\section{Introduction}
In the zero temperature limit, quantum fluids behave at the macroscopic scale as a single coherent quantum state, the superfluid \cite{DonnellyLivreVortices}.  Compared to classical fluids, the quantum coherence of superfluids creates a strong additional constraint on the velocity field, namely to be irrotational. Rotational motion can only appear when the macroscopic coherence of the wave function is broken by topological defects called quantum vortices. In that case, the circulation of the velocity around the quantum vortex has a fixed value ($\kappa \simeq 10^{-7}$m$^2$s$^{-1}$ in $^4$He). Turbulence in superfluids can be thought of as an intricate process of distortion, reconnection and breaking of those topological singularities \cite{BarenghiSkrbekSreenivasan_IntroPNAS2014}, but in such a way that the system seems to mimic the classical turbulence at large scales \cite{spectra:PNAS2014}. This has been particularly obvious in the velocity spectra probed with a variety of anemometers, in highly turbulent flows \cite{Maurer1998,salort2010turbulent,Salort:EPL2012,rusaouen2017intermittency} or in the measurement of vortex bundles using parietal pressure probes \cite{Rusaouen:parietalEPL2017}. In some sense, quantum turbulence is an irreducible model, or to say it differently, is a kind of "skeleton" for all types of turbulence.

At finite temperature, the quantum fluid is not a pure superfluid: it behaves as if it experienced friction with a background viscous fluid, called the ``normal fluid''. The relative mass density of the superfluid $\rho_s/\rho$ (where $\rho$ is the total mass density) decreases from one at 0~K to zero at the superfluid transition temperature ($T_\lambda \simeq 2.18$~K in $^4$He). The presence of a finite normal fluid fraction allows for propagation of temperature waves - a property referred to as ``second sound"- which opens the rare opportunity to probe directly the presence of the quantum vortices \cite{DonnellyPhysicsToday1984}.

This is done in the present article, where the statistics of superfluid vortex lines density $\mathcal{L}$ are locally measured by ``second sound tweezer" (see the description in paragraph``probes"), over one and a half decade of the inertial scales, and over a wide range of $\rho_s/\rho$ spanning from 0.16 to 0.81. Surprisingly, the result does not corroborate the widespread idea that the large scales of quantum turbulence reproduce those of classical turbulence: the measured  spectra of $\mathcal{L}$ (see Fig. \ref{fig:spectres})  differs from classical-like enstrophy spectra \cite{baudet1996spatial,ishihara2003spectra}. Besides, it also differs from  the only\footnote{
Literature also reports experimental \cite{Bradley:PRL2008} and numerical \cite{FujiyamaJLTP2010, BaggaleyPRB2011,Baggaley_VLD:PRL2012,BaggaleyPRL2015,tsepelin2017visualization}  spectra of the vortex line density spatially integrated across the whole flow. Still, spectra of such ``integral'' quantities differ in nature from the spectra of local quantities, due to strong filtering effects of spatial fluctuations.}
previous direct measurement of $\mathcal{L}$ with second sound tweezers \cite{roche2007vortex} at $\rho_s/\rho \simeq 0.84$.

The measurement of the vortex lines density provides one of the very few constraints for the disputed modeling of the small scales of quantum turbulence. Even after intense numerical \cite{salort2011mesoscale,Baggaley_Coherentvortexstructures_EPL2012}  and theoretical \cite{RocheInterpretation:EPL2008,Nemirovskii:PRB2012,boue2015energyVorticity} studies,  the statistics of quantum vortices show that even the large scales of quantum flows can still be surprising.

\section{Experimental setup}

\begin{figure}
\begin{centering}
\includegraphics[height=10cm]{figure1.pdf}
\par\end{centering}
\caption{Sketch of the flow and the experimental setup with
probes. \label{fig:schema_toupie}}
\end{figure}

The experimental setup has been described in details in a previous publication
\cite{rusaouen2017intermittency},
and we only review in this section the major modifications. The setup
consists in a wind tunnel inside a cylindrical cryostat (see
Fig. \ref{fig:schema_toupie}) filled with He-II.
The flow is continuously
powered by a centrifugal pump located at the top of the tunnel. At
the bottom, an optimized 3D-printed conditioner ensures a smooth
entry of the fluid, without boundary layer detachment, inside a pipe of $\Phi=76$ mm inner diameter. Spin motion is broken by radial screens built in the conditioner. The fluid
is then ``cleaned'' again by a 5-cm-long and $3$-mm-cell honeycomb. The mean flow velocity $U$ is measured with a Pitot tube located $130$ mm upstream the pipe outlet. We allow a maximal mean velocity $U=1.3$ m/s inside the pipe to avoid any cavitation effect with the pump.

The main new element compared to the previous design
is a mono-planar grid located $177$ mm upstream the probes to generate
turbulence. The grid has a $M=17$ mm mesh with square bars of thickness $b=4$ mm, which gives a porosity of $\beta=(1-b/M)^{2}\approx0.58$.

The choice to position the probes at a distance $\sim 10M$ downstream the grid is the result of a compromise between the desire to have a ``large'' turbulence intensity, and the necessity to leave enough space for turbulence to develop between the grid and the probes. According to \cite{vita2018generating}, this distance is enough to avoid near-field effects of the grid. However, we emphasize that our main experimental results (Fig. \ref{fig:spectres}-\ref{fig:histogrammes}) do not depend on perfect turbulent isotropy and homogeneity. In-situ measurements of the mean vortex line density can be used to indirectly (via Eq. \ref{eq:tau}) give an estimation of the turbulence intensity $\tau=u^\mathrm{rms}/U \simeq 12-13\%$ (where $u^\mathrm{rms}$ is the standard deviation of longitudinal velocity component). We present the results later in Fig. \ref{fig:scalingU}.
For comparison, Vita and co. \cite{vita2018generating} report a turbulence intensity around $\tau=9\%$ percents at $10M$ in a classical grid flow of similar porosity. The difference between both values of $\tau$ could originate from a prefactor uncertainty in Eq. (\ref{eq:tau}) or from differences in flow design (e.g. the absence of a contraction behind the honeycomb). This difference has no important consequences for the measurement of quantum vortex statistics.

The longitudinal integral length scale of the flow $H\simeq 5.0$~mm is assessed by fitting velocity spectra (see bottom panel of Fig. \ref{fig:spectres}) with the von K\'arm\'an formula (eg. see \cite{vita2018generating}). For comparison, the integral scale reported for the similar grid in \cite{vita2018generating}, once rescaled by the grid size, gives a nearby estimate of $7.4$ mm.

The Reynolds number $Re$ defined with $u^\mathrm{rms} H$ and the kinematic
viscosity $1.8\times10^{-8}$~m$^2$s$^{-1}$ of liquid He just above $T_\lambda$, is $Re=3.3\times10^4$ for $U=1$ m/s. Using standard homogeneous isotropic turbulence formula, the Taylor scale Reynolds number is $R_\lambda=\sqrt{15Re}\approx 700$ (for $\tau=12\%$ and $H=5$ mm). This gives an indication of turbulence intensity of the flow below $T_\lambda$.

Temperature of the helium bath is set via pressure regulation gates.
The exceptional thermal conductivity of He-II ensures an homogeneous
temperature inside the bath for $T<T_{\lambda}$. Two  Cernox
thermometers, one located just above the pump, the other one on the
side of the pipe close to the probes, allow for direct monitoring
of $T$.

\section{Probes}

Our probes are micro-fabricated second sound tweezers
of the millimeter size according to the same principle as in \cite{roche2007vortex}.
As displayed in the inset of Fig. \ref{fig:probes}, the tweezers are composed
of one heating plate and one thermometer plate facing each other and
thus creating a resonant cavity for thermal waves. The heating
plate generates a stationary thermal wave of the order of $0.1$
mK between the plates, the amplitude of which can be recorded by the
thermometer plate. Two major improvements have been done compared
to the tweezers in \cite{roche2007vortex} : first, the length of
the arms supporting the plates has been increased to \textbf{$14$}
mm to avoid blockage effects due to the stack of silicon wafers (about 1.5 mm thick) downstream the cavity. Second,
two notches are done in the arms to avoid interference due to additional
reflections of the thermal wave on the arms. Further details will be given in a future publication.
\begin{figure}
\begin{centering}
\includegraphics[height=6cm]{figure2.png}
\par\end{centering}
\caption{Ring with probes. The inset is a zoom on the heating and the thermometer plates of a second sound tweezers. The Pitot tube is not used in the present experiment. \label{fig:probes}}
\end{figure}

In the presence of He flow, a variation of the amplitude and phase
of the thermal wave can be observed. This variation is due
to two main physical effects. The presence of quantum vortex lines
inside the cavity causes an attenuation of the wave \cite{DonnellyPhysicsToday1984,varga2019} with a very
minor phase shift \cite{miller1978velocity}. This attenuation can be very accurately modelized by
a bulk dissipation coefficient inside the cavity denoted $\xi_{L}$. The second effect is a ballistic advection
of the wave out of the cavity. It is related to both an attenuation of
the temperature oscillation and an important phase shift. Depending
on the flow mean velocity $U$, the size of the tweezers, and the
frequency of the wave, one of these two effects can overwhelm the
other. We have thus designed two models of tweezers: one model to
take advantage of the first effect to measure the vortex lines density (VLD), and the other one
to take advantage of the second effect to measure the velocity.

The two largest tweezers displayed in Fig. \ref{fig:probes} are designed to measure the quantum vortex
lines density. The plates size is $l=1$ mm and the gaps
between the plates are $D=1.32$ mm and $D=0.83$ mm respectively.
The plates face each other with positioning accuracy of a few micrometers.
The tweezers are oriented parallel to the flow (see Fig. \ref{fig:probes}, the mean flow is directed from top to bottom)
to minimize the effect of ballistic advection of the wave.

The smallest tweezers displayed in Fig. \ref{fig:probes} are designed to be mainly sensitive to the velocity fluctuations
parallel to the mean flow. The two plates have a size $l=250$ $\mu$m,
and are separated by a gap of $D=0.431$ mm. The tweezers are oriented
perpendicular to the mean flow (see Fig. \ref{fig:probes})
with an intentional lateral shift of the heater and the thermometer
of about $l/2$. This configuration is expected to maximize the sensitivity
to ballistic advection, and thus to velocity fluctuations. To second order however, the probe still keeps sensitivity to the quantum vortices produced both by turbulence and by the intense heating of the plates, that's why we were not able to calibrate it reliably. The (uncalibrated) spectrum of this probe (see bottom panel of Fig. \ref{fig:spectres}) is only used to estimate the integral length scale. The role of this probe is also to prove that the signal statistics of the largest tweezers are not due to velocity fluctuations.

\section{Method}
Figure \ref{fig:methode} displays a resonance of a large tweezers
at frequency $f_{0}=15.2$ kHz, for increasing values of the mean velocity. The temperature oscillation $T$ measured by the thermometer
is demodulated by a Lock-in amplifier NF LI5640. $T$ can be accurately fitted by a classical
Fabry-Perot formula
\begin{equation}
T=\frac{A}{\sinh\left(i\frac{2\pi(f-f_{0})D}{c_{2}}+\xi D\right)} \label{eq:FP formula}
\end{equation}
where $i^2=-1$, $f_{0}$ is the resonant frequency for which the wave locally reaches its maximal
amplitude, $c_{2}$ is the second sound velocity, $A$ is a parameter
to be fitted, and $\xi$ is related to the energy loss of
the wave in the cavity. The top panel of Fig. \ref{fig:methode}
displays the amplitude of the thermal wave (in mK) as a function
of the frequency, and the bottom panel shows the same signal in phase and quadrature. When the
frequency is swept, the signal follows a curve close to a circle crossing
the point of coordinates $(0,0)$. Fig. \ref{fig:methode} clearly shows that the
resonant peak shrinks more and more when $U$ increases, which is
interpreted as attenuation of the wave inside the cavity. The red points
display the attenuation of the signal at constant value of $f$. It can
be seen on the bottom panel that the variation of the signal is close
to a pure attenuation, that is, without phase shift.
\begin{figure}
\begin{centering}
\includegraphics[height=4.5cm]{figure3a.pdf}\\
\includegraphics[height=5cm]{figure3b.pdf}
\par\end{centering}
\caption{\textbf{Top: }second sound resonance of one of the large tweezers around $15.2$
kHz. The value of $U$ increases from top curve to bottom curve. The vertical axis gives the amplitude
of the thermal wave in K. \textbf{Bottom:} representation of the
same resonance in phase and quadrature.\label{fig:methode}}
\end{figure}
$\xi$ can be decomposed as
\begin{equation}
\xi=\xi_{0}+\xi_{L}\label{eq:xi_dec}
\end{equation}
 where $\xi_{0}$ is the attenuation factor when $U=0$ m/s and $\xi_{L}$
is the additional attenuation created by the presence of quantum vortex
lines inside the cavity. $\xi_{L}$ is the signal of interest as
it can be directly related to the vortex lines density (VLD) using
the relation
\begin{eqnarray}
\xi_{L} & = & \frac{B\kappa L_{\perp}}{4c_{2}},\label{eq:VLD}\\
L_{\perp} & = & \frac{1}{\mathcal{V}}\int\sin^{2}\theta(l){\rm d}l\label{eq:VLD def}
\end{eqnarray}
where $B$ is the first Vinen coefficient, $\kappa\approx9.98\times10^{-8}$
m$^{2}$/s is the quantum of circulation, $\mathcal{V}$ is the cavity
volume, $l$ is the curvilinear absciss along the vortex line, $\theta(l)$
is the angle between the vector tangent to the line and the direction
perpendicular to the plates. We note that the summation is weighted by the distribution of the second sound nodes and antinodes inside the cavity and does not exactly corresponds to a uniform average but we neglect this effect in the following. Our aim is to measure both the average value
and the fluctuations of $L_{\perp}$, as a function of $U$ and the superfluid fraction.

The method goes as follows: first, we choose a resonant frequency
$f_{0}$ where the amplitude of the signal has a local maximum and
we fix the frequency of the heating to this value $f_{0}$. Then we
vary the mean velocity $U$ and we record the response of the thermometer
plate in phase and quadrature. The measurements
show that the velocity-induced displacement in the complex plane follows a straight line in a direction
$\overrightarrow{e}$ approximately orthogonal to the resonant curve.
Expressions (\ref{eq:FP formula}-\ref{eq:xi_dec}) give $\xi_{L}$
from the measured amplitude $T$ by\cite{roche2007vortex}
\begin{equation}
\xi_{L}=\frac{1}{D}\rm{asinh}\left(\frac{A}{T}\right)-\xi_{0}.\label{eq:displacement}
\end{equation}

The colored dots of Fig. \ref{fig:fluctuations} illustrate the fluctuations of the signal
in phase and quadrature, for different values of $U$. The average signal
moves in the direction of the attenuation axis. The figure also shows
a part of the resonant curve for $U=0$. The fluctuations have two
components in the plane, both associated with different physical
phenomena. Fluctuations in the direction tangent to the resonant curve
can be interpreted as a variation of the acoustic path $\frac{2\pi(f-f_{0})D}{c_{2}}$
without attenuation of the wave. Those fluctuations can occur for example
because the two arms of the tweezers vibrate with submicron amplitude,
or because the temperature variations modify the second sound velocity
$c_{2}$. To isolate only the fluctuations associated to attenuation by
the quantum vortices, we split the signal into a component along the attenuation axis, and another one along the acoustic path axis.
We then convert the displacement along the attenuation axis into vortex line density (VLD) using
expressions (\ref{fig:methode}-\ref{eq:displacement}).

\begin{figure}
\begin{centering}
\includegraphics[height=7cm]{figure4.pdf}
\par\end{centering}
\caption{Fluctuations of the thermal wave in phase and quadrature. The colored
clouds show the fluctuations of the signal, for different values of
$U$. The blue curve shows the resonance for $U=0$ m/s. The fluctuations tangent
to the resonant curve are created by a variation of the acoustic path.
The quantum vortices are associated to attenuation of the wave and create
a displacement along the attenuation axis. \label{fig:fluctuations}}
\end{figure}

\section{Results}

As a check of the validity of our approach, we measured the average
response of the second sound tweezers as a function of the mean velocity
$U$. According to literature\cite{Babuin:EPL2014}, we were expecting the scaling $\left\langle L_{\perp}\right\rangle^2 \propto U^{3}$, with a prefactor related to the flow main characteristics.  The function $\left\langle L_{\perp}\right\rangle $ was thus
measured for a range $0.4<U<1.25$ m/s with a time averaging over
$300$ ms, at the three different temperatures $1.65$ K, $1.99$ K and $2.14$ K.

An effective superfluid viscosity $\nu_\mathrm{eff}$ is customarily defined in quantum turbulence by $\epsilon = \nu_\mathrm{eff} ( \kappa \mathcal{L} )^2$ where $\epsilon$ is the dissipation and $\mathcal{L}=3\left\langle L_{\perp}\right\rangle/2$ is the averaged VLD (we assume isotropy of the tangle)\cite{Vinen:JLTP2002}. For large $R_\lambda$ homogeneous isotropic flows, we also have $\epsilon \simeq 0.79\, U^3\tau^3/H$ (eg see \cite{pope:book} p.245), which entails
\begin{equation}
    \tau^3 \simeq 2.85  \frac{ \nu_\mathrm{eff} H \kappa^2\left\langle L_{\perp}\right\rangle^2}{U^3} \label{eq:tau}
\end{equation}
Using Eq. (\ref{eq:tau}), we compute the turbulence intensity as a function of $U$, for the three considered temperatures. The result is displayed in Fig. \ref{fig:scalingU}. The figure shows that the turbulence intensity reaches a plateau of about $12\%$ above $0.8$ m/s, a value in accordance with the turbulence intensity of $9\%$ reported in \cite{vita2018generating} for a grid turbulence with similar characteristics. The figure also confirms that the expected scaling $\left\langle L_{\perp}\right\rangle^2 \propto U^{3}$ is reached in our experiment for the range of velocities $U>0.8$ m/s.

The temperature-dependent viscosity $\nu_\mathrm{eff}$ in Eq. (\ref{eq:tau}) has been measured in a number of experiments (e.g. see compilations in \cite{Babuin:EPL2014,boue2015energyVorticity,gao2018dissipation}). Still, the uncertainty on its value exceeds a factor 2. For the temperatures $1.65$ K and $1.99$ K, we used the average values $0.2 \kappa$ and $0.25\kappa$. By lack of reference experimental value of $\nu_\mathrm{eff}$ above $2.1$ K, we determined it by collapsing the $\tau (U)$ datasets obtained at $2.14$ K with the two others. We found the value $\nu_\mathrm{eff}\approx 0.5\kappa$ at $2.14$ K.

 Assuming isotropy of the vortex tangle, the value of $\mathcal{L} $ gives
a direct order of magnitude of the inter-vortex spacing $\delta=1/\sqrt{\mathcal{L}}$. We find $\delta\approx5$
$\mu m$ at 1.65 K and a mean velocity of 1 m/s. This shows the large scale separation between the inter-vortex spacing and the flow integral scale $H$, a confirmation of an intense turbulent regime.
\begin{figure}
\begin{centering}
\includegraphics[height=6cm]{figure5.pdf}
\caption{Indirect measurement of the turbulence intensity $\tau=u^{\rm{rms}}/U$ as a function of $U$ using Eq. (\ref{eq:tau}). The three different symbols correspond to three values of the mean temperature.   \label{fig:scalingU}}
\par\end{centering}
\end{figure}

Fig. \ref{fig:spectres} presents the main result of this letter.
We display on the top panel the VLD power spectral density $P_L(f)$ of $L_\perp/\left\langle L_{\perp}\right\rangle$. With this definition, the VLD turbulence intensity $L_{\perp}^{\rm{rms}}/\left\langle L_{\perp} \right\rangle$ is directly given by the integral of $P_L(f)$. We have measured the VLD fluctuations at the temperatures $T=1.65$ K and
superfluid fraction $\rho_{S}/\rho=81\%$, $T=1.99$ K and $\rho_{S}/\rho=47\%$,
$T=2.14$ K and $\rho_{S}/\rho=16\%$. At each temperature, the measurement was done for
at least two different mean velocities.

The first striking result is the collapse of all the spectra
independently of the temperature, when properly rescaled
using $f/U$ as coordinate (and $P_L(f)\times U$ as power spectral density to keep the integral constant).
The VLD spectrum does not depend on the superfluid fraction
even for vanishing superfluid fractions, when $T$ comes very close to $T_{\lambda}$. Only one measurement with one of the large tweezers at $T=1.650$ K has given a slight deviation from the master curve of the VLD spectra: it is displayed as the thin grey curve in Fig. \ref{fig:spectres}. We have no explanation for this deviation but did not observe this particular spectrum with the second tweezers, and neither at any other temperature.

Second, the VLD spectrum has no characteristic
power-law decay. We only observe that the spectrum follows an exponential decay approximately above $f/U>100$ m$^{-1}$. This strongly contrasts with the velocity spectrum obtained with the small second sound tweezers anemometer (see bottom panel),
which displays all the major features expected for a velocity
spectrum in classical turbulence: it has a sharp transition from a plateau at large scale to a power law scaling close
to $-5/3$ in the inertial scales of the turbulent cascade. Actually, it can be seen that the spectral decrease is less steep than $-5/3$, which can be due either to non-perfect isotropy and homogeneity, or more likely because the signal has some second-order corrections in addition to its dependence on velocity fluctuations. A fit of the transition using the von K\'arm\'an expression (see \cite{vita2018generating}) gives the value $H=5$ mm for the longitudinal integral scale. As a side remark, the apparent cut-off above $10^3$ m$^{-1}$ is an instrumental frequency cut-off of the tweezers.

 We find a value of the VLD turbulent intensity close to 20\%,
which is significantly higher than the velocity turbulence intensity. We also checked that we obtain the same VLD spectrum using different
resonant frequencies $f_{0}$.

Our measurements are limited by two characteristic frequencies. First,
the tweezers average the VLD over a cube of side $l$, which means
that our resolution cannot exceed $f/U>1/l$. For the large tweezers,
this sets a cut-off scale of $10^{3}$ m$^{-1}$, much larger than
the range of inertial scales presented in top panel of Fig. \ref{fig:spectres}.
Second, the frequency bandwidth of the resonator decreases when the
quality factor of the second sound resonance increases. This again sets a cut-off scale given by
$f/U=\xi_{0}c_{2}/(2U)$. The worst configuration corresponds to the
data obtained at 2.14 K and $U=1.2$ m/s where the cut-off scale is about
$600$ m$^{-1}$. For this reason, the VLD spectra of
Fig. \ref{fig:spectres} are conservatively restricted to $f/U<300$ m$^{-1}$ which allows to resolve about one and a half decade of inertial scales.

Figure \ref{fig:histogrammes} displays some typical PDF of the rescaled
VLD fluctuations $L_{\perp}/\left\langle L_{\perp} \right\rangle$ in semilogarithmic scale, for the three considered
temperatures. The PDF have been vertically shifted by one decade from each other
for readability. The figure shows a strong asymmetry at all temperatures,
with a nearly Gaussian left wing, and an exponential right wing. Contrary to the VLD spectra, the PDF do not accurately collapse on a single master curve at different velocities and temperatures: yet, they remain very similar when the temperature and the mean velocity are changed, and their strongly asymetric shape seems to be a robust feature.
\begin{figure}
\begin{centering}
\includegraphics[height=9cm]{figure6.pdf}\\
\par\end{centering}
\caption{\textbf{Top:} Power spectral density of the projected vortex line density (VLD) $L_{\perp}$, obtained
with the large second sound tweezers, for different values of $U$
and temperatures. All measured spectra collapse using the scaling
$f/U$ and $P_L(f)\times U$. The fluctuations have been rescaled by the mean
value of the VLD such that the integral of the above curves directly
give the VLD turbulence intensity.
\textbf{Bottom:} Power spectral density of the uncalibrated velocity signal obtained from the second
sound tweezers anemometer, for two values of $U$ at 1.65 K.
The spectra collapse using the scaling $f/U$ for the frequency
and $P_U(f)/U$ for the spectral density. The straight line displays
the $-5/3$ slope which is expected for a classical velocity spectrum
in the inertial range of the turbulent cascade. The dotted line is a fit using the von K\'arm\'an expression (see \cite{vita2018generating}) to find the integral scale $H$.
\label{fig:spectres}}
\end{figure}
By contrast, the dotted curve in Fig. \ref{fig:histogrammes} displays one PDF of the small tweezers anemometer at $1.65$ K, for which the mean has been shifted and the variance rescaled. It can be seen that the general shape of this latter PDF is much more symmetric and closer to a Gaussian as expected for a PDF of velocity fluctuations.

\begin{figure}
\begin{centering}
\includegraphics[height=6cm]{figure7.pdf}
\par\end{centering}
\caption{Normalized probability distributions of the VLD fluctuations obtained
at three temperatures. The PDF have been shifted by one decade from
each other for readability. By comparison, the dotted black curve displays a rescaled PDF obtained with the small tweezers measuring velocity. \label{fig:histogrammes}}
\end{figure}

\section{Discussion and conclusion}

In the present paper, we have investigated the temperature dependence of the statistics of the local density of vortex lines (VLD) in quantum turbulence. About one and a half decade of inertial scales of the turbulent cascade was resolved. We measure the VLD mean value and deduce from Eq. (\ref{eq:tau}) the turbulence intensity (Fig. \ref{fig:scalingU}), we report the VLD power spectrum (Fig. \ref{fig:spectres}), and the VLD probability distribution (Fig. \ref{fig:histogrammes}). Whereas the VLD mean value at different temperatures confirms previous numerical \cite{salort2011mesoscale,Babuin:EPL2014} and experimental studies \cite{Babuin:EPL2014}, the  spectral and PDF studies are completely new. Only one measurement of the VLD fluctuations had been done previously around 1.6K \cite{roche2007vortex} but in a wind tunnel with a very specific geometry and a non-controlled turbulence production. In the present work, we have used a grid turbulence, which is recognized as a reference flow with well-documented turbulence characteristics.

To conclude, we discuss below the three main findings:
\begin{enumerate}
    \item A master curve of the VLD spectra, independent of temperature and mean velocity.
    \item The observed master curve does not correspond to previously reported spectra in the context of highly turbulent classical flows.
    \item A global invariant shape of the strongly skewed PDF.
\end{enumerate}

The mean VLD gives the inter-vortex spacing, and thus tells how many quantum vortices are created in the flow, whereas the PDF and spectra tell how those  vortices are organized in the flow. From 2.14K to 1.65K, our results confirm that the inter-vortex spacing only weakly decreases, by less than 23\% for a 5-times increase of the superfluid fraction. In other words, the superfluid fraction has a limited effect on the creation of quantum vortices. The current understanding of the homogeneous isotropic turbulence in He-II is that the superfluid and normal fluid are locked together at large and intermediate scales where they undergo a classical Kolmogorov cascade \cite{spectra:PNAS2014}. The experimental evidences are based on the observation of classical velocity statistics
using  anemometers measuring the barycentric velocity of the normal and superfluid components. Here, the temperature-independence of (normalized) VLD spectra supports this general picture, by reminiscence of a similar property of He-II velocity spectra.

In contrast to velocity, the observed VLD master curve has an unexpected shape in the inertial range, at odd with the spectra reported  as ``compatible with'' a $f^{-5/3}$ scaling in \cite{roche2007vortex}. The probe is sensitive to the total amount of vorticity in the scales smaller than the probe spatial resolution, and thus keeps track of the small scales fluctuations. A close classical counterpart of VLD is enstrophy, because its 1-D spectrum is also related to the velocity spectrum at smaller scales (eg. see \cite{antonia1996note}). However, the experimental \cite{baudet1996spatial} and numerical (e.g. \cite{ishihara2003spectra}) enstrophy spectra reported so far in three-dimensional classical turbulence strongly differ from the present VLD spectra. We have no definite explanation for this difference. It could originate from remanent quantum vortices pinned on the grid, that cause additional energy injection in the inertial range, in which case the peculiarity of our spectra would be specific to the type of forcing. Otherwise, it could be a more fundamental property associated with the microscopic structure of the vortex tangle that, together with the observed temperature-independence of the spectra, would be very constraining to develop mathematical closures for the continuous description of He-II (eg. see \cite{nemirovskii2020closure}).

As a discussion of the third statement,
we compare the PDF with those of numerical simulations done in classical turbulence. The absolute value of vorticity can be seen as a classical counterpart to the VLD. The work of Iyer and co. \cite{yeung2015extreme} for example, displays some enstrophy PDF from high resolution DNS, that can be compared to the PDF of Fig. \ref{fig:histogrammes}.
At small scale, the enstrophy PDF are strongly asymmetric and will ultimately converge to a Gaussian distribution when averaged over larger and larger scales. Although our tweezers average the VLD over a size much larger than the inter-vortex spacing, they are small enough to sense short-life intense vortical events, typical of small scale phenomenology in classical turbulence. Thus, the strong asymmetry of the PDF supports the analogy between VLD and enstrophy (or its square root) and shows the relevance of VLD statistics to explore the small scales of quantum turbulence.

A side result of the present work is to obtain the relative values of
the empirical coefficient $\nu_\mathrm{eff}=\epsilon(\kappa \mathcal{L})^{-2}$ at the three considered temperatures. Models and simulations predict that $\nu_\mathrm{eff}$ should steeply increase close to $T_\lambda$ (see \cite{Babuin:EPL2014,boue2015energyVorticity,gao2018dissipation} and ref. within),
in apparent contradiction with the only  systematic experimental exploration \cite{stalp2002}. We found in Fig. \ref{fig:scalingU} that the effective viscosity $\nu_\mathrm{eff}$ is twice larger at 2.14K than at 1.99K.
To the best of our knowledge, our estimate $\nu_\mathrm{eff}(2.14 K) \simeq 2\,(\pm 0.25)\times \nu_\mathrm{eff}(1.99K)$ is the first experimental hint of such an effective viscosity increase.

\acknowledgments
We warmly  thank B. Chabaud for support in upgrading the wind-tunnel and P. Diribarne, E. Lévêque and B. Hébral for their comments.
We thank K. Iyer with his co-authors for sharing data on the statistics of spatially averaged enstrophy analyzed in \cite{iyerNJP2019}.
Financial support from grants ANR-16-CE30-0016 (Ecouturb) and ANR-18-CE46-0013 (QUTE-HPC).
\bibliographystyle{eplbib}


\section{Introduction}
 \label{sec:introduction}

 We study  {\sc greedy}\xspace  routing over uni-dimensional  metrics\footnote{The
 principles  of this work  can  be  extended  to higher  dimensional
 spaces. We focus on one-dimension   for simplicity.}  defined  over
 $n$  nodes lying in   a ring.  {\sc greedy}\xspace  routing   is the strategy  of
 forwarding a message along  that  out-going edge that minimizes   the
 {\it distance} remaining to the destination:

 \begin{mydefinition}{Greedy Routing}
   In a graph $(V,E)$ with a given distance function $\delta: V \times
   V \rightarrow \mathcal{R}^+$, {\sc greedy}\xspace routing entails the following
   decision: Given a target node $t$, a node $u$ with neighbors $N(u)$
   forwards  a  message  to its  neighbor   $v  \in  N(u)$ such   that
   $\delta(v,t) = \min_{x \in N(u)} \delta(x,t)$.
 \end{mydefinition}

 \noindent
 Two {\it natural} distance metrics  over $n$ nodes placed in a circle
 are the clockwise-distance and the absolute-distance between pairs of
 nodes:
 \begin{eqnarray*}
   \delta_{clockwise}(u, v) & = &
   \begin{cases}
      v-u   & v\geq u\\
      n+v-u & \text{otherwise}
   \end{cases}\\
   \delta_{absolute}(u, v)  & = &
   \begin{cases}
      \min \{ v-u, n+u-v \} & v\geq u\\
      \min \{ u-v, n+v-u \} & \text{otherwise}
   \end{cases}
 \end{eqnarray*}

 \noindent
 In this paper, we study the following related problems for the
 above distance metrics:

 \begin{center}
 \begin{minipage}{0.95\textwidth} \it
   \squishlist

   \item[I.]  Given integers  $d$ and  $\Delta$, what  is  the largest
      graph that satisfies two constraints: the out-degree of any node
      is at most  $d$, and the length of the  longest {\sc greedy}\xspace route is
      at most $\Delta$ hops?

   \item[II.] Given  integers $d$ and  $n$, design a network  in which
      each node has out-degree at most $d$ such that the length of the
      longest {\sc greedy}\xspace route is minimized.

   \squishend
 \end{minipage}
 \end{center}

 \subsection*{Summary of results}

 \begin{enumerate}

 \item  We  construct  a   family  of  network  topologies,  the  {\em
    Papillon\xspace}\footnote{Our   constructions   are   variants  of   the
    well-known butterfly family, hence  the name Papillon\xspace.}, in which
    {\sc  greedy}   routes  are  asymptotically   optimal.   For  both
    $\delta_{clockwise}$   and   $\delta_{absolute}$,  Papillon\xspace   has
    {\sc greedy}\xspace routes of  length $\Delta = \Theta(\log n  / \log d)$ hops
    in  the  worst-case  when  each  node  has  $d$  out-going  links.
    Papillon\xspace is  the first construction  that achieves asymptotically
    optimal worst-case {\sc greedy}\xspace routes.

 \item Upon further investigation:, two properties of  Papillon\xspace emerge:
     (a) {\sc greedy}\xspace routing does  not send messages along shortest paths,
     and (b)  Edge congestion with {\sc greedy}\xspace routing  is not  uniform --
     some edges are used more often  than others. We exhibit the first
     property  by identifying routing  strategies that result in paths
     shorter than those  achieved  by {\sc greedy} routing.   In fact,
     one of these strategies guarantees uniform edge-congestion.

 \item Finally, we consider another distance function
   $\delta_{xor}(u,  v)$, defined as   the number of  bit-positions in
   which $u$ and $v$ differ. $\delta_{xor}$ occurs naturally, e.g., in
   hypercubes, and  {\sc greedy}\xspace routing  with $\delta_{xor}$  routes along
   shortest  paths in them.  We construct  a variant of Papillon\xspace that
   supports  asymptotically optimal routes  of length $\Theta(\log n /
   \log  d)$  in  the worst-case,  for  {\sc greedy}\xspace  routing with distance
   function $\delta_{xor}$.

 \end{enumerate}

 \section{Related Work}
 \label{sec:related}

 {\sc greedy}\xspace  routing is  a fundamental  strategy in  network  theory.  It
 enjoys numerous advantages.  It  is completely decentralized, in that
 any node  takes routing decisions  locally and independently.   It is
 oblivious,  thus  message  headers  need  not be  written  along  the
 route. It is inherently fault tolerant, as progress toward the target
 is guaranteed so  long as some links are available.   And it has good
 locality behavior  in that every  step decreases the distance  to the
 target.   Finally,  it  is   simple  to  implement,  yielding  robust
 deployments.   For  these  reasons,  {\sc greedy}  routing  has  long
 attracted  attention in  the research  of network  design.  Recently,
 {\sc greedy}\xspace routing  has witnessed increased research  interest in the
 context of  decentralized networks.  Such networks  arise in modeling
 social networks  that exhibit the ``small world  phenomenon'', and in
 the design of overlay networks for peer-to-peer (P2P) systems. We now
 summarize known results pertaining to {\sc greedy}\xspace routing on a circle.

 \subsection*{The Role of the Distance Function}

 Efficient graph constructions are  known that support {\sc greedy}\xspace routing
 with   distance   function   other  than   $\delta_{clockwise}$,
 $\delta_{absolute}$ and $\delta_{xor}$.
 For de Bruijn networks,  the traditional
 routing algorithm  (which routes almost always  along shortest paths)
 corresponds  to {\sc greedy}\xspace routing  with $\delta(u,  v)$ defined  as the
 longest suffix of $u$ that is also the prefix of $v$.  For a 2D grid,
 shortest  paths correspond  to  {\sc greedy}\xspace routing  with $\delta(u,  v)$
 defined  as  the  Manhattan  distance  between  nodes  $u$  and  $v$.

 For {\sc greedy}\xspace routing on a circle, the best-known constructions have $d
 = \Theta(\log n)$ and $\Delta  =  \Theta(\log n)$. Examples  include:
 Chord~\cite{chord:sigcomm01}           with         distance-function
 $\delta_{clockwise}$,  a   variant   of   Chord  with ``bidirectional
 links''~\cite{ganesan:soda04}           and         distance-function
 $\delta_{absolute}$, and   the   hypercube  with    distance function
 $\delta_{xor}$. In this paper, we improve upon all of these
 constructions by showing how to route in
 $\Theta(\log n / \log d)$ hops in the worst case with
 $d$ links per node.

 \subsection*{{\sc greedy}\xspace Routing in Deterministic Graphs}

 The  \textsf{Degree-Diameter  Problem},  studied  in  extremal  graph
 theory, seeks  to identify the largest graph  with diameter $\Delta$,
 with each node having  out-degree at most $d$ (see Delorme~\cite{ddp}
 for a survey).  The best  constructions for large $\Delta$ tend to be
 sophisticated~\cite{bermond:92,comellas:92,exoo:01}.    A  well-known
 upper bound  is $N(d, \Delta)  = 1 +  d + d^2  + \cdots +  d^\Delta =
 \frac{d^{\Delta+1}  - 1}{d-1}$,  also known  as the  Moore  bound.  A
 general lower  bound is $d^\Delta + d^{\Delta-1}$,  achieved by Kautz
 digraphs~\cite{kautz:68,kautz:69}, which are  slightly superior to de
 Bruijn  graphs~\cite{debruijn:46} whose  size is  only
 $d^\Delta$. Thus  it is   possible to route
 in $O(\log  n / \log  d)$ hops in  the worst-case with  $d$ out-going
 links  per  node.  Whether  {\sc greedy}\xspace  routes  with distance  functions
 $\delta_{clockwise}$  or  $\delta_{absolute}$  can achieve  the  same
 bound, is the question we have addressed in this paper.

 {\sc greedy}\xspace routing  with distance function  $\delta_{absolute}$ has been
 studied for  Chord~\cite{ganesan:soda04}, a popular  topology for P2P
 networks.  Chord  has $2^b$ nodes,  with out-degree $2b-1$  per node.
 The longest {\sc greedy}\xspace route takes  $\Floor{b/2}$ hops.  In terms of $d$
 and $\Delta$, the  largest-sized Chord network has $n  = 2^{2\Delta +
 1}$ nodes.  Moreover, $d$ and $\Delta$ cannot be chosen independently
 -- they  are  functionally  related.    Both  $d$  and  $\Delta$  are
 $\Theta(\log n)$.   Analysis of {\sc greedy}\xspace routing of  Chord leaves open
 the  following question:

 \smallskip
 \centerline{\it For  {\sc greedy}\xspace routing  on a  circle, is
 $\Delta = \Omega(\log n)$ when $d = O(\log n)$?}
 \smallskip

 Xu {\it et al.}\xspace~\cite{xu:infocom03} provide a partial answer to the above question
 by   studying   {\sc greedy}\xspace   routing  with   distance   function
 $\delta_{clockwise}$ over  \emph{uniform} graph topologies.   A graph
 over $n$ nodes placed in a circle is said to be uniform if the set of
 clockwise   offsets  of   out-going  links   is  identical   for  all
 nodes. Chord  is an example of  a uniform graph.  Xu  {\it et al.}\xspace show that
 for  any uniform  graph  with  $O(\log n)$  links  per node,  {\sc greedy}\xspace
 routing  with  distance  function  $\delta_{clockwise}$  necessitates
 $\Omega(\log n)$ hops in the worst-case.

 Cordasco {\it et al.}\xspace~\cite{fchord:sirocco04} extend the result of Xu
 {\it et al.}\xspace~\cite{xu:infocom03}  by  showing   that  {\sc greedy}\xspace  routing  with
 distance function  $\delta_{clockwise}$ in  a uniform graph  over $n$
 nodes satisfies the inequality $n \leq  F(d + \Delta + 1)$, where $d$
 denotes the  out-degree of each node,  $\Delta$ is the  length of the
 longest  {\sc greedy}\xspace  path, and  $F(k)$  denotes  the $k^{th}$  Fibonacci
 number.  It  is well-known that  $F(k) = [\phi^k /  \sqrt{5}]$, where
 $\phi  = 1.618\ldots$  is  the  Golden ratio  and  $[x]$ denotes  the
 integer closest to  real number $x$.  It follows  that $1.44 \log_2 n
 \leq d  + \Delta  + 1$.  Cordasco {\it et al.}\xspace show  that the  inequality is
 strict  if $|d  -  \Delta| >  1$. For  $|d  - \Delta|  \leq 1$,  they
 construct uniform  graphs based upon Fibonacci  numbers which achieve
 an optimal tradeoff between $d$ and $\Delta$.

 \medskip

 The  results  in ~\cite{ganesan:soda04,xu:infocom03,fchord:sirocco04}
 leave open  the question whether there exists  any graph construction
 that permits {\sc greedy}\xspace routes of  length $\Theta(\log n / \log d)$ with
 distance  function  $\delta_{clockwise}$ and/or  $\delta_{absolute}$.
 Papillon\xspace  provides  an  answer  to  the problem  by  constructing  a
 non-uniform graph --- the set of clockwise offsets of out-going links
 is different for different nodes.

 \subsection*{{\sc greedy}\xspace Routing in Randomized Graphs}

 {\sc greedy}\xspace  routing  over  nodes  arranged  in a  ring  with  distance
 function $\delta_{clockwise}$  has recently been  studied for certain
 classes of {\it randomized} graph constructions. Such graphs arise in
 modeling social networks that exhibit the ``small world phenomenon'',
 and in the design of overlay networks for P2P systems.

 In   the  seminal  work   of  Kleinberg   \cite{kleinberg:stoc00},  a
 randomized  graph was  constructed in  order to  explain  the ``small
 world         phenomenon'',        first         identified        by
 Milgram~\cite{milgram:pt67}. The phenomenon refers to the observation
 that individuals are able to  route letters to unknown targets on the
 basis  of knowing  only their  immediate social  contacts.  Kleinberg
 considers  a set  of nodes  on  a uniform  two-dimensional grid.   It
 proposes  a  link  model in  which  each  node  is connected  to  its
 immediate grid  neighbors, and in  addition, has a single  long range
 link drawn  from a normalized  harmonic distribution with  power $2$.
 In the resulting graph, {\sc greedy}\xspace  routes have length at most $O(\log^2
 n)$ hops in expectation; this  complexity was later shown to be tight
 by Barri{\`e}re {\it et al.}\xspace in \cite{barriere:disc01}.

 Kleinberg's construction has  found  applications  in the design   of
 overlay  routing   networks    for     Distributed Hash    Tables.
 Symphony~\cite{symphony:usits03}  is   an  adaptation of  Kleinberg's
 construction in a single  dimension.  The idea  is to place $n$ nodes
 in a virtual circle and to equip each  node with $d \geq 1$ out-going
 links.  In the resulting network, the  average path length of {\sc greedy}\xspace
 routes      with    distance     function  $\delta_{clockwise}$    is
 $O(\frac{1}{d}\log^2 n)$ hops.  Note that unlike Kleinberg's network,
 the space here is virtual and  so are the distances  and the sense of
 {\sc greedy}\xspace routing.  The same  complexity  was achieved with a  slightly
 different        Kleinberg-style     construction     by       Aspnes
 {\it et al.}\xspace~\cite{aspnes:podc02}.  In the   same paper, it  was  also shown
 that  any     symmetric,   randomized   degree-$d$      network   has
 $\Omega(\frac{\log^2 n}{d\log\log n})$ {\sc greedy}\xspace routing complexity.

 Papillon outperforms all of the above randomized constructions, using
 degree $d$ and achieving  $\Theta(\log n/\log d)$ routing.  It should
 be  possible to  randomize Papillon  along similar  principles to the
 Viceroy\cite{viceroy:podc02} randomized construction of the butterfly
 network, though we do not pursue this direction here.

 \subsection*{Summary of Known Results}

 With $\Theta(\log n)$ out-going  links per node, several graphs  over
 $n$ nodes  in a circle support  {\sc greedy}\xspace routes  with $\Theta(\log n)$
 {\sc greedy} hops.  Deterministic graphs with this property include:
 (a) the original  Chord~\cite{chord:sigcomm01} topology with distance
 function $\delta_{clockwise}$,   (b)  Chord with  edges   treated  as
 bidirectional~\cite{ganesan:soda04}    with    distance      function
 $\delta_{absolute}$.   This is also    the known lower  bound  on any
 uniform    graph  with  distance     function    $\delta_{clockwise}$
 \cite{xu:infocom03}.    Randomized   graphs  with  the  same tradeoff
 include  randomized-Chord~\cite{gummadi:sigcomm03,zhang:sigmetrics03}
 and Symphony~\cite{symphony:usits03}  -- both with  distance function
 $\delta_{clockwise}$.   With   degree   $d  \le   \log n$,   Symphony
 \cite{symphony:usits03} has  {\sc greedy}\xspace routes of length $\Theta((\log^2
 n)/ d)$    on  average.  The   network  of  \cite{aspnes:podc02} also
 supports {\sc greedy}\xspace routes of length $O((\log^2 n)/d)$ on average , with
 a   gap   to  the   known  lower    bound   on   their   network   of
 $\Omega(\frac{\log^2 n}{d\log\log n})$.

 The above results are  somewhat discouraging, because routing that is
 \textbf{non}-{\sc  greedy}  can  achieve  much  better  results.   In
 particular, networks  of degree $2$  with hop complexity  $O(\log n)$
 are  well known,  e.g.,  the Butterfly  and  the de  Bruijn (see  for
 example \cite{leighton:92} for exposition material).  And networks of
 logarithmic  degree  can  achieve  $O(\log n/  \log\log  n)$  routing
 complexity (e.g., take the  degree-$\log_2 n$ de Bruijn).  Routing in
 these networks is non-{\sc greedy} according to any one of our
 metrics ($\delta_{clockwise}$, $\delta_{absolute}$, and
 $\delta_{xor}$).

 The Papillon\xspace\  construction demonstrates  that we can  indeed design
 networks  in which  {\sc greedy}  routing  along  these metrics   has
 asymptotically optimal  routing  complexity.  Our  contribution  is a
 family  of networks that extends the  Butterfly network family, so as
 to  facilitate  efficient {\sc  greedy} routing.  With  $d$ links per
 node, {\sc    greedy}  routes are  $\Theta(\log  n/\log   d)$  in the
 worst-case, which is asymptotically  optimal.  For  $d = o(\log  n)$,
 this  beats the  lower  bound of  \cite{aspnes:podc02}  on symmetric,
 randomized  greedy routing networks  (and  it meets it for  $d=O(\log
 n$).  In the specific case of $d=\log n$, our greedy routing achieves
 $O(\log n/\log \log n)$ average route length.

\subsection*{{\sc greedy}\xspace with {\sc lookahead}\xspace}

 Recent  work~\cite{manku:stoc04} explores the surprising advantages of
 {\sc greedy}\xspace with   {\sc lookahead}\xspace in  randomized graphs  over  $n$ nodes  in a
 circle.  The  idea behind {\sc lookahead}\xspace  is  to take neighbor's neighbors
 into  account to make  routing  decisions.  It  shows that greedy with
{\sc lookahead}\xspace achieves  $O(\log^2 n/ d \log d)$  expected route length in
Symphony~\cite{symphony:usits03}.    For   other networks  which  have
$\Theta(\log     n)$      out-going   links      per    node,    e.g.,
randomized-Chord~\cite{gummadi:sigcomm03,zhang:sigmetrics03},
randomized-hypercubes~\cite{gummadi:sigcomm03},
skip-graphs~\cite{aspnes:soda03}  and  SkipNet~\cite{skipnet:usits03},
average path length  is $\Theta(\log n  / \log  \log  n)$ hops.  Among
these networks, Symphony and randomized-Chord use {\sc greedy}\xspace routing with
distance function $\delta_{clockwise}$. Other networks use a different
distance  function  (none of them uses  $\delta_{xor}$).   For each of
these networks,  with  $O(\log n)$ out-going   links per node,  it was
established   that   plain  {\sc greedy}\xspace   (\emph{without} {\sc lookahead}\xspace)   is
sub-optimal and achieves $\Omega(\log n)$ expected route lengths.  The
results suggest that {\sc lookahead} has significant impact on {\sc greedy}\xspace
 routing.

 Unfortunately,  realizing    {\sc greedy}\xspace routing   with   {\sc lookahead}\xspace on a
 degree-$k$ network implies that $O(k^2)$  nodes need to be considered
 in each hop,  while plain {\sc greedy}\xspace needs  to consider only $k$ nodes.
 For $k= \log_2 n$, this implies a $O(\log n)$ overhead for {\sc lookahead}\xspace
 routing in every hop.

 Papillon\xspace demonstrates that  it is possible to  construct a graph  in
 which  each node has degree  $d$  and in which {\sc greedy}\xspace \emph{without}
 1-{\sc lookahead}\xspace has routes of  length $\Theta(\log n  / \log d)$  in the
 worst case, for the metrics $\delta_{clockwise}$, $\delta_{absolute}$
 and $\delta_{xor}$.  Furthermore, for all $d = o(\log n)$, plain {\sc
   greedy} on  our network design beats  even the  results obtained in
 \cite{manku:stoc04} with $1$-{\sc lookahead}.

 \subsection*{Previous Butterfly-based Constructions}

 \noindent

 Butterfly networks have been used  in the context of routing networks
 for DHTs as follows:
 \begin{enumerate}

 \item Deterministic butterflies have been proposed for DHT routing by
    Xu  {\it et al.}\xspace~\cite{xu:infocom03},  who  subsequently developed  their
    ideas into  Ulysses~\cite{ulysses:icnp03}.  Papillon\xspace for distance
    function  $\delta_{clockwise}$  has  structural similarities  with
    Ulysses -- both are butterfly-based networks.  The key differences
    are as  follows: (a) Ulysses  does not use  $\delta_{absolute}$ as
    its distance  function, (b) Ulysses does not  use {\sc greedy}\xspace routing,
    and  (c)  Ulysses uses  more  links  than  Papillon\xspace for  distance
    function  $\delta_{clockwise}$  --   additional  links  have  been
    introduced  to ameliorate  non-uniform edge  congestion  caused by
    Ulysses' routing algorithm. In contrast, the {\sc congestion-free}
    routing algorithm  developed in \S\ref{sec:improved}  obviates the
    need    for    any   additional    links    in   Papillon\xspace    (see
    Theorem~\ref{thm:congestion_free_clockwise}).

 \item Viceroy~\cite{viceroy:podc02}  is a \emph{randomized} butterfly
    network  which routes  in  $O(\log n)$  hops  in expectation  with
    $\Theta(1)$      links      per      node.      Mariposa      (see
    reference~\cite{dipsea:2004} or~\cite{manku:podc03}) improves upon
    Viceroy by providing routes of length  $O(\log n / \log d)$ in the
    worst-case,  with  $d$  out-going  links per  node.   Viceroy  and
    Mariposa are different from  other randomized networks in terms of
    their design philosophy.
 The Papillon\xspace\ topology borrows elements of the geometric embedding of the
 butterfly in a circle from Viceroy \cite{viceroy:podc02} and from
 \cite{manku:podc03}, while extending them for {\sc greedy} routing.

 \end{enumerate}

 \section{Papillon\xspace}
 \label{sec:papillon}

 We  construct  two variants   of   butterfly networks, one  each  for
 distance-functions $\delta_{clockwise}$ and $\delta_{absolute}$.  The
 network has $n$ nodes arbitrarily positioned  on a ring. We label the
 nodes from $0$ to  $n-1$ according to their  order on the  ring.  For
 convenience,  $x \bmod n$  always represents an  element lying in the
 range $[0, n-1]$ (even when $x$ is negative, or greater than $n-1$).

 \begin{mydefinition}{Papillon\xspace for $\delta_{clockwise}$}
   ${\mathcal B}_{clockwise}(\kappa,m)$  is a directed  graph, defined
   for any pair of integers $\kappa,m \geq 1$
   \begin{enumerate}

   \item Let $n =  \kappa^m m$.

   \item  Let $\ell(u) \equiv  (m-1) -  (u \bmod  m)$.  Each  node has
      $\kappa$ links. For node $u$,  these directed links are to nodes
      $(u + x) \bmod n$, where
      $ x \in  \{1 + im\kappa^{\ell(u)}\ |\ i \in [0, \kappa -1 ] \}$.

      We  denote  the  link  with   node  $(u+1)  \bmod  n$  as  $u$'s
      \emph{``short  link''}.  The other  $\kappa-1$ links  are called
      $u$'s \emph{``long links''}.

   \end{enumerate}
 \end{mydefinition}

 \begin{mydefinition}{Papillon\xspace for $\delta_{absolute}$}
   ${\mathcal  B}_{absolute}(k,m)$ is a  directed graph,  defined for
   any pair of integers $k,m \geq 1$,
   \begin{enumerate}

   \item Let $n =(2k+1)^m m$.

   \item  Let $\ell(u) \equiv  (m-1) -  (u \bmod  m)$.  Each  node has
      $2k+2$ out-going links.  Node  $u$ makes $2k+1$ links with nodes
      $(u + x) \bmod n$, where
      $ x  \in  \{1  +   im(2k+1)^{\ell(u)}\ |\ i \in [-k, +k] \}$.
      Node $u$ also makes an out-going link with node $(u+x) \bmod n$,
      where $x = -m+1$.

      We denote  the link with node $(u+1) \bmod n$
      as  $u$'s \emph{``short  link''}.   The other  $2k+1$ links  are
      called $u$'s \emph{``long links''}.

    \end{enumerate}
 \end{mydefinition}

 In both ${\mathcal B}_{clockwise}$ and ${\mathcal B}_{absolute}$, all
 out-going links  of node   $u$ are incident    upon nodes with  level
 $(\ell(u) - 1) \bmod  m$.   In ${\mathcal B}_{clockwise}$, the  short
 links   are    such   that each    hop    diminishes  the   remaining
 \emph{clockwise} distance by at least one. Therefore, {\sc greedy}\xspace routing
 is guaranteed   to take  a finite   number  of  hops.   In ${\mathcal
   B}_{absolute}$, not  every  {\sc greedy}\xspace  hop  diminishes the  remaining
 \emph{absolute} distance.  However,  {\sc greedy}\xspace routes are  still finite
 in length, as we show in the proof of Theorem~\ref{thm:absolute}.

 \begin{theorem} \label{thm:clockwise}
   {\sc greedy}\xspace routing in ${\mathcal  B}_{clockwise}$ with distance
   function $\delta_{clockwise}$ takes $3m-2$ hops in
   the worst-case.  The average is less than $2m-1$ hops.
 \end{theorem}

 \begin{proof}
  For any node $u$, we define
  $
  \text{SPAN}(u)\equiv \{  v \  |\ 0  \leq \delta_{clockwise}(u,  v)  < m
  \kappa^{\ell(u)+1} \}.
  $
  Let  $t$ and $u$ denote the target node and the current node,
  respectively. Routing
  proceeds in (at most) three phases:
  \begin{center}
    \begin{tabular}{lll}
      Phase I: & $t \not\in \text{SPAN}(u)$ & (at most $m-1$ hops)\\
      Phase II: & $t \in \text{SPAN}(u)$ and
            $\delta_{clockwise}(u,t)  \ge  m$ & (at most $m$ hops)\\
      Phase III: & $t \in \text{SPAN}(u)$ and
            $\delta_{clockwise}(u,t) <  m$ & (at most $m-1$ hops)
    \end{tabular}
  \end{center}
  We now prove upper bounds on the number of hops in each phase.
  \begin{enumerate}

  \item[I.]
     The out-going  links of $u$ are  incident upon nodes at
     level $(\ell(u) - 1) \bmod  m$. So eventually, the level of the
     current node
     $u$ will be $m-1$. At  this point,
     $t \in \text{SPAN}(u)$ because $\text{SPAN}(u)$ includes
     \emph{all} the
     nodes.  Thus Phase 1 lasts for at most $m-1$ hops
     ($\frac{m-1}{2}$ hops on
     average).

  \item[II.]
     {\sc greedy}\xspace will  forward  the
     message  to some node  $v$  such that  $t  \in \text{SPAN}(v)$  and
     $\ell(v)=\ell(u)-1$.  Eventually, the current node $u$ will
     satisfy the property   $\ell(u) = 0$.  This node will  forward
     the message  to some node
     $v$ with $\ell(v) = m-1$ such that $\delta_{clockwise}(v,t) < m$,
     thereby terminating this phase of  routing. There are at most $m$
     hops in this phase (at most $m$ on average as well).

  \item[III.]
     In this phase, {\sc greedy}\xspace  will  decrease  the clockwise
     distance by  exactly one in each hop by following the
     short-links. Eventually, target $t$ will be reached. This phase
     takes  at  most  $m-1$  hops  ($\frac{m-1}{2}$  hops  on
     average).
  \end{enumerate}

  The worst-case  route length is  $3m-2$.
  On average, routes are at most $2m-1$ hops long.
 \end{proof}

 \begin{theorem} \label{thm:absolute}
   {\sc greedy}\xspace routing  in ${\mathcal B}_{absolute}$ with distance
   function $\delta_{absolute}$ takes  $3m-2$ hops in
   the worst-case.  The average is less than $2m-1$ hops.
 \end{theorem}
 \begin{proof}
  For any node $u$, we define
  \[
  \text{SPAN}(u)\equiv  \{ v \ |\  \delta_{absolute}(u, v) =  | c +
  m\sum_{i=0}^{\ell(u)} (2k+1)^i d_i |,\ c \in [0, m-1],\ d_i \in [-k,
  +k] \,\}.
  \]

  Let  $t$ and $u$ denote the target node and the current node,
  respectively.

  Routing  proceeds in (at most) three phases:
  \begin{center}
    \begin{tabular}{lll}
      Phase I: & $t \not\in \text{SPAN}(u)$ & (at most $m-1$ hops)\\
      Phase II: & $t \in \text{SPAN}(u)$ and
            $\delta_{absolute}(u,t)  \ge  m$ & (at most $m$ hops)\\
      Phase III: & $t \in \text{SPAN}(u)$ and
            $\delta_{absolute}(u,t) <  m$ & (at most $m-1$ hops)
    \end{tabular}
  \end{center}
  We now prove upper bounds on the number of hops in each phase.

  \begin{enumerate}

  \item[I.] All out-going  links of node $u$ are  incident upon nodes at
     level $(\ell(u) - 1) \bmod  m$. So eventually, the current node
     $u$ will satisfy the property $\ell(u) = m - 1$. At this point,
     $t \in \text{SPAN}(u)$ because $\text{SPAN}(u)$ includes
     \emph{all} nodes. Thus Phase I lasts at most  $m-1$ hops
     (at most $\frac{m-1}{2}$ hops on average).

  \item[II.]
     Phase  2 terminates if target node $t$  is reached, or if
     $\delta_{absolute}(u, t)  < m$.
      Node
     $u$ always forwards the  message  to some  node  $v$ such that
     $t \in \text{SPAN}(v)$  and  $\ell(v)  = \ell(u)  -  1$. So
     eventually, either target $t$ is reached, or the current node
     $u$ satisfies the property
     $\ell(u) =  0$.  At this point, if node $u$  forwards the  message to
      node $v$, then it is guaranteed that $\ell(v) =
     m-1$ and  $\delta_{absolute}(v,t) <  m$,
     thereby terminating Phase II. There are at most $m$
     hops in this phase (at most $m$ on average as well).

  \item[III.]     The target node $t$ is reached in at most $m-1$
    hops (the existence of  the ``back
     edge''  that connects node  $u$ to  node $(u  + 1  - m)  \bmod n$
     guarantees this).  This  phase takes at most $m-1$  hops (at most
     $\frac{m-1}{2}$ hops on average).
  \end{enumerate}

  The worst-case  route length is  $3m-2$.
  On average, routes are at most $2m-1$ hops long.
 \end{proof}

 \bigskip

 Routes  in both ${\mathcal  B}_{clockwise}$ and  ${\mathcal B}_{absolute}$
 are  at  most $3m-2$  hops,  which is  $O(\log  (\kappa^m  m) /  \log
 \kappa)$  and $O(\log  ((2k+1)^m  m) /  \log (2k+2))$,  respectively.
 Given degree $d$ and diameter $\Delta$, the size of Papillon\xspace is $n =
 2^{O(\Delta)}\Delta$ nodes.   Given degree $d$ and  network size $n$,
 the longest route has length $\Delta = O(\log n / \log d)$.

 \section{Improved Routing Algorithms for Papillon\xspace}
 \label{sec:improved}

 {\sc greedy}\xspace  routing does  not route  along shortest-paths  in ${\mathcal
 B}_{clockwise}$ and  ${\mathcal B}_{absolute}$.  We  demonstrate this
 constructively below,  where we study a routing  strategy called {\sc
 hypercubic-routing} which achieves shorter path lengths than {\sc greedy}\xspace.

 \subsection*{Hypercubic Routing}

 \begin{theorem} \label{thm:fast_clockwise}
   There exists  a routing strategy for  ${\mathcal B}_{clockwise}$ in
   which routes take $2m-1$ hops  in the worst-case. The average is at
   most $1.5m$ hops.
 \end{theorem}

 \begin{proof}
   Consider  the  following   {\sc  hypercubic-routing}  algorithm  on
   ${\mathcal B}_{clockwise}$.   Let $s$ be  the source node,  $t$ the
   target,  and  let  $dist =  \delta_{clockwise}(s,  t)  =  c +  m  +
   m\sum_{i=0}^{i=m-1} \kappa^i  d_i$ with $0  \leq c <m$ and  $0 \leq
   d_i <  \kappa$ ($dist$ has exactly one  such representation, unless
   $dist \leq m$ in which case routing takes $< m$ hops).

   Phase I: Follow  the short-links to ``fix'' the  $c$-value to zero.
   This takes at most $m-1$ hops (at most $0.5m$ hops on average).

   Phase II: In exactly $m$ hops, ``fix'' the $d_i$'s in succession to
   make  them  all  zeros:  When  the  current node  is  $u$,  we  fix
   $d_{\ell(u)}$ to zero by following the appropriate long-link, i.e.,
   by    shrinking   the    clockwise    distance   by    $d_{\ell(u)}
   \kappa^{\ell(u)} m  + 1$.   The new node  $v$ satisfies  $\ell(v) =
   (\ell(u)+m-1) (\bmod~m)$.  When each $d_i$ is zero, we have reached
   the target.

   Overall,  the worst-case  route  length is  $2m-1$.  Average  route
   length is at most $1.5m$.
 \end{proof}

 \begin{theorem} \label{thm:fast_absolute}
   There exists  a routing  strategy for ${\mathcal  B}_{absolute}$ in
   which routes take $2m-1$ hops  in the worst-case. The average is at
   most $1.5m$ hops.
 \end{theorem}
 \begin{proof}
   Let $s$  be the  source node,  $t$ the   target.

   Phase  I: Follow  the short-links  in the  clockwise  direction, to
   reach a  node $s'$ such that  $\ell(s') = \ell(t)$.   This takes at
   most $m-1$  hops (at most  $0.5m$ hops on average).   The remaining
   distance can be expressed as $m + m\sum_{i=0}^{i=m-1} (2k+1)^i d_i$
   where $-k \leq d_i \leq k$. There is a unique such representation.

   Phase II: In exactly $m$ hops, ``fix'' the $d_i$'s in succession to
   make  them  all  zeros:  When  the  current node  is  $u$,  we  fix
   $d_{\ell(u)}$ by following the appropriate long-link, i.e.,
   by traveling  distance $1  + d_{\ell(u)} (2k+1)^{\ell(u)}  m$ along
   the circle  (this distance is positive or  negative, depending upon
   the sign of $d_{\ell(u)}$).  The  new node $v$ satisfies $\ell(v) =
   (\ell(u)-1) (\bmod~m)$.   When each $d_i$ is zero,  we have reached
   the target.

   Overall,  the worst-case  route  length is  $2m-1$.  Average  route
   length is at most $1.5m$.
 \end{proof}

 \bigskip

 Note that the  edges that connect node $u$ to  node $(u+1-m) \bmod n$
 are redundant for {\sc hypercubic-routing} since they are never used.
 However,  these edges  play  a  crucial role  in  {\sc greedy}\xspace routing  in
 ${\mathcal  B}_{absolute}$ (to  guide the  message to  the  target in
 Phase 3).

 \subsection*{Congestion-Free Routing}

 Theorems~\ref{thm:fast_clockwise}  and ~\ref{thm:fast_absolute} prove
 that {\sc greedy}\xspace  routing is sub-optimal  in the constants.  {\sc
 hypercubic-routing}, as described above, is faster than
 {\sc greedy}\xspace. However, it causes {\it edge-congestion}
 because short-links  are used more often than  long-links.  Let $\pi$
 denote the ratio of maximum and  minimum loads on edges caused by all
 $n  \choose   2$  pairwise  routes.    {\sc  hypercubic-routing}  for
 ${\mathcal  B}_{clockwise}$  consists of  two  phases  (see Proof  of
 Theorem~\ref{thm:fast_clockwise}).   The  load  due  to Phase  II  is
 uniform  -- all  edges  (both short-links  and  long-links) are  used
 equally.  However, Phase  I uses only short-links, due  to which $\pi
 \not= 1$.  We now modify the routing scheme slightly to obtain $\pi =
 1$ for both ${\mathcal B}_{clockwise}$ and ${\mathcal B}_{absolute}$.

 \begin{theorem} \label{thm:congestion_free_clockwise}
   There  exists  a  congestion-free  routing strategy  in  ${\mathcal
   B}_{clockwise}$ that takes  $2m - 1$ hops in  the worst-case and at
   most $1.5m$ hops on average, in which $\pi = 1$.
 \end{theorem}
 \begin{proof}
   The  theorem is proved  constructively, by  building a  new routing
   strategy  called {\sc  congestion-free}. This  routing  strategy is
   exactly the same as {\sc hypercubic-routing}, with a small change.

   Let $s$ be the source node,  $t$ the target. Let $c = (t+m-s) \bmod
   m$, the difference in levels between $\ell(s)$ and $\ell(t)$.

   Phase I: For $c$ steps, follow any out-going link, chosen uniformly
   at  random.   We thus  reach  a node  $s'$  such  that $\ell(s')  =
   \ell(t)$.

   Phase II: The remaining  distance is $dist = \delta_{clockwise}(s',
   t)  = m+  m\sum_{i=0}^{i=m-1}  \kappa^i  d_i$ with  $0  \leq d_i  <
   \kappa$.  Continue  with Phase  II of the  {\sc hypercubic-routing}
   algorithm       for      ${\mathcal       B}_{clockwise}$      (see
   Theorem~\ref{thm:fast_clockwise}).

   It is easy to see that in this case, all outgoing links (short- and
   long-)  are used with  equal probability  along the  route.  Hence,
   $\pi = 1$.
 \end{proof}

 \begin{theorem} \label{thm:congestion_free_absolute}
   There  exists  a  congestion-free  routing strategy  in  ${\mathcal
   B}_{absolute}$ that  takes $2m - 1$  hops in the  worst-case and at
   most $1.5m$ hops on average, in which $\pi = 1$.
 \end{theorem}
 \begin{proof}
   We will  ignore the  edges that connect  node $u$ to  node $(u+1-m)
   \bmod  n$   (recall  that  these   edges  are  not  used   in  {\sc
   hypercubic-routing}                   described                  in
   Theorem~\ref{thm:fast_absolute}). We will ensure  $\pi = 1$ for the
   remainder of the edges.

   {\sc  congestion-free} routing follows  the same  idea as  that for
   ${\mathcal                                           B}_{clockwise}$
   (Theorem~\ref{thm:congestion_free_clockwise}):   Let  $s$   be  the
   source  node, $t$  the  target.  Let  $c  = (t+m-s)  \bmod m$,  the
   difference in levels between  $\ell(s)$ and $\ell(t)$.  In Phase I,
   for $c$  steps, we follow  any out-going link, chosen  uniformly at
   random.  We thus reach a  node $s'$ such that $\ell(s') = \ell(t)$.
   In  Phase   II,  we   continue  as  per   Phase  II  of   the  {\sc
   hypercubic-routing}   algorithm   for   ${\mathcal   B}_{absolute}$
   (Theorem~\ref{thm:fast_absolute}).

   \medskip

   An alternate {\sc congestion-free} routing algorithm for ${\mathcal
   B}_{absolute}$  that  routes deterministically  is  based upon  the
   following idea: We express any integer  $a \in [-k, +k]$ as the sum
   of   two   integers:   $a'    =   \Floor{(k+a)/2}$   and   $a''   =
   -\Floor{(k-a)/2}$. It is  easy to verify that $a =  a' + a''$.  Now
   if we list all pairs $\langle a', a''\rangle$ for $a \in [-k, +k]$,
   then each integer in the  range $[-k, +k]$ appears exactly twice as
   a member of some pair.

   Let $s$ be the source node, $t$ the target.  Let $c = (t+m-s) \bmod
   m$, the difference in  levels between $\ell(s)$ and $\ell(t)$.  The
   remaining distance  is $dist = c +  m+ m\sum_{i=0}^{i=m-1} (2k+1)^i
   d_i$ with $-k \leq d_i \leq  k$ (there is a unique way to represent
   $dist$ in this fashion).

   Phase I: For $c$ steps, if  the current node is $u$, then we follow
   the  edge  corresponding to  $d_{\ell(u)}'$,  i.e.,  the edge  that
   covers  distance   $1  +  md_{\ell(u)}'(2k+1)^{\ell(u)}$   (in  the
   clockwise or the anti-clockwise  direction, depending upon the sign
   of $d_{\ell(u)}'$). At the end of  this phase, we reach a node $s'$
   such that $\ell(s') = \ell(t)$.

   Phase II:  Continue with Phase  II of the  {\sc hypercubic-routing}
   algorithm           for          ${\mathcal          B}_{absolute}$
   (Theorem~\ref{thm:fast_absolute}), for exactly $m$ steps.

   Due to the  decomposition of integers in $[-k,  +k]$ into pairs, as
   defined  above, all  outgoing  links (short-  and  long-) are  used
   equally.  Hence, $\pi = 1$.
 \end{proof}

 \bigskip

 {\bf Notes}: In the context  of the current Internet, out-going links
 correspond to full-duplex TCP connections.  Therefore, the undirected
 graph corresponding to ${\mathcal  B}_{absolute}$ is of interest.  In
 this  undirected  graph, it  is  possible  to devise  congestion-free
 routing with  $\pi =  1$, maximum path  length $m +  \Floor{m/2}$ and
 average route-length at most $1.25m$.   This is achieved by making at
 most $\Floor{m/2}$ initial random steps  either in the down or the up
 direction, whichever gets to a node with level $\ell(t)$ faster.

 \section{Papillon\xspace with Distance Function $\delta_{xor}$}
 \label{sec:xor}

 In this Section, we define a variant of Papillon\xspace in which {\sc greedy}\xspace
 routing with distance function $\delta_{xor}$ results in worst-case
 route length $\Theta(\log n / \log d)$, with $n$ nodes, each having
 $d$ out-going links. For integers $s$ and $t$, $\delta_{xor}(s, t)$
 is defined as the number of bit-positions in which the binary
 representations of $s$ and $t$ differ.

 \begin{mydefinition}{Papillon\xspace for $\delta_{xor}$}
   ${\mathcal B}_{xor}(\lambda, m)$  is a directed  graph, defined
   for any pair of integers $\lambda, m \geq 1$ where $\lambda$ is a
   power of two.
   \begin{enumerate}

   \item The  network has $n = m\lambda^m$ nodes labeled  from $0$ to
      $n-1$.

   \item Let $u$ denote a node. Let $\ell(u)$ denote the unique
     integer $x \in [0, m-1]$ that satisfies $x\lambda^m \leq u <
     (x+1)\lambda^m$. The node $u$ makes links with nodes with labels
     \[
      ((\ell(u) + 1) \bmod m)\lambda^m + i\lambda^{\ell(u)},
      \quad \mathrm{where}\ \
      0 \leq i < \lambda.
     \]
   Thus, if $(u, v)$ is an edge, then $\ell(v) =
   (\ell(u) + 1) \bmod m$.
   \end{enumerate}
 \end{mydefinition}

 \begin{theorem} \label{thm:xor}
   {\sc greedy}\xspace routing  in ${\mathcal B}_{xor}$ with distance function
   $\delta_{xor}$ takes  $2m-1$ hops in
   the worst-case.  The average is at most $1.5m$ hops.
 \end{theorem}
 \begin{proof}
   Let the current node be $s$. Let $t$ denote the target node.
   Then $s \oplus t$, the bit-wise exclusive-OR of $s$ and $t$, can
   uniquely be expressed as $c + \sum_{i=0}^{i=m-1}
   \lambda^i d_i$, where $c \geq 0$ and $0 \leq d_i < \lambda$.
   Routing proceeds in two phases. In Phase I, each of the $d_i$ is
   set to zero. This takes at most $m$ steps (at most $m$ on
   average). In Phase II, the most significant
   $\Ceiling{\log_2 m}$ bits of $s \oplus t$ are set to zero,
   thereby
   reaching the target.   This phase takes at most $m-1$ hops (at most
   $\frac{m-1}{2}$ on average).
 \end{proof}

 \section{Summary}
 \label{summary}

 We presented Papillon\xspace, a  variant of  multi-butterfly networks  which supports
 asymptotically optimal {\sc greedy}\xspace routes of  length $O(\log n / \log d)$
 with distance functions $\delta_{clockwise}$, $\delta_{absolute}$ and
 $\delta_{xor}$,
 when each  node makes  $d$ out-going links,  in an  $n$-node network.
 Papillon\xspace  is the  first construction  with this  property.

 \medskip

 Some questions that remain unanswered:
 \begin{enumerate}

 \item {\it  Is it possible to  devise graphs in  which {\sc greedy}\xspace routes
    with      distance      function     $\delta_{clockwise}$      and
    $\delta_{absolute}$  are  along  shortest-paths?  }   As  Theorems
    ~\ref{thm:fast_clockwise} and ~\ref{thm:fast_absolute} illustrate,
    {\sc greedy}\xspace routing  on Papillon\xspace  do not route  along shortest-paths.
    Is this property inherent in {\sc greedy}\xspace routes?

 \item {\it What is the  upper-bound for the Problem of Greedy Routing
    on  the Circle?  }   Papillon\xspace furnishes  a lower-bound,  which is
    asymptotically     optimal.      However,     constructing     the
    largest-possible graph  with degree $d$ and  diameter $\Delta$, is
    still an interesting combinatorial problem.

 \end{enumerate}

\newcommand{\etalchar}[1]{$^{#1}$}