognition.

\section{Proposed Metric}
This paper highlights the limitations of the previous metrics and also proposes a new metric, Tree-Edit-Distance Based Similarity with IOU \textit{(TEDS-IOU)}, for evaluating table structure recognition algorithms. The paper also demonstrates how \textit{TEDS-IOU} addresses the limitations of existing metrics.

\begin{figure}[t]
\begin{center}
  \includegraphics[width=0.5\linewidth]{example.png}
   \vspace{-10pt}
  \caption{Original table and an example prediction for the same. For Adjacency relation (Text), the characters can be considered as representing the text inside cells. For Adjacency relation (IOU), characters can be considered as labels representing cells.}
  \label{fig:digitalization}
  \vspace{-16pt}
 \end{center}
\end{figure}

Table \ref{tab1} describes the limitations of the commonly used metrics in table structure recognition literature. For example, in figure \ref{fig:digitalization}, even though the predicted table missed one entire row and $4$ empty cells, in terms of adjacency relations, the only extra relation in the predicted table is the \{C, A, Horizontal\}, where `Horizontal' is the direction of relation. This only affects precision but the recall is still $100$\% which clearly should have been penalised. Also, in the case of the IOU based metric, lets assume label mapping, i.e. cell represented by ``C" in ground truth is a mapped to the ``C" cell in predicted table using IOU thresholds. We still have that same extra relation \{``C", ``A", Horizontal\}, where `Horizontal' is the direction, which demonstrates the inability to capture empty cells and mis-alignments. We should note that metric is still better then the text-based version, since it does not rely on comparing text. Accurately detecting and recognizing text (OCR) is a separate field in itself, while in table structure recognition, we are primarily interested in localizing the cell boundaries and assign text to them.

\begin{table}[t]
\caption{ Existing metrics in literature and their limitations}\label{tab1}
\begin{tabular}{p{0.32\textwidth}|p{0.66\textwidth}}
\hline
\textbf{Metric} & \textbf{Limitations}\\
\hline
{Adjacency Relation (Text)} & {Doesn't handle empty cells, misalignment of cells beyond immediate neighbours \& text dependent}\\
{Adjacency Relation (IOU)} & {Doesn't handle empty cells, misalignment of cells beyond immediate neighbours}\\
{TEDS (Text)} & {Text dependent but less strict due to Levenshtein distance}\\
\hline
\end{tabular}
 \vspace{-16pt}
\end{table}
TEDS (Text) metric solved the shortcomings of previous metrics with regard to empty cells and multi-hop mis-alignments \cite{zhong2020image}. In TEDS, all cells, with or without text are considered, thereby also including empty cells as part of computation. So, TEDS (text) will penalise the absence of a row and all the alignment mismatches when comparing ground truth table against predicted table in figure \ref{fig:digitalization}. But it computes the edit distance between cells' texts as compared to the exact match in Adjacency Relation (Text).

Table structure recognition algorithms aim at predicting the location (bounding boxes) of cells and their logical relation with one another, irrespective of the text in the cell. Therefore, the evaluation metric should not penalize an algorithm for inaccuracies in text. With this observation, this paper propose TEDS (IOU) which replaces the string edit distance between cells' text with the IOU distance between their bounding boxes. This effectively, removes dependency on text or OCR, while also preserving the benefits of the original TEDS (text) metric. Specifically, we compute TEDS (IOU) as follows: cost of insertion \& deletion operations is 1 unit; while substituting a node $n_s$ with $n_t$ - cost of edit is 1 unit if either $n_s$ or $n_t$ is not $<td>$, cost of edit is 1 unit if both $n_s$ \& $n_t$ is $<td>$ and the column span or row span of $n_s$ \& $n_t$ is different, otherwise, cost of edit is $1 - IOU(n_s.bbox,  n_t.bbox)$. Finally,
  \begin{equation}
  TEDS\_IOU(T_a, T_b) = 1 - \frac{EditDistIOU(T_a, T_b)}{max(|T_a|, |T_b|)}
  \end{equation}
TEDS (IOU) $\in [0,1]$, the higher the better. $|.|$ denotes cardinality. IOU distance $(IOU_d = 1 - IOU)$ being a Jaccard index \cite{kosub2019note}, is a metric as it satisfies:
\begin{enumerate}
\item $IOU_d(A, B) = 0 \iff A = B  \qquad \qquad \qquad \qquad \qquad \; \;$ $Identity$
\item $IOU_d(A, B) = IOU_d(B, A) \qquad  \qquad \qquad \qquad \qquad \; \; \; \; \; \;$ $Symmetry$
\item $IOU_d(A, C) <= IOU_d(A, B) + IOU_d(B, C) \qquad \qquad \; \,$ $Triangle\;Inequality$
\end{enumerate}

\begin{figure}[t]
\begin{center}
  \includegraphics[width=0.8\linewidth]{test_table.png}
   \vspace{-10pt}
  \caption{(a) is a table from PubTabNet dataset. In (b), red lines denote the predicted structure and blue lines depict the true structure.}
  \label{test}
  \vspace{-16pt}
 \end{center}
\end{figure}

To demonstrate the effectiveness of the proposed TEDS (IOU) metric, we compute the all four metrics for the predicted table in figure \ref{test}(b). In the example above, we had known OCR issues where it was unable to recognize the $\pm$ symbol (it got recognized as +) and all the cells with``NA" were detected as empty. Adjacency Relation (Text) got a very poor score of $13.7$ F1 due to the exact text match constraint. Adjacency Relation (IOU), being text independent, is more robust and achieves a Weighted Avg. F1 of $59.8$. TEDS (text) matches text through edit distances, therefore, for it, only the ``NA" cells gave high edit distance (of 1) and it scores $71.6$ on this table. TEDS (IOU) being text independent and computing the IOU distance between cells, assigns a higher score of $80.6$ which seems to be the most representative one of the prediction.

\section{Discussion \& Future Work}

We proposed a new metric for table structure recognition and demonstrated its benefits against existing metrics. As future steps, we plan to compare these metrics across different datasets and models. A possible extension of this work can be to introduce different thresholds for the IOU as in Adjacency Relation (IOU), instead of using absolute numbers.
\vspace{-10pt}

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