# COMPUTING THE PLANAR $\beta$-SKELETON DEPTH ††thanks: Partially supported by
NSERC Canada

David Bremner and Rasoul Shahsavarifar22footnotemark: 2 ,
This paper is submitted to Journal of Computational Geometry. Faculty of
Computer Science, University of New Brunswick,
Fredericton, NB, Canada<EMAIL_ADDRESS>
###### Abstract

For $\beta\geq 1$, the _$\beta$ -skeleton depth_
($\operatorname{\textit{SkD}}_{\beta}$) of a query point $q\in\mathbb{R}^{d}$
with respect to a distribution function $F$ on $\mathbb{R}^{d}$ is defined as
the probability that $q$ is contained within the _$\beta$ -skeleton influence
region_ of a random pair of points from $F$. The $\beta$-skeleton depth of
$q\in\mathbb{R}^{d}$ can also be defined with respect to a given data set
$S\subseteq\mathbb{R}^{d}$. In this case, computing the $\beta$-skeleton depth
is based on counting all of the $\beta$-skeleton influence regions, obtained
from pairs of points in $S$, that contain $q$. The $\beta$-skeleton depth
introduces a family of depth functions that contains _spherical depth_ and
_lens depth_ for $\beta=1$ and $\beta=2$, respectively. The straightforward
algorithm for computing the $\beta$-skeleton depth in dimension $d$ takes
$O(dn^{2})$. This complexity of computation is a significant advantage of
using the $\beta$-skeleton depth in multivariate data analysis because unlike
most other data depths, the time complexity of the $\beta$-skeleton depth
grows linearly rather than exponentially in the dimension $d$. The main
results of this paper include two algorithms. The first one is an optimal
algorithm that takes $\Theta(n\log n)$ for computing the planar spherical
depth, and the second algorithm with the time complexity of
$O(n^{\frac{3}{2}+\epsilon})$ is for computing the planar $\beta$-skeleton
depth, $\beta>1$. By reducing the problem of Element Uniqueness, we prove that
computing the $\beta$-skeleton depth requires $\Omega(n\log n)$ time. Some
geometric properties of $\beta$-skeleton depth are also investigated in this
paper. These properties indicate that _simplicial depth_
($\operatorname{\textit{SD}}$) is linearly bounded by $\beta$-skeleton depth
(in particular, $\operatorname{\textit{SkD}}_{\beta}\geq\frac{2}{3}SD$;
$\beta\geq 1$). To illustrate this relationship, the results of some
experiments on random point sets are provided. In these experiments, the
bounds of $\operatorname{\textit{SphD}}\geq 2\operatorname{\textit{SD}}$ and
$\operatorname{\textit{LD}}\geq 1.2\operatorname{\textit{SphD}}$ are achieved.

##  1 Introduction

The rank statistic tests play an important role in univariate non-parametric
statistics. If one attempts to generalize the rank tests to the multivariate
case, the problem of defining a multivariate order statistic will occur. It is
not clear how to define a multivariate order or rank statistic in a meaningful
way. One approach to overcome this problem is to use the notion of data depth.
Data depth measures the centrality of a point in a given data set in non-
parametric multivariate data analysis. In other words, it indicates how deep a
point is located with respect to the data set.

Over the last decades, various notions of data depth such as _halfspace depth_
[11, 19, 21], _simplicial depth_ [14], _Oja depth_ [17], _regression depth_
[18], and others have emerged as powerful tools for non-parametric
multivariate data analysis. Most of them have been defined to solve specific
problems in data analysis. They are different in application, definition, and
geometry of their central regions (regions with the maximum depth). Regarding
the planar data depth functions, some research on the algorithmic aspects of
them can be found in [2, 3, 5, 6, 7, 8, 13, 16, 18].

In 2006, Elmore, Hettmansperger, and Xuan [9] defined another notion of data
depth named _spherical depth_. It is defined as the probability that point $q$
is contained in a closed random hyperball with the diameter
$\overline{x_{i}x_{j}}$, where $x_{i}$ and $x_{j}$ are two random points from
a common distribution function $F$. These closed hyperballs are known as
influence regions of the spherical depth function. In 2011, Liu and Modarres
[15], modified the definition of influence region, and defined _lens depth_.
Each lens depth influence region is defined as the intersection of two
hyperballs $B(x_{i},\|x_{i},x_{j}\|)$ and $B(x_{j},\|x_{i},x_{j}\|)$. These
influence regions of spherical depth and lens depth are the multidimensional
generalization of _Gabriel circles_ and lunes in the definition of the
_Gabriel Graph_ [10] and _Relative Neighbourhood Graph_ [20], respectively. In
2017, Yang [23], generalized the definition of influence region, and
introduced a familly of depth functions called $\beta$-skeleton depth, indexed
by a single parameter $\beta\geq 1$. The influence region of $\beta$-skeleton
depth is defined to be the intersection of two hyperballs given by
$B(c_{i},\frac{\beta}{2}\|x_{i},x_{j}\|)$ and
$B(c_{j},\frac{\beta}{2}\|x_{i},x_{j}\|)$, where $c_{i}$ and $c_{j}$ are some
combinations of $x_{i}$ and $x_{j}$. Spherical depth and lens depth can be
obtained from $\beta$-skeleton depth by considering $\beta=1$ and $\beta=2$,
respectively. The $\beta$-skeleton depth has some nice properties including
symmetry about the center, maximality at the centre, vanishing at infinity,
and monotonicity. Depending on whether Euclidean distance or Mahalanobis
distance is used to construct the influence regions, the $\beta$-skeleton
depth can be orthogonally invariant or affinely invariant. All of these
properties are explored in [9, 15, 22, 23].

Although we focus on the planar case here, a notable characteristic of the
$\beta$-skeleton depth is that its time complexity grows linearly in the
dimension $d$ while for most other data depths the time complexity grows
exponentially. To the best of our knowledge, the current best algorithm for
computing the $\beta$-skeleton depth is the straightforward algorithm which
takes $O(dn^{2})$.

In this paper, we present an optimal algorithm for computing the spherical
depth ($\beta=1$) in $\mathbb{R}^{2}$ that takes $\Theta(n\log n)$ time. We
also introduce an $O(n^{\frac{3}{2}+\epsilon})$ algorithm for computing the
planar $\beta$-skeleton depth, $\beta>1$. Furthermore, we reduce the problem
of Element Uniqueness to prove that computing the $\beta$-skeleton depth
($\beta\geq 1$) of a query point requires $\Omega(n\log n)$ time. We also
investigate some geometric properties of $\beta$-skeleton depth. These
properties lead us to bound the simplicial depth, spherical depth, and lens
depth of a point in terms of one another. Finally, some experiments are
provided to illustrate the relationships between $\beta$-skeleton depth and
simplicial depth.

##  2 $\beta$-skeleton Depth

######  Definition 1.

The $\beta$-skeleton influence region of $x_{i}$ and $x_{j}$
($S_{\beta}(x_{i},x_{j})$) for $0\leq\beta<\infty$ is defined as follows:

  * •

for $\beta=0$, $S_{\beta}(x_{i},x_{j})$ is equivalent to the line segment
$\overline{x_{i}x_{j}}$.

  * •

for $0<\beta<1$, $S_{\beta}(x_{i},x_{j})$ is the intersection of two balls
with the radius ${\|x_{i}-x_{j}\|}/{2\beta}$ such that the boundaries contain
both points $x_{i}$ and $x_{j}$.

  * •

for $1\leq\beta<\infty$, the lune based version of $S_{\beta}(x_{i},x_{j})$ is
defined as:

$S_{\beta}(x_{i},x_{j})=B(c_{i},r)\cap B(c_{j},r),$ (1)

where $r=\frac{\beta}{2}\|x_{i}-x_{j}\|$,
$c_{i}=\frac{\beta}{2}x_{i}+(1-\frac{\beta}{2})x_{j}$, and
$c_{j}=(1-\frac{\beta}{2})x_{i}+\frac{\beta}{2}x_{j}$.

Figure 1 illustrates the $\beta$-skeleton influence regions for different
values of $\beta$.

######  Note 1.

It seems that for $0<\beta<1$, the $\beta$-skeleton influence region in
dimension $d>2$ is not well defined because the two balls in Definition 1 are
not unique. Since the $\beta$-skeleton depth is defined for
$1\leq\beta<\infty$ [23], we ignore the $\beta<1$ in our study of the
$\beta$-skeleton depth and its influence region.

######  Note 2.

In literature, the ball based version of $S_{\beta}(x_{i},x_{j})$,
$1\leq\beta<\infty$ is also defined. In this case, the
$S_{\beta}(x_{i},x_{j})$ is given by the union of the balls, instead of the
intersection of them in Equation (1). For example, the hatched area in Figure
1 denotes the ball based version of the $S_{2}(x_{i},x_{j})$. Since the
definition of the $\beta$-skeleton depth is given based on the lune based
$S_{\beta}(x_{i},x_{j})$ alone, by $S_{\beta}(x_{i},x_{j})$ we only mean its
lune based version hereafter in this paper.

Figure 1: The $\beta$-skeleton influence regions defined by $x_{i}$ and
$x_{j}$ for $\beta$=0, 0.5, 1, 2, 3, and $\beta\rightarrow\infty$, where
$c=\frac{x_{i}+x_{j}}{2}$, $c_{i}=\frac{3}{2}x_{i}+(1-\frac{3}{2})x_{j}$, and
$c_{j}=(1-\frac{3}{2})x_{i}+\frac{3}{2}x_{j}$

######  Definition 2.

For integers $i$ and $j$ ($i<j$), we use the shorter notation
$\\{x_{i}..x_{j}\\}$ to represent the set $\\{x_{i},x_{i+1},...,x_{j}\\}$.

######  Definition 3.

For a given data set $S=\\{x_{1}..x_{n}\\}$ of points in general positions in
$\mathbb{R}^{d}$ and the parameter $\beta$, the $\beta$-skeleton is defined as
a graph $(S,E)$, such that $x_{i}x_{j}\in E$ if and only if no point in
$S\backslash\\{x_{i},x_{j}\\}$ belongs to $S_{\beta}(x_{i},x_{j})$, $1\leq
i,j\leq n$.

######  Definition 4.

The $\beta$-skeleton depth is defined as the probability that a point is
contained within the $\beta$-skeleton influence region of two random vectors
from a common distribution. For a distribution function $F$ on
$\mathbb{R}^{d}$, and a vector $q$ in $\mathbb{R}^{d}$, the $\beta$-skeleton
depth of $q$ with respect to $F$ is defined by equation (2), where $x_{i}$ and
$x_{j}$ are two random observations from $F$.

$\operatorname{\textit{SkD}}_{\beta}(q;F)=P[q\in
S_{\beta}(x_{i},x_{j})];\;\beta\geq 1$ (2)

######  Definition 5.

Let $S=\\{x_{1}..x_{n}\\}$ be a set of points in $\mathbb{R}^{d}$. The
$\beta$-skeleton depth of a point $q\in\mathbb{R}^{d}$ with respect to $S$, is
defined as a proportion of the $\beta$-skeleton influence regions of
$S_{\beta}(x_{i},x_{j}),1\leq i<j\leq n$ that contain $q$. Using the indicator
function $I$, this definition can be represented by Equation (3).

$\operatorname{\textit{SkD}}_{\beta}(q;S)=\frac{1}{{n\choose 2}}\sum_{1\leq
i<j\leq n}{I(q\in S_{\beta}(x_{i},x_{j})})$ (3)

Referring to Equation (1), it can be verified that $\\{q\in
S_{\beta}(x_{i},x_{j})\\}$ is equivalent to the inequality of
${\beta\|x_{i}-x_{j}\|}/{2}\geq\max\\{\|q-c_{i}\|,\|q-c_{j}\|\\}$, where
$\|c_{i}-c_{j}\|=\|(1-\beta)(x_{i}-x_{j})\|=(\beta-1)\|x_{i}-x_{j}\|$ for
$\beta\geq 1$. To compute $\operatorname{\textit{SkD}}_{\beta}(q;S)$ in a
straightforward way, it is sufficient to check this inequality for all $1\leq
i,j\leq n$. As such, the computational complexity of the $\beta$-skeleton
depth in $\mathbb{R}^{d}$ is $O(dn^{2})$.

In [15, 22, 23], it is proved that the $\beta$-skeleton depth functions
satisfy the data depth framework provided by Zuo and Serfling [24] because
these depth functions are monotonic, maximized at the center, and vanishing at
infinity. The $\beta$-skeleton depth functions are also orthogonally
(affinely) invariant if the Euclidean (Mahalanobis) distance is used to
construct the influence regions of $\beta$-skeleton depth influence regions.

###  2.1 Spherical Depth and Lens Depth

As we discussed in Section 2, the $\beta$-skeleton depth is a family of
statistical depth functions that includes the spherical depth when $\beta=1$,
and the lens depth when $\beta=2$. From the equations (1) and (3), the
definitions of spherical depth ($\operatorname{\textit{SphD}}$) and lens depth
($\operatorname{\textit{LD}}$) of a query point $q$ with respect to a given
data set $S$ in $\mathbb{R}^{d}$ are as follows:

$\operatorname{\textit{SphD}}(q;S)=\frac{1}{{n\choose 2}}\sum_{1\leq i<j\leq
n}{I(q\in\operatorname{\textit{Sph}}(x_{i},x_{j})})$ (4)

$\operatorname{\textit{LD}}(q;S)=\frac{1}{{n\choose 2}}\sum_{1\leq i<j\leq
n}{I(q\in L(x_{i},x_{j})}),$ (5)

where the influence regions $\operatorname{\textit{Sph}}(x_{i},x_{j})$ and
$L(x_{i},x_{j})$ are equal to $S_{1}(x_{i},x_{j})$ and $S_{2}(x_{i},x_{j})$,
respectively.
Figure 2 shows the spherical depth and lens depth of points in the plane with
respect to a set of three points $p_{1},p_{2}$, and $p_{3}$ in
$\mathbb{R}^{2}$.

Figure 2: Spherical depth (left figure) and lens depth (right figure) of
points in the plane

##  3 Algorithms

The current best algorithm for computing the $\beta$-skeleton depth of a point
$q\in\mathbb{R}^{d}$ with respect to a data set
$S=\\{x_{1}..x_{n}\\}\subseteq\mathbb{R}^{d}$ is the brute force algorithm.
This naive algorithm needs to check all of the $n\choose 2$ $\beta$-skeleton
influence regions obtained from the data points to figure out how many of them
contain $q$. Checking all of such influence regions causes the naive algorithm
to take $\Theta(dn^{2})$. In this section, we present an optimal algorithm for
computing the planar spherical depth ($\beta=1$) and an algorithm to compute
the planar $\beta$-skeleton depth when $\beta>1$. In these algorithms, we need
to solve some halfspace and some circle range counting problems, where all of
the halfspaces have one common point. The circles also have the same
characteristic. In the spherical depth algorithm, we have the halfspace range
counting problems alone whereas in computing the $\beta$-skeleton depth,
$\beta>1$ we need to solve both circle and halfspace range counting problems.

###  3.1 Optimal Algorithm for Computing the Planar Spherical Depth of a Query
Point

Instead of checking all of the spherical influence regions, we focus on the
geometric aspects of such regions in $\mathbb{R}^{2}$. The geometric
properties of these regions lead us to develop an $\Theta(n\log n)$ algorithm
for the computation of planar spherical depth of $q\in\mathbb{R}^{2}$.

######  Lemma 1.

For arbitrary points $a$, $b$, and $t$ in $\mathbb{R}^{2}$,
$t\in\operatorname{\textit{Sph}}(a,b)$ if and only if $\angle
atb\geq\frac{\pi}{2}$.

###### Proof.

If $t$ is on the boundary of $\operatorname{\textit{Sph}}(a,b)$ _Thales’
Theorem_ 111Thales’ Theorem also known as the _Inscribed Angle Theorem_ : If
$a$, $b$, and $c$ are points on a circle where $\overline{ac}$ is a diameter
of the circle, then $\angle abc$ is a right angle. suffices as the proof in
both directions. For the rest of the proof, by
$t\in\operatorname{\textit{Sph}}(a,b)$ we mean $t\in
int\operatorname{\textit{Sph}}(a,b)$.
$\Rightarrow$) For $t\in\operatorname{\textit{Sph}}(a,b)$, suppose that
$\angle atb<\frac{\pi}{2}$ (proof by contradiction). We continue the line
segment $\overline{at}$ to cross the boundary of the
$\operatorname{\textit{Sph}}(a,b)$. Let $t^{\prime}$ be the crossing point
(see the left figure in Figure 3). Since $\angle atb<\frac{\pi}{2}$, then,
$\angle btt^{\prime}$ is greater than $\frac{\pi}{2}$. Let $\angle
btt^{\prime}=\frac{\pi}{2}+\epsilon_{1};\epsilon_{1}>0$. From the Thales’
Theorem, we know that $\angle at^{\prime}b$ is a right angle. The angle
$tbt^{\prime}=\epsilon_{2}>0$ because $t\in\operatorname{\textit{Sph}}(a,b)$.
Summing up the angles in $\bigtriangleup tt^{\prime}b$, as computed in (6),
leads to a contradiction. So, this direction of proof is complete.

$\angle tt^{\prime}b+\angle t^{\prime}bt+\angle
btt^{\prime}\geq\frac{\pi}{2}+\epsilon_{2}+\frac{\pi}{2}+\epsilon_{1}>\pi$ (6)

$\Leftarrow$) If $\angle atb=\frac{\pi}{2}+\epsilon_{1};\epsilon_{1}>0$, we
prove that $t\in\operatorname{\textit{Sph}}(a,b)$. Suppose that
$t\notin\operatorname{\textit{Sph}}(a,b)$ (proof by contradiction). Since
$t\notin\operatorname{\textit{Sph}}(a,b)$, at least one of the line segments
$\overline{at}$ and $\overline{bt}$ crosses the boundary of
$\operatorname{\textit{Sph}}(a,b)$. Without loss of generality, assume that
$\overline{at}$ is the one that crosses the boundary of
$\operatorname{\textit{Sph}}(a,b)$ at the point $t^{\prime}$ (see the right
figure in Figure 3). Considering Thales’ Theorem, we know that $\angle
at^{\prime}b=\frac{\pi}{2}$ and consequently, $\angle
bt^{\prime}t=\frac{\pi}{2}$. The angle $\angle t^{\prime}bt=\epsilon_{2}>0$
because $t\notin\operatorname{\textit{Sph}}(a,b)$. If we sum up the angles in
the triangle $\bigtriangleup tt^{\prime}b$, the same contradiction as in (6)
will be implied. ∎

Figure 3: Point $t$ and spherical influence region $Sph(a,b)$

#### Algorithm 1:

Using Lemma 1, we present an algorithm to compute the spherical depth of a
query point $q\in\mathbb{R}^{2}$ with respect to
$S=\\{x_{1}..x_{n}\\}\subseteq\mathbb{R}^{2}$. This algorithm is summarized in
the following steps. The pseudocode of this algorithm is provided in the
Appendix.

  * •

Translating the points: Suppose that $T$ is a translation by $(-q)$. We apply
$T$ to translate $q$ and all data points into their new coordinates.
Obviously, $T(q)=O$.

  * •

Sorting the translated data points: In this step we sort the translated data
points based on their angles in their polar coordinates. After doing this
step, we have $S_{T}$ which is a sorted array of the translated data points.

  * •

Calculating the spherical depth: For the $i^{th}$ element in $S_{T}$, we
define $O_{i}$ and $N_{i}$ as follows:

$\begin{split}&O_{i}=\left\\{j:x_{j}\in
S_{T}\>,\frac{\pi}{2}\leq|\theta(x_{i})-\theta(x_{j})|\leq\frac{3\pi}{2}\right\\}\\\
&N_{i}=\\{1,2,...,n\\}\setminus O_{i}.\end{split}$ (7)

Thus the spherical depth of $q$ with respect to $S$, can be computed by:

$\operatorname{\textit{SphD}}(q;S)=\operatorname{\textit{SphD}}(O;S_{T})=\frac{1}{2}\sum_{1\leq
i\leq n}|O_{i}|.$ (8)

To present a formula for computing $|O_{i}|$, we define $f(i)$ and $l(i)$ as
follows:

$f(i)=\begin{cases}\min N_{i}-1&\text{if
$\frac{\pi}{2}<\theta(x_{i})\leq\frac{3\pi}{2}$}\\\ \min
O_{i}&\text{otherwise}\end{cases}$ $l(i)=\begin{cases}\max N_{i}+1&\text{if
$\frac{\pi}{2}<\theta(x_{i})\leq\frac{3\pi}{2}$}\\\ \max
O_{i}&\text{otherwise.}\end{cases}$

Figure 4 illustrates $O_{i}$, $N_{i}$, $f(i)$, and $l(i)$ in two different
cases. Considering the definitions of $f(i)$ and $l(i)$,

$|O_{i}|=\begin{cases}f(i)+(n-l(i)+1)&\text{if
$\frac{\pi}{2}<\theta(x_{i})\leq\frac{3\pi}{2}$}\\\
l(i)-f(i)+1&\text{otherwise.}\end{cases}$

This allows us to compute $|O_{i}|$ using a pair of binary searches.

Figure 4: $\theta(x_{i})\in(\frac{\pi}{2},\frac{3\pi}{2}]$ (left figure), and
$\theta(x_{i})\notin(\frac{\pi}{2},\frac{3\pi}{2}]$ (right figure)

#### Time complexity of Algorithm 1:

The first procedure in the algorithm takes $O(n)$ to translate $q$ and all
data points into the new coordinate system. The second procedure takes
$O(n\log n)$ time. In this procedure, the loop iterates $n$ times, and the
sorting algorithm takes $O(n\log n)$. Due to using binary search for every
$O_{i}$, the running time of the last procedure is also $O(n\log n)$. The rest
of the algorithm contributes some constant time. In total, the running time of
the algorithm is $O(n\log n)$.

#### Coordinate system:

In practice it may be preferable to work in the Cartesian coordinate system.
Sorting by angle can be done using some appropriate right-angle tests
(determinants). Regarding the other angle comparisons, they can be done by
checking the sign of dot products.

###  3.2 Algorithms for Computing the Planar $\beta$-skeleton Depth ($\beta>1$
) of a Query Point

As illustrated in Figure 1, $S_{\beta}(x_{i},x_{j});\beta>1$ forms some
lenses, and $S_{\infty}(x_{i},x_{j})$ forms some slabs for different $x_{i}$
and $x_{j}$ in $S\subseteq\mathbb{R}^{2}$. Using some geometric properties of
such lenses and slabs, we prove Lemma 2. This lemma along with some results in
range counting problems studied by Agrawal in [1] help us to compute
$\operatorname{\textit{SkD}}_{\beta}(q;S);\beta>1$ in
$O(n^{\frac{3}{2}+\epsilon})$ time, where $q$ and $S$ are in $\mathbb{R}^{2}$.

######  Definition 6.

For an arbitrary non-zero point $a\in\mathbb{R}^{2}$ and parameter $\beta\geq
1$, $\ell(p)$ is a line that is perpendicular to $\overrightarrow{a}$ at the
point $p=p(a,\beta)={(\beta-1)a}/{\beta}$. This line forms two halfspaces
$H_{o}(p)$ and $H_{a}(p)$. The one that includes the origin is $H_{o}(p)$ and
the other one that includes $a$ is $H_{a}(p)$.

######  Definition 7.

For a disk $B(c,r)$ with the center $c=c(a,\beta)={\beta a}/{2(\beta-1)}$ and
radius $r=\|c\|$, $B_{o}(c,r)$ is the intersection of $H_{o}(p)$ and $B(c,r)$,
and $B_{a}(c,r)$ is the intersection of $H_{a}(p)$ and $B(c,r)$, where
$\beta>1$ and $a$ is an arbitrary non-zero point in $\mathbb{R}^{2}$.

Figure 5 is an illustration of these definitions for different values of
parameter $\beta$.

Figure 5: The $H_{o}(p)$ and $B(c,r)$ defined by $a\in\mathbb{R}^{2}$ for
$\beta=1,\;1.5,\;2,\;\text{and}\;\beta\rightarrow\infty$, where
$A=H_{o}(p)\setminus\\{intB_{o}(c,r)\\}$

######  Lemma 2.

For arbitrary non-zero points $a$, $b$ in $\mathbb{R}^{2}$ and parameter
$\beta>1$, $b\in H_{o}(p)\setminus\\{intB_{o}(c,r)\\}$ if and only if the
origin $O=(0,0)$ is contained in $S_{\beta}(a,b)$, where $c={\beta
a}/{2(\beta-1)}$, $r=\|c\|$, and $p={(\beta-1)a}/{\beta}$.

###### Proof.

First, we show that $B_{o}(c,r)$ is a well-defined set meaning that $\ell(p)$
intersects $B(c,r)$. We compute $d(c,\ell(p))$, the distance of $c$ from
$\ell(p)$, and prove that this value is not greater than $r$. It can be
verified that $d(c,\ell(p))=d(c,p)$. Let $k={\beta}/{2(\beta-1)}$; the
following calculations complete this part of the proof.

$\displaystyle d(c,p)$ $\displaystyle=d(\frac{\beta
a}{2(\beta-1)},\frac{(\beta-1)a}{\beta})=d(ka,\frac{1}{2k}a)=(k-\frac{1}{2k})\sqrt{({a_{x}}^{2}+{a_{y}}^{2})}=(\frac{2k^{2}-1}{2k})\|a\|$
$\displaystyle\leq\frac{2k^{2}}{2k}\|a\|=k\|a\|=r.$

We recall Definition 1 for $\beta>1$,
$S_{\beta}(a,b)=B(c_{a},\frac{\beta}{2}\|a-b\|)\cap
B(c_{b},\frac{\beta}{2}\|a-b\|)$, where
$c_{a}=\frac{\beta}{2}a+(1-\frac{\beta}{2})b$ and
$c_{b}=\frac{\beta}{2}b+(1-\frac{\beta}{2})a$. Using this definition,
following equivalencies can be derived from $O\in S_{\beta}(a,b)$.

$O\in S_{\beta}(a,b)\Leftrightarrow\frac{\beta\|a-b\|}{2}\geq
max\\{\|c_{a}\|,\|c_{b}\|\\}\Leftrightarrow\beta\|a-b\|\geq
max\\{\|\beta(a-b)+2b\|,\|\beta(b-a)+2a\|\\}\Leftrightarrow$
$\beta^{2}\|a-b\|^{2}\geq
max\\{\|\beta(a-b)+2b\|^{2},\|\beta(b-a)+2a\|^{2}\\}\Leftrightarrow 0\geq
max\\{b^{2}(1-\beta)+\beta\overrightarrow{a}.\overrightarrow{b},a^{2}(1-\beta)+\beta\overrightarrow{a}.\overrightarrow{b}\\}.$

By solving these inequalities for $(\beta-1)/\beta$ which is equal to $1/2k$,
we have:

$\frac{1}{2k}\geq
max\left\\{\frac{\overrightarrow{a}.\overrightarrow{b}}{\|a\|^{2}},\frac{\overrightarrow{a}.\overrightarrow{b}}{\|b\|^{2}}\right\\}.$
(9)

For a fixed point $a$, the inequalities in Equation (9) determine one
halfspace and one disk given by (10) and (11), respectively.

$\frac{1}{2k}\geq\frac{\overrightarrow{a}.\overrightarrow{b}}{\|a\|^{2}}\Leftrightarrow\overrightarrow{a}.\overrightarrow{b}\leq\frac{1}{2k}\|a\|^{2}$
(10)
$\frac{1}{2k}\geq\frac{\overrightarrow{a}.\overrightarrow{b}}{\|b\|^{2}}\Leftrightarrow
b^{2}-2k\overrightarrow{a}.\overrightarrow{b}\geq 0\Leftrightarrow
b^{2}-2k\overrightarrow{a}.\overrightarrow{b}+k^{2}a^{2}\geq
k^{2}a^{2}\Leftrightarrow\left(b-ka\right)^{2}\geq\left(k\|a\|\right)^{2}$
(11)

The proof is complete because for a point $a$, the set of all points $b$
containing in the feasible region defined by Equations (10) and (11) is equal
to $H_{o}(p)\setminus\\{intB_{o}(c,r)\\}$. ∎

#### Algorithm 2:

Using Lemma 2, we present an algorithm to compute the $\beta$-skeleton depth
of $q\in\mathbb{R}^{2}$ with respect to
$S=\\{x_{1}..x_{n}\\}\subseteq\mathbb{R}^{2}$. This algorithm is summarized in
two steps. Pseudocode for this algorithm can be find in the Appendix.

  * •

Translating the points: This step is exactly the same step as in Algorithm 1.

  * •

Calculating the $\beta$-skeleton depth: Suppose that $a=(a_{x},a_{y})$ is an
element in $S^{\prime}$ (translated $S$). We consider a disk and a line as
follows:

$B(c,r):\left(x-ka_{x}\right)^{2}+\left(y-ka_{y}\right)^{2}=\left(k\|a\|\right)^{2}$
$\ell(p):a_{x}x+a_{y}y=\frac{1}{2k}\|a\|,$

where $k$, $c$, $r$, and $p$ are defined in Lemma 2. From Theorem 1.2 proved
in [1], we can compute $|H_{o}(p)|$ with $O(n)$ storage, $O(n\log n)$ expected
preprocessing time, and $O(n^{\frac{1}{2}+\epsilon})$ query time, where
$|H_{o}(p)|$ is the number of all elements of $S^{\prime}$ that are contained
in $H_{o}(p)$. For the elements of $S^{\prime}$, $|intB_{o}(c,r)|$ which is
defined as the number of elements containing in the interior of $B_{o}(c,r)$
can also be computed with the same storage, expected preprocessing time, and
query time. We recall that $B_{o}(c,r)$ is the intersection of halfspace
$H_{o}(p)$ and disk $B(c,r)$, where $p$, $c$, and $r$ are some functions of
$a$. Finally, $\operatorname{\textit{SkD}}_{\beta}(q,S)$ which is equal to
$\operatorname{\textit{SkD}}_{\beta}(O;S^{\prime})$ can be computed by
Equation (12).

$\operatorname{\textit{SkD}}_{\beta}(O;S^{\prime})=\frac{1}{2}\sum_{a\in
S^{\prime}}(|H_{o}(p)|-|intB_{o}(c,r)|)$ (12)

Note that $H_{o}(p)$ and $B_{o}(c,r)$, referring to Definitions 6 and 7, can
be obtained from $\ell(p)$ and $B(c,r)$, respectively in constant time.

Lemma 2 and Algorithm 2 are valid for $\beta>1$. However, the case $\beta=1$
(Algorithm 1 for spherical depth) can also be included if we replace
$intB_{o}(c,r)$ with an empty set for $\beta=1$. In this case, we need to
solve the halfspace range counting problems alone and therefore,

$\operatorname{\textit{SphD}}(q;S)=\operatorname{\textit{SphD}}(O;S^{\prime})=\frac{1}{2}\sum_{a\in
S^{\prime}}|H_{o}(p)|.$

#### Time complexity of Algorithm 2:

The Translating procedure as is discussed in Algorithm 1, takes $O(n)$ time.
With the $O(n\log n)$ expected preprocessing time, the second procedure takes
$O(n^{\frac{3}{2}+\epsilon})$ time. In this procedure, the loop iterates $n$
times, and the range counting algorithms take $O(n^{\frac{1}{2}+\epsilon})$
time. The expected preprocessing time $O(n\log n)$ is required to obtain a
data structure for the aforementioned range counting algorithms. The rest of
the algorithm take some constant time per loop iteration, and therefore the
total expected running time of the algorithm is $O(n^{\frac{3}{2}+\epsilon})$.

##  4 Lower Bounds for Computing the $\beta$-skeleton Depth of a Point in the
Plane

We reduce the problem of Element Uniqueness222 Element Uniqueness problem:
Given a set $A=\\{a_{1}..a_{n}\\}$, is there a pair of indices $i,j$ with
$i\neq j$ such that $a_{i}=a_{j}$? to the problem of computing the
$\beta$-skeleton depth. It is known that the question of Element Uniqueness
has a lower bound of $\Omega(n\log n)$ in the algebraic decision tree model of
computation proposed in [4].

######  Theorem 1.

Computing the spherical depth of a query point in the plane takes
$\Omega(n\log n)$ time.

###### Proof.

We show that finding the spherical depth allows us to answer the question of
Element Uniqueness. Suppose that $A=\\{a_{1}..a_{n}\\}$, for $n\geq 2$ is a
given set of real numbers. We suppose all of the numbers to be positive
(negative), otherwise we shift the points onto the positive X-axis. For every
$a_{i}\in A$ we construct four points $x_{i}$, $x_{n+i}$, $x_{2n+i}$, and
$x_{3n+i}$ in the polar coordinate system as follows:

$x_{(kn+i)}=\left(r_{i},\theta_{i}+\frac{k\pi}{2}\right);\;0\leq k\leq 3,$

where $r_{i}=\sqrt{1+{a_{i}^{2}}}$ and $\theta_{i}=\tan^{-1}(1/a_{i})$. Thus
we have a set $S$ of $4n$ points $x_{kn+i}$, for $1\leq i\leq n$. See Figure
6. The Cartesian coordinates of the points can be computed by:

$x_{(kn+i)}=\left[{\begin{array}[]{cc}0&-1\\\ 1&0\\\
\end{array}}\right]^{k}\left({\begin{array}[]{cc}a_{i}\\\ 1\\\
\end{array}}\right);\;k=0,1,2,3.$

We select the query point $q=(0,0)$, and present an equivalent form of
Equation (7) for $O_{j}$ as follows:

$O_{j}=\left\\{x_{k}\in S\mid\angle
x_{j}qx_{k}\geq\frac{\pi}{2}\right\\},\>1\leq j\leq 4n,$ (13)

We compute $\operatorname{\textit{SphD}}(q;S)$ in order to answer the Element
Uniqueness problem. Suppose that every $x_{j}\in S$ is a unique element. In
this case, $|O_{j}|=2n+1$ because, from (13), it can be figured out that the
expanded $O_{j}$ is as follows:

$O_{j}=\begin{cases}\\{x_{n+1}..x_{n+j},x_{2n+1}..x_{3n},x_{3n+j}..x_{4n}\\};&j\in\\{1,...,n\\}\\\
\\{x_{2n+1}..x_{n+j},x_{3n+1}..x_{4n},x_{j-n}..x_{n}\\};&j\in\\{n+1,...,2n\\}\\\
\\{x_{3n+1}..x_{n+j},x_{1}..x_{n},x_{j-n}..x_{2n}\\};&j\in\\{2n+1,...,3n\\}\\\
\\{x_{1}..x_{j-3n},x_{n+1}..x_{2n},x_{j-n}..x_{3n}\\};&j\in\\{3n+1,...,4n\\}.\end{cases}$

Referring to Lemma 1 and Equation (8),

$\operatorname{\textit{SphD}}(q;S)=\frac{1}{2}\sum_{1\leq j\leq
4n}(2n+1)=4n^{2}+2n.$

Now suppose that there exist some $i\neq j$ such that $x_{i}=x_{j}$ in $S$. In
this case, from Equation (13), it can be seen that:

$|O_{(kn+i)\bmod 4n}|=|O_{(kn+j)\bmod 4n}|=2n+2,$

where $k=0,1,2,3$ (see Figure 6). As an example, for $k=0$,
$|O_{j}|=|O_{i}|=2n+2$ because the expanded form of these two sets is as
follows: (without loss of generality, assume $i<j<n$)

$O_{i}=O_{j}=\\{x_{n+1}..x_{n+j},x_{2n+1}..x_{3n},x_{3n+i},x_{3n+j},x_{3n+j+1}..x_{4n}\\}.$

Lemma 1 and Equation (8) imply that:

$\operatorname{\textit{SphD}}(q;S)\geq\frac{1}{2}(8+\sum_{1\leq j\leq
4n}(2n+1))=4n^{2}+2n+4.$

Therefore the elements of $A$ are unique if and only if the spherical depth of
$(0,0)$ with respect to $S$ is $4n^{2}+2n$. This implies that the computation
of spherical depth requires $\Omega(n\log n)$ time. It is necessary to mention
that the only computation in the reduction is the construction of $S$ which
takes $O(n)$ time. At the end of the proof, we mention that the reduction does
not depend on the sorted order of the elements. ∎

Figure 6: A representation of $A$, $S$, and duplications in these sets

######  Note 3.

Instead of four copies of the elements of $A$, we could consider two copies of
such elements to construct $S$. However, the depth calculation becomes more
complicated in this case.

######  Lemma 3.

Suppose that $S$ and $L_{j}$ are two sets defined as follows:

$S=\\{(b_{i},0),(b_{i},\pi/3)\mid b_{i}>0,1\leq i\leq
n\\}\;\text{and}\;L_{j}=\\{x_{k}\in S\mid q\in L(x_{j},x_{k})\\}\;,1\leq j\leq
2n.$

For a unique element $x_{j}$ in $S$, $L_{j}=\\{x_{(n+j)\bmod 2n}\\}$.

###### Proof.

Suppose that $L_{j}=\\{x_{k},x_{(n+j)\bmod 2n}\\}$ for some $x_{k}\in S$
($k\neq j$). We prove that such $x_{k}$ does not exist. If $\angle
x_{j}Ox_{k}=0$, it is obvious that $O\notin L(x_{j},x_{k})$ and $x_{k}$ cannot
be an element of $L_{j}$. For the case $\angle x_{j}Ox_{k}=\pi/3$, we assume
that $O\in L(x_{j},x_{k})$ which means that

$d(x_{j},x_{k})\geq max\\{d(O,x_{k}),d(O,x_{j})\\}=max\\{b_{k},b_{j}\\}.$ (14)

From the cosine formula333Cosine formula: For a triangle $\bigtriangleup abc$,
$|ab|^{2}=|ac|^{2}+|bc|^{2}-2|ac||bc|cos(\angle bca).$ in triangle
$\bigtriangleup x_{i}Ox_{j}$, we have

$d^{2}(x_{j},x_{k})=b^{2}_{j}+b^{2}_{k}-2b_{k}b_{j}cos(\pi/3).$ (15)

Equations (14) and (15) imply that

$d(x_{j},x_{k})\geq b_{k}\Rightarrow b^{2}_{j}-b_{j}b_{k}\geq 0\Rightarrow
b_{j}-b_{k}\geq 0\Rightarrow b_{j}\geq b_{k}$

and

$d(x_{j},x_{k})\geq b_{j}\Rightarrow b^{2}_{k}-b_{j}b_{k}\geq 0\Rightarrow
b_{k}-b_{j}\geq 0\Rightarrow b_{k}\geq b_{j}.$

This means that $b_{k}=b_{j}$ which contradicts the assumption of $x_{k}\neq
x_{j}.$ ∎

######  Theorem 2.

Computing the lens depth of a query point in the plane takes $\Omega(n\log n)$
time.

###### Proof.

Suppose that $B=\\{b_{1}..b_{n}\\}$, for $n\geq 2$ is a given set of real
numbers. Without loss of generality, we let these numbers to be positive (see
the proof of Theorem 1). For $1\leq i\leq n$, we construct set
$S=\\{x_{i},x_{n+i}\\}$ of $2n$ points in the polar coordinate system such
that $x_{i}=(b_{i},0)$ and $x_{n+i}=(b_{i},\pi/3)$. See Figure 7. We select
the query point $q=(0,0)$, and define $L_{j}$ as follows:

$L_{j}=\left\\{x_{k}\in S\mid q\in L(x_{j},x_{k})\right\\},\>1\leq j\leq 2n.$
(16)

Using Equation (16), the unnormalized form of Equation (5) can be presented
by:

$\operatorname{\textit{LD}}_{S}(q)={n\choose
2}\cdot\operatorname{\textit{LD}}(q;S)=\frac{1}{2}\sum_{1\leq j\leq
2n}|L_{j}|.$ (17)

We solve the problem of Element Uniqueness by computing
$\operatorname{\textit{LD}}_{S}(q)$. Suppose that every $x_{j}\in S$ is a
unique element. In this case, it can be verified that $L_{j}=\\{x_{(n+j)\bmod
2n}\\}$ (see Lemma 3). Equation (17) implies that:

$\operatorname{\textit{LD}}_{S}(q)=\frac{1}{2}\sum_{1\leq j\leq 2n}1=n.$

Now assume that there exists some $i\neq j$ such that $x_{i}=x_{j}$ in $S$. In
this case,

$L_{j}=L_{i}=\\{x_{(n+i)\bmod 2n},x_{(n+j)\bmod
2n}\\}\;\text{and}\;L_{n+i}=L_{n+j}=\\{x_{i},x_{j}\\}.$

As such,

$\operatorname{\textit{LD}}_{S}(q)=\frac{1}{2}\sum_{1\leq j\leq
2n}|L_{j}|=n+2.$

For the case of having more duplications among the elements of $S$,

$\operatorname{\textit{LD}}_{S}(q)=\frac{1}{2}\sum_{1\leq j\leq
2n}|L_{j}|=n+2c,$ (18)

where $c$ is the number of duplications. Therefore the elements of $S$ are
unique if and only if $c=0$ in Equation (18). This implies that the
computation of lens depth requires $\Omega(n\log n)$. Note that all of the
other computations in this reduction take $O(n)$. ∎

######  Note 4.

This reduction technique can be generalized to prove Theorem 3. It is enough
to choose the rotation angle $\theta=cos^{-1}(1-1/\beta)$ in the construction
of $S=\\{(b_{i},0),(b_{i},\theta)\\}$. For the case of
$\beta\rightarrow\infty$, where $\theta=0$, we construct $S$ as
$S=\\{(b_{i},0),(b_{i},1)\\}$. Note that we use the real RAM model of
computation, where we can compute the square root of a real number in constant
time.

######  Theorem 3.

For $\beta>1$, computing the $\beta$-skeleton depth of a query point in the
plane requires $\Omega(n\log n)$ time.

Figure 7: A representation of $B$, $S$, and duplications in these sets

##  5 Relationships Among Spherical Depth, Lens Depth, and Simplicial Depth

######  Theorem 4.

For $S=\\{x_{1}..x_{n}\\}\subset\mathbb{R}^{d}$ and $q\in\mathbb{R}^{d}$,
$\operatorname{\textit{LD}}(q;S)\geq\operatorname{\textit{SphD}}(q;S)$.

###### Proof.

From Definition 1 for the spherical and lens influence regions of any
arbitrary pair of points $x_{i}$ and $x_{j}$ in $S$, it can be seen that
$\operatorname{\textit{Sph}}(x_{i},x_{j})$ is contained in $L(x_{i},x_{j})$.
Hence Equation (19) is sufficient to complete the proof.

$\operatorname{\textit{SphD}}(q;S)=\sum_{\\{x_{i},x_{j}\\}\subset
S}{I(q\in\operatorname{\textit{Sph}}(x_{i},x_{j}))}\leq\sum_{\\{x_{i},x_{j}\\}\subset
S}{I(q\in L(x_{i},x_{j}))}=\operatorname{\textit{LD}}(q;S)$ (19)

∎

######  Definition 8.

The simplicial depth of $q\in\mathbb{R}^{d}$ with respect to a data set
$S=\\{x_{1}..x_{n}\\}\subset\mathbb{R}^{d}$ is defined by:

$\operatorname{\textit{SD}}(q;S)=\frac{1}{{n\choose
d+1}}\sum_{\\{x_{1}..x_{d+1}\\}\subset
S}{I(q\in\operatorname{\textit{Conv}}\\{x_{1}..x_{d+1}\\})},$ (20)

where $\operatorname{\textit{Conv}}\\{x_{1}..x_{d+1}\\}$ is a closed simplex
formed by $d+1$ points of $S$.

######  Definition 9.

For a point $q\in\mathbb{R}^{2}$ and a data set
$S=\\{x_{1}..x_{n}\\}\subset\mathbb{R}^{2}$, we define $B_{in}(q;S)$ to be the
set of all closed spherical influence regions, out of $n\choose 2$ possible of
them, that contain $q$. We also define $S_{in}(q;S)$ to be the set of all
closed simplices, out of $n\choose 3$ possible defined by $S$, that contain
$q$.

######  Lemma 4.

Suppose that $q$ is a point in a given convex polygon $H$ obtained from a data
set $S$ in $\mathbb{R}^{2}$. $q$ is covered by the union of spherical
influence regions defined by $S$.

###### Proof.

It can be seen that there is at least one triangle, defined by the vertices of
$H$, that contains $q$. We prove that the union of the spherical influence
regions defined by such triangle contains $q$. See Figure 8. We prove this
statement by contradiction. Suppose that $q$ is covered by none of
$\operatorname{\textit{Sph}}(a,b)$, $\operatorname{\textit{Sph}}(a,c)$, and
$\operatorname{\textit{Sph}}(b,c)$. Therefore, Lemma 1 implies that none of
the angles $\angle aqb$, $\angle aqc$, and $\angle bqc$ is greater than or
equal to $\frac{\pi}{2}$ which is a contradiction because at least one of
these angles should be at least $\frac{2\pi}{3}$ in order to get $2\pi$ as
their sum. ∎

######  Lemma 5.

Suppose that $S=\\{a,b,c\\}$ is a set of points in $\mathbb{R}^{2}$. For every
$q\in\mathbb{R}^{2}$, if $|S_{in}(q;S)|=1$, then $|B_{in}(q;S)|\geq 2$.

Another form of Lemma 5 is that if $q\in\bigtriangleup abc$, then $q$ falls
inside at least two spherical influence regions out of
$\operatorname{\textit{Sph}}(a,b)$, $\operatorname{\textit{Sph}}(c,b)$, and
$\operatorname{\textit{Sph}}(a,c)$. The equivalency between these two forms of
the lemma is clear. We prove just the first one.

###### Proof.

From Lemma 4, $|B_{in}(q;S)|\geq 1$. Suppose that $|B_{in}(q;S)|=1$. If $q$ is
one of the vertices of $\bigtriangleup abc$, it is clear that
$|B_{in}(q;S)|\geq 2$. Without loss of generality, we suppose that $q$ falls
in $int\operatorname{\textit{Sph}}(a,b)$. For the rest of the proof, we focus
on the relationships among the angles $\angle aqb$, $\angle cqa$, and $\angle
cqb$ (see Figure 8). Since $q$ is inside $\bigtriangleup abc$, $\angle
aqb\leq\pi$. Consequently, at least one of $\angle cqa$ and $\angle cqb$ is
greater than or equal to $\pi/2$. So, Lemma 1 implies that $q$ is in at least
one of $int\operatorname{\textit{Sph}}(a,c)$ and
$int\operatorname{\textit{Sph}}(b,c)$. Hence, $|B_{in}(q;S)|=1$ contradicts
$|S_{in}(q;S)|=1$ which means that $|B_{in}(q;S)|\geq 2$. As an illustration,
in Figure 8, for the points in the hatching area $|B_{in}(q;S)|=3$. ∎

Figure 8: Triangle $abc$ contains point $q$

######  Lemma 6.

For a data set $S=\\{x_{1}..x_{n}\\}\subset\mathbb{R}^{2}$,

$\frac{|B_{in}(q;S)|}{|S_{in}(q;S)|}\geq\frac{2}{n-2}.$

###### Proof.

Suppose that $\operatorname{\textit{Sph}}(x_{i},x_{j})\in B_{in}(q;S)$. There
exist at most $(n-2)$ triangles in $S_{in}(q;S)$ such that
$\overline{x_{i}x_{j}}$ is an edge of them. We consider $\bigtriangleup
x_{i}x_{j}x_{k}$ to be one of such triangles (see Figure 9 as an
illustration). Referring to Lemma 5, $q$ belongs to at least one of
$\operatorname{\textit{Sph}}(x_{i},x_{k})$ and
$\operatorname{\textit{Sph}}(x_{j},x_{k})$. Similarly, there exist at most
$(n-2)$ triangles in $S_{in}(q;S)$ such that $x_{i}x_{k}$ (respectively
$x_{j}x_{k}$) is an edge of them. In the process of computing the
$|S_{in}(q;S)|$, triangle $\bigtriangleup x_{i}x_{j}x_{k}$ is counted at least
two times, once for $\operatorname{\textit{Sph}}(x_{i},x_{j})$ and another
time for $\operatorname{\textit{Sph}}(x_{i},x_{k})$ (or
$\operatorname{\textit{Sph}}(x_{j},x_{k})$ ). Consequently, for every sphere
area in $B_{in}(q;S)$, there exist at most $\frac{(n-2)}{2}$ distinct
triangles, triangles with only one common side, in $S_{in}(q;S)$. As a result,
Equation (21) can be obtained.

$\frac{(n-2)}{2}|B_{in}(q;S)|\geq|S_{in}(q;S)|\Rightarrow\frac{|B_{in}(q;S)|}{|S_{in}(q;S)|}\geq\frac{2}{(n-2)}$
(21)

∎

Figure 9: $q\in\operatorname{\textit{Sph}}(x_{i},x_{j})$

######  Theorem 5.

For a data set $S=\\{x_{1}..x_{n}\\}$ and a query point $q$ in
$\mathbb{R}^{2}$,
$\operatorname{\textit{SphD}}(q;S)\geq\frac{2}{3}\operatorname{\textit{SD}}(q;S)$.

###### Proof.

From Equation (4) and Definitions 8 and 9, the ratio of spherical depth and
simplicial depth can be calculated as follows:

$\frac{\operatorname{\textit{SphD}}(q;S)}{\operatorname{\textit{SD}}(q;S)}=\frac{\frac{|B_{in}(q;S)|}{{n\choose
2}}}{\frac{|S_{in}(q;S)|}{{n\choose
3}}}=\frac{(n-2)|B_{in}(q;S)|}{3|S_{in}(q;S)|}.$ (22)

Equation (22) and Lemma 6 imply that

$\frac{\operatorname{\textit{SphD}}(q;S)}{\operatorname{\textit{SD}}(q;S)}\geq\frac{2}{3}\Rightarrow\operatorname{\textit{SphD}}(q;S)\geq\frac{2}{3}\operatorname{\textit{SD}}(q;S).$

∎

##  6 Experiments

To support Theorems 4 and 5, we compute the spherical depth, lens depth, and
the simplicial depth of the points in three random sets $Q_{1}$, $Q_{2}$, and
$Q_{3}$ with respect to data sets $S_{1}$, $S_{2}$, and $S_{3}$, respectively.
The elements of $Q_{i}$ and $S_{i}$ are some randomly generated points (double
precision floating point) within the square $\\{(x,y)|x,y\in[-10,10]\\}$. The
results of our experiments are summarized in Table 1. Every cell in the table
represents the corresponding depth of $q_{i}$ with respect to data set
$S_{i}$, where $q_{i}\in Q_{i}$. The cardinalities of $Q_{i}$ and $S_{i}$ are
as follows: $|Q_{1}|=100$, $|S_{1}|=750$, $|Q_{2}|=750$, $|S_{2}|=2500$,
$|Q_{3}|=2500$, $|S_{3}|=10000$. As can be seen in Table 1, there are some
gaps between obtained experimental bounds for random points and the
theoretical bounds. These gaps motivate us to do more research in this area.

Table 1: Summary of experimental results | $(q_{1};S_{1})$ | $(q_{2};S_{2})$ | $(q_{3};S_{3})$
---|---|---|---
| Min | Max | Min | Max | Min | Max
$\operatorname{\textit{SD}}$ | 0.00 | 0.25 | 0.00 | 0.25 | 0.00 | 0.24
$\operatorname{\textit{SphD}}$ | 0.01 | 0.50 | 0.00 | 0.50 | 0.00 | 0.50
$\operatorname{\textit{LD}}$ | 0.05 | 0.61 | 0.05 | 0.61 | 0.04 | 0.61
$\frac{\operatorname{\textit{SphD}}}{\operatorname{\textit{SD}}}$ | 2.00 | $\infty$ | 2.00 | $\infty$ | 2.03 | $\infty$
$\frac{\operatorname{\textit{LD}}}{\operatorname{\textit{SD}}}$ | 2.43 | $\infty$ | 2.44 | $\infty$ | 2.44 | $\infty$
$\frac{\operatorname{\textit{LD}}}{\operatorname{\textit{SphD}}}$ | 1.21 | 8.11 | 1.22 | 23.16 | 1.22 | 157.16

##  7 Conclusions

In this paper, we developed an optimal $\Theta(n\log n)$ algorithm to compute
the spherical depth of a bivariate query point with respect to a given data
set in $\mathbb{R}^{2}$. We also proposed an $O(n^{\frac{3}{2}+\epsilon})$
algorithm for computing the planar $\beta$-skeleton depth where $\beta>1$. To
obtain a lower bound for computing the $\beta$-skeleton depth ($\beta\geq 1$),
the Element Uniqueness problem, which requires $\Omega(n\log n)$ time, is
reduced to the problem of computing the planar $\beta$-skeleton depth of
$O=(0,0)$. In addition to the time complexity, the main advantage of the first
algorithm is its simplicity of implementation. We also investigated some
geometric properties which lead us to find some theoretical relationships
(i.e. $\operatorname{\textit{SphD}}\geq\frac{2}{3}SD$ and
$\operatorname{\textit{LD}}\geq\operatorname{\textit{SphD}}$) among spherical
depth, lens depth, and simplicial depth. Finally, some experimental results
(i.e. $\operatorname{\textit{SphD}}\geq 2SD$ and
$\operatorname{\textit{LD}}\geq 1.2\operatorname{\textit{SphD}}$) are
provided. More research on this topic is needed to figure out if the real
bounds are closer to the experimental bounds or to the current theoretical
bounds.

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## Appendix

######  Note 5.

To avoid some unusual notations in the pseudocode of Algorithm 1, we use the
variables $a$ and $b$ instead of $f(i)$ and $l(i)$, respectively in the text.

Algorithm 1 Computing the spherical depth of points in the plane

1:Data set $S$ and Query point $q$

2:$\operatorname{\textit{SphD}}(q;S)$

3:

4:procedure Translating points

5:input: $S$

6:output: Translated data set $S^{\prime}$

7: for each $x_{i}\in S$ do

8: $x_{i}\leftarrow(x_{i}-q)$

9: return $S^{\prime}$

10:

11:procedure Sorting around $T(q)$

12:input: $T(q)$ and $S^{\prime}$

13:output: Sorted array $S_{T}$

14: for each $x_{i}\in S^{\prime}$ do

15: Compute $\theta(x_{i})$, where $\theta(x_{i})$ is the angle of $x_{i}$ in
polar coordinate system

16: Using an $O(n\log n)$ sorting algorithm, sort $x_{i}$ based on
$\theta(x_{i})$ in counterclockwise order

17: return $S_{T}$

18:

19:procedure Depth calculation

20:input: $S_{T}$

21:output: Depth value of $\operatorname{\textit{SphD}}(q;S)$

22: Initialize $\operatorname{\textit{SphD}}(q;S)=0$

23: for each $x_{i}\in S_{T}$ do

24: Initialize $a=0$ and $b=n+1$

25: Using two _binary search calls_ , update the values of $a$ and $b$

26:

27: if ($0<\theta(x_{i})\leq\frac{\pi}{2}$)

28: $a=min\\{j:\>\theta(x_{j})-\theta(x_{i})\geq\frac{\pi}{2}\\}$

29:
$b=max\\{j:\>\frac{\pi}{2}\leq\theta(x_{j})-\theta(x_{i})\leq\frac{3\pi}{2}\\}$

30: else-if ($\frac{\pi}{2}<\theta(x_{i})\leq\frac{3\pi}{2}$)

31: $a=max\\{j:\>\theta(x_{i})-\theta(x_{j})\geq\frac{\pi}{2}\\}$

32: $b=min\\{j:\>\theta(x_{j})-\theta(x_{i})\geq\frac{\pi}{2}\\}$

33: else

34: $a=min\\{j:\>\theta(x_{i})-\theta(x_{j})\leq\frac{3\pi}{2}\\}$

35:
$b=max\\{j:\>\frac{\pi}{2}\leq\theta(x_{i})-\theta(x_{j})\leq\frac{3\pi}{2}\\}$

36:

37: if $(a=0\;\;\text{and }\;\;b=n+1)$

38: $\hskip 2.84544pt|O_{i}|=0$

39: else Compute $|O_{i}|$

40:
$|O_{i}|=\begin{cases}a+(n-b+1)\;;\frac{\pi}{2}<\theta(x_{i})\leq\frac{3\pi}{2}\\\
b-a+1\hskip 25.6073pt;\text{otherwise.}\end{cases}$

41:

42:
$\operatorname{\textit{SphD}}(q;S)\leftarrow\operatorname{\textit{SphD}}(q;S)+\frac{1}{2}|O_{i}|$

43: return $\operatorname{\textit{SphD}}(q;S)$

44:End;

Algorithm 2 Computing the $\beta$-skeleton depth of points in the plane

1:Data set $S$, Query point $q$, Parameter $\beta>1$

2:$\operatorname{\textit{SkD}}_{\beta}(q;S)$

3:

4:procedure Translating points

5: Call Translating procedure from Algorithm 1.

6:

7:procedure Depth calculation

8:input: Translated data set $S^{\prime}$

9:output: Depth value of $\operatorname{\textit{SkD}}_{\beta}(q;S)$

10: Initialize $\operatorname{\textit{SkD}}_{\beta}(q;S)=0$

11: for each $a\in S^{\prime}$ do

12: Using two $O(n^{\frac{1}{2}+\epsilon})$ range counting algorithms, compute
$|H_{o}(p)|$ and $|intB_{o}(c,r)|$

13:

14:
$\operatorname{\textit{SkD}}_{\beta}(q;S)\leftarrow\operatorname{\textit{SkD}}_{\beta}(q;S)+\frac{1}{2}(|H_{o}(p)|-|intB^{o}(c,r)|)$

15: return $\operatorname{\textit{SkD}}_{\beta}(q;S)$

16:End;


Time evolution of two-point function in expanding universes

Chanyong Parka ***e-mail<EMAIL_ADDRESS>Hanse Kima †††e-mail :
<EMAIL_ADDRESS>and Kyungchan Choa ‡‡‡e-mail<EMAIL_ADDRESS>

a Department of Physics and Photon Science, Gwangju Institute of Science and
Technology
Gwangju 61005, Korea

ABSTRACT

By applying the braneworld model, we investigate the time evolution of two-
point functions holographically in expanding universes. In order to explain
the FLRW cosmologies, we take into account a $p$-brane gas geometry. When we
take the boundary of a $p$-brane gas geometry as the braneworld we are living
in, $p$-branes are detected as $(p-1)$-dimensional objects to an observer
living in the braneworld. These objects lead to the FLRW cosmology which is
related to the radial motion of the braneworld in the $p$-brane gas geometry.
Utilizing this $p$-brane gas geometry, we investigate the time-dependent two-
point functions of FLRW cosmologies. For $p=4$, we reproduce the known two-
point functions of the dS space and find a new analytic two-point function for
$p=2$ which corresponds to an expanding universe with uniformly distributed
cosmic strings. Moreover, we show that the two-point function suppresses in
the late time era with a specific power that is associated with the equation
of state parameter of the uniformly distributed matter.

##  1 Introduction

According to the AdS/CFT correspondence Maldacena (1998); Gubser  _et al._
(1998); Witten (1998a, b), nonperturbative quantum field theory (QFT)
involving all quantum effects can be described by a one-dimensional higher
gravity theory. Using this holographic techniques in the last two decades,
there were many studies trying to figure out various strongly interacting
systems appearing in nuclear and condensed matter physics Bergman  _et al._
(2007); Casalderrey-Solana  _et al._ (2014); Hartnoll  _et al._ (2010); Preis
_et al._ (2011); Rozali  _et al._ (2008); Davis  _et al._ (2008); Baggioli
(2016); Charmousis  _et al._ (2010); Hong  _et al._ (2008); Hashimoto  _et
al._ (2009); Franco  _et al._ (2010). Another important feature of the
holography is that the extra dimension of a bulk geometry can be regarded as
an energy scale measuring the dual QFT. This implies that a bulk geometry
further explains the renormalization group (RG) flow of a dual QFT which can
specify the IR properties. Applying the holographic method, in this work, we
investigate the time evolution of nonperturbative quantum correlations in
various expanding universes.

The holography provides a fascinating and useful tool to understand strongly
interacting systems. Some important quantities of strongly interacting system
can be realized as simple geometrical objects in the dual geometry. For
example, the $q\bar{q}$-potential and quantum entanglement entropy can be
evaluated minimal surface extending to the dual geometry Maldacena (1998); Rey
and Yee (2001); Herzog (2007); Karch  _et al._ (2006); Park (2010); Ryu and
Takayanagi (2006a, b); Casini (2008); Myers and Sinha (2011); Casini  _et al._
(2011); Kim and Park (2017); Park (2015). Moreover, it was proposed that
correlation functions of QFT are also evaluated by a geodesic length
connecting two boundary operators in the holographic setup Susskind and Witten
(1998); Balasubramanian and Ross (2000); Louko  _et al._ (2000); Solodukhin
(1999); D’Hoker  _et al._ (1999)

$\displaystyle\left\langle
O(\tau_{1},\vec{x}_{1})O(\tau_{2},\vec{x}_{2})\right\rangle=e^{-\Delta\,L(\tau_{1},\vec{x}_{1};\tau_{2},\vec{x}_{2})/R},$
(1.1)

where $\Delta$ and $L(\tau_{1},\vec{x}_{1};\tau_{2},\vec{x}_{2})$ indicate a
conformal dimension and geodesic length, respectively. It was shown that the
holographic proposal (1.1) can reproduce the known correlation functions of
conformal filed theory (CFT) defined in a flat, dS and AdS space Kim  _et al._
(2023). Moreover, the holographic formula (1.1) was exploited to look into
quantum correlation in medium composed of various matter Park  _et al._
(2024); Park (2023a); Georgiou and Zoakos (2022); Rodriguez-Gomez and Russo
(2021a, b); Fuertes and Moroz (2009); Krishna and Rodriguez-Gomez (2021);
Volovich and Wen (2009).

The holography is also applied to the cosmological model. In the standard
Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, the Friedmann equation
leads to an expanding universe as a solution where the rate of expansion is
influenced by the equation of state parameter of components filling the
universe. In the holography studies, this FLRW cosmology can be realized in
two different ways, the dS boundary Koh  _et al._ (2020); Park (2021); Barcelo
and Visser (2000) and braneworld models Randall and Sundrum (1999a, b); Kraus
(1999); Park and Sin (2000); Park (2023b, 2020). For the dS boundary model, we
consider a $(d+1)$-dimensional AdS space which has a $d$-dimensional dS
boundary. Following the AdS/CFT correspondence, the classical gravity in a
$(d+1)$-dimensional AdS space can describe nonperturbative quantum physics in
an eternally inflating universe. After considering a three-dimensional AdS
space as a toy model, time-dependent quantum entanglement entropy in a two-
dimensional inflating universe has been investigated by using the dS boundary
model Koh  _et al._ (2020); Park (2021); Barcelo and Visser (2000). For the dS
boundary model, it is difficult to describe the FLRW cosmologies expanding by
a power-law, because their dual geometries are not known yet. To detour this
problem, one can take into account the braneworld model where the FLRW
cosmologies are described by the radial motion of a braneworld in the
$p$-brane gas geometry Park (2023b, 2020). Applying the braneworld model, the
leading time evolution of entanglement entropy in the FLRW cosmology has been
studied Park (2021, 2023b, 2020); Kim  _et al._ (2023).

By applying the holographic technique, in the present work, we investigate
time-dependent correlation functions in expanding universes. The correlation
functions in cosmology give us important information about the history and
structure of our universe, like the power spectrum and non-Gaussianity Kinney
(2009). For expanding universes in the radiation and matter dominated era, it
is not easy to calculate two-point function on the QFT side because it is
defined in a nontrivial curved spacetime. Assuming that one kind of matter
with a equation of state parameter $w$ is uniformly distributed in the
universe, it was known that the universe in the standard FLRW cosmology
expands by

$\displaystyle a(t)\sim t^{\frac{2}{3(1+w)}},$ (1.2)

where $w$ means an equation of state parameter. In this expanding universe, we
try to evaluate two-point function of two local operators by using the
holographic principle. For CFT, two-point function suppresses by a power-law.
This is not the case in medium. In general, the medium breaks the conformal
symmetry and gives rise to the screening effect, which reduces the two-point
function exponentially in the long distance limit. In expanding universes,
there is an additional effect reducing a two-point function. The expansion of
the universe leads to the increase of the physical distance between two
operators. Therefore, one expects that the two-point function decreases with
time in expanding universes.

To clarify time dependence of two-point functions in expanding universes, we
investigate two-point functions holographically in the braneworld model. To do
so, we first show how the braneworld model can reproduce the known two-point
function in the inflating universe, which was derived in the dS boundary model
Kim  _et al._ (2023). After that, we consider a braneworld model describing
the FLRW cosmologies and calculate two-point functions in expanding universes.
We find that two-point function in the late time era suppresses by

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim
t^{-\frac{4\Delta}{3(1+w)}},$ (1.3)

where $\Delta$ means a conformal dimension of an operator $O$. We also check
this result with the exact numerical simulation.

The rest of this paper is organized as follows. In Sec. 2, we briefly review
the braneworld cosmology in a five-dimensional $p$-brane gas geometry and
rederive the known four-dimensional FLRW cosmologies. In Sec. 3, we consider
an eternal inflation for $p=4$ and reproduce the known two-point function of a
dS space in the braneworld model. We also show that the two-point function in
the late time era suppresses exponentially. In Sec. 4, we further take into
account FLRW cosmologies with $p\neq 4$. By calculating the two-point function
in expanding universes, we study how the two-point functions suppresses in
expanding universes. Lastly, we finish this work with some concluding remarks
in Sec. 5.

##  2 FLRW cosmology in the braneworld model

To study correlation functions in expanding universe, let us first discuss how
to describe the standard cosmological in the holographic setup. In the
standard cosmology, the scale factor relying on the matter is determined by
the Friedmann equation Kinney (2009)

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\kappa^{2}}{3}\frac{\rho_{0}}{a^{3(1+w)}},$
(2.1)

where $\rho_{0}$ and $w$ are the matter’s density and equation of state
parameter, respectively. The Friedmann equation allows the following scale
factor

$\displaystyle a(\tau)\sim t^{\frac{2}{3(1+w)}},$ (2.2)

When $n$-dimensional objects are uniformly distributed in a four-dimensional
expanding universe, the equation of state parameter is determined as Park
(2021); Kim  _et al._ (2023); Park  _et al._ (2024).

$\displaystyle w=-\frac{n}{3}.$ (2.3)

In order to the standard cosmology, we take into account a braneworld model in
a $p$-brane gas geometry. We assume that $p$-branes are uniformly distributed
in a five-dimensional AdS space and that they extend to the radial direction
of the bulk. Denoting the radial direction as $r$, the gravitational
backreaction of a $p$-brane gas leads to the following $p$-brane gas geometry
Park (2020); Koh  _et al._ (2020); Park (2021, 2023b)

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
g_{\tau\tau}d\tau^{2}+g_{rr}dr^{2}+g_{ij}\
dx^{i}dx^{j}=-\frac{r^{2}}{R^{2}}f(r)\
d\tau^{2}+\frac{R^{2}}{r^{2}f(r)}dr^{2}+\frac{r^{2}}{R^{2}}\
\delta_{ij}dx^{i}dx^{j},$ (2.4)

where $\tau$ indicates a bulk time and the metric factor $f(r)$ is given by

$\displaystyle f(r)=1-\frac{\rho_{p}}{r^{4-p}},$ (2.5)

We assume that two five-dimensional $p$-brane gas geometries are bordered by a
four-dimensional hypersurface, which we call a braneworld (see more details in
Ref. Park (2020, 2021)). In this case, the radial motion of the braneworld is
governed by the junction equation

$\displaystyle\pi^{(+)}_{\mu\nu}-\pi^{(-)}_{\mu\nu}=T_{\mu\nu},$ (2.6)

where $(\pm)$ indicate two $p$-brane gas geometries on the both sides of the
braneworld. Assuming that two $p$-brane gas geometries are $Z_{2}$-symmetric
for convenience, the canonical momenta of the boundary metric
$\gamma_{\mu\nu}$ are given by

$\displaystyle\pi^{(+)}_{\mu\nu}=-\pi^{(-)}_{\mu\nu}=\frac{1}{2\kappa^{2}}\left(K_{\mu\nu}-\gamma_{\mu\nu}K\right),$
(2.7)

where $K_{\mu\nu}=\nabla_{\mu}n_{\nu}$ means an extrinsic curvature at the
braneworld with a unit normal vector $n_{\nu}$. When we parameterize the
braneworld’s energy-momentum tensor as

$\displaystyle T_{\mu\nu}=\frac{\sigma}{2\kappa^{2}}\ \gamma_{\mu\nu},$ (2.8)

the junction equation in the spatial section is rewritten as

$\displaystyle K_{ij}-\gamma_{ij}K=\frac{\sigma}{2}\ \gamma_{ij},$ (2.9)

where $K=K_{\mu\nu}\gamma^{\mu\nu}$.

Defining a cosmological time $t$ in the braneworld as

$\displaystyle
dt^{2}=g_{\tau\tau}d\tau^{2}-g_{rr}dr^{2}=\left[g_{\tau\tau}-g_{rr}\left(\frac{dr}{d\tau}\right)^{2}\right]d\tau^{2},$
(2.10)

the induced metric on the braneworld is reduced to

$\displaystyle ds^{2}_{B}=-dt^{2}+\frac{r(t)^{2}}{R^{2}}\
\delta_{ij}dx^{i}dx^{j},$ (2.11)

where $r(t)$ means the radial position of the braneworld. This is the FLRW-
type metric in which the scale factor is defined as

$\displaystyle a(t)=r(t)/R.$ (2.12)

In the braneworld model, therefore, the radial motion of the braneworld is
reinterpreted as an expanding universe. Rewriting the previous junction
equation in terms of the cosmological time $t$, we obtain

$\displaystyle\dot{r}^{2}=\frac{\sigma^{2}\,r^{2}}{36}-\frac{1}{g_{rr}}=\frac{(\sigma^{2}-\sigma_{c}^{2})\,r^{2}}{36}+\frac{\rho_{p}}{R^{2}}\frac{1}{r^{2-p}},$
(2.13)

with a critical tension $\sigma_{c}=6/R$, where the dot means a derivative
with respect to the cosmological time $t$. This junction equation corresponds
to the Friedmann equation in the braneworld.

  * •

For $\sigma\neq\sigma_{c}$ with $\rho_{p}=0$ or $\sigma=\sigma_{c}$ with
$\rho_{4}\neq 0$, the junction equation is reduced to

$\displaystyle\left(\frac{\dot{r}}{r}\right)^{2}=\frac{\sigma^{2}-\sigma_{c}^{2}}{36}\quad{\rm
or}\quad\frac{\rho_{4}}{R^{2}},$ (2.14)

where the right-hand side can be regarded as the cosmological constant
allowing inflation

$\displaystyle r(t)=R\,e^{Ht}.$ (2.15)

with the following Hubble constant, $H=\sqrt{\sigma^{2}-\sigma_{c}^{2}}/6$ or
$\sqrt{\rho_{4}}/R$. This describes an eternal inflation with $w=-1$.

  * •

For $\sigma=\sigma_{c}$ with $p\neq 4$, the junction equation reduces to the
following form

$\displaystyle\left(\frac{\dot{r}}{r}\right)^{2}=\frac{\rho_{p}}{R^{2}}\frac{1}{r^{2-p}},$
(2.16)

where we utilize the relation in (2.17). Since one living in the braneworld
cannot see the radial direction of the bulk space, one detects the bulk
$p$-branes as $(p-1)$-dimensional objects. Therefore, the bulk $p$-branes map
to $(p-1)$-dimensional objects in the braneworld which give rise to the
following equation of state parameter

$\displaystyle w=-\frac{p-1}{3}.$ (2.17)

Using this relation, we see that the junction equation in the braneworld model
is equivalent to the Friedmann equation in (2.1).

##  3 Two-point function in an eternally inflating universe

For the FLRW cosmology described in the braneworld model, we investigate two-
point function. When a universe has a positive cosmological constant, its
background geometry is described by a dS space and results in an eternal
inflation. On this dS background geometry, the correlation function between
two local operators with a conformal dimension $\Delta$ is given by

$\displaystyle\left\langle
O\left(T_{1},x_{1}\right)O\left(T_{2},x_{2}\right)\right\rangle\sim\left(\frac{T_{1}T_{2}}{-|T_{1}-T_{2}|{}^{2}+|x_{1}-x_{2}|{}^{2}}\right)^{\Delta},$
(3.1)

where $T_{i}$ is a conformal time in the comoving frame and
$x_{i}=|\vec{x}_{i}|$. In the holographic setup, one can rederive the same
correlator in the boundary dS model Kim  _et al._ (2023). If one wants to
further study two-point functions of other FRLW cosmologies, the boundary dS
model is not applicable because the bulk geometry allowing the FRLW cosmology
at the boundary is not known. In this case, we can exploit the braneworld
model where the FRLW cosmology is realized by the radial motion of the
braneworld. We first study a two-point function of a dS space in the
braneworld model.

To describe an eternal inflation in the braneworld model, we take into account
the case of $\sigma\neq\sigma_{c}$ with $\rho_{p}=0$. Introducing a new radial
coordinate $z=R^{2}/r$ for convenience, the metric of a $(d+1)$-dimensional
AdS space is rewritten as

$\displaystyle ds^{2}=\frac{R^{2}}{z^{2}}\left(-d\tau^{2}+d\vec{x}\cdot
d\vec{x}+dz^{2}\right).$ (3.2)

In terms of the new radial coordinate, the junction equation in (2.14) is
rewritten as

$\displaystyle\frac{dz}{dt}=-Hz\quad{\rm with}\quad
H=\frac{\sigma^{2}-\sigma_{c}^{2}}{6},$ (3.3)

where $t$ is the cosmological time in (2.10). Using the chain rule, the
cosmological time $t$ is related to the bulk time $\tau$

$\displaystyle\frac{dt}{d\tau}=\frac{R}{z\,\sqrt{1+H^{2}R^{2}}}.$ (3.4)

Combining two relations in (3.3) and (3.4) gives rise to the relation between
$z$ and $\tau$

$\displaystyle\frac{d\tau}{dz}=-\frac{\sqrt{1+H^{2}R^{2}}}{HR}.$ (3.5)

Solving these relations, we obtain

$\displaystyle z(t)=z_{0}\,e^{-H(t-t_{0})}\quad{\rm
and}\quad\tau(z)=-\frac{\sqrt{1+H^{2}R^{2}}}{HR}z,$ (3.6)

where $t_{0}$ is introduced as an appropriate initial (or reference) time and
$z_{0}$ corresponds to the radial position of the braneworld at $t=t_{0}$.

To describe the dS space in the comoving frame, we introduce a conformal time

$\displaystyle T=\frac{z}{HR}=\frac{1}{H}\ e^{-H(t-t_{0})}.$ (3.7)

In the comoving frame, then the induced metric becomes in the braneworld

$\displaystyle
ds^{2}_{B}=\frac{1}{H^{2}T^{2}}\left(-dT^{2}+\delta_{ij}dx^{i}dx^{j}\right).$
(3.8)

This is exactly the same as a four-dimensional dS metric. Using the solutions
in (3.6), we can see that the conformal time satisfies the following relation

$\displaystyle-dT^{2}=-d\tau^{2}+dz^{2}.$ (3.9)

We suppose that two local operators are located at
$\left\\{\eta_{1},x_{1},z_{1}\right\\}$ and
$\left\\{\eta_{2},x_{2},z_{2}\right\\}$ where $z_{i}$ are the braneworld
position at given Euclidean time, $\eta_{i}=i\tau_{i}$. Since a geodesic curve
connecting two operators is associated with a two-point function in the
braneworld, we evaluate a geodesic length in the bulk AdS space (3.2)

$\displaystyle
L(\eta_{1},x_{1},z_{1};\eta_{2},x_{2},z_{2})=R\int_{\eta_{1}}^{\eta_{2}}d\eta\
\frac{\sqrt{1+\dot{x}^{2}+\dot{z}^{2}}}{z},$ (3.10)

where the dot indicates a derivative with respect to $\eta$. Regarding the
geodesic length as an action, it has two conserved charges

$\displaystyle P=\frac{R\,v}{z_{t}\sqrt{1+v^{2}}}\quad{\rm and}\quad
H=-\frac{R}{z_{t}\sqrt{1+v^{2}}},$ (3.11)

where $z_{t}$ means a turning point at which $\dot{z}=0$ and $\dot{x}=v$

Comparing $P$ with $H$, we find that $\dot{x}=v$ must be constant. After
solving it, the operator’s position is determined as a function of the bulk
time

$\displaystyle x(\eta)=v\,\eta+c_{1}.$ (3.12)

Imposing the boundary conditions, $x(\eta_{1})=x_{1}$ and $x(\eta_{2})=x_{2}$,
the integral constant and velocity are determined as

$\displaystyle c_{1}=-v\eta_{1}+x_{1}\quad{\rm and}\quad v=\frac{\Delta
x}{\Delta\eta},$ (3.13)

where $\Delta x=|x_{1}-x_{2}|$ and $\Delta\eta=|\eta_{1}-\eta_{2}|$. From
(3.11), we also determine the relation between $z$ and $\eta$

$\displaystyle\dot{z}=\frac{\sqrt{1+v^{2}}\,\sqrt{z_{t}^{2}-z^{2}}}{z},$
(3.14)

which describes the radial motion of the braneworld. Solving this equation
with the boundary conditions, $z(\eta_{1})=z_{1}$ and $z(\eta_{2})=z_{2}$, we
arrive at

$\displaystyle z(\eta)=\sqrt{z_{t}^{2}-(1+v^{2})\,(\eta-c_{2})^{2}}.$ (3.15)

with the following constants

$\displaystyle
c_{2}=\eta_{1}+\frac{\sqrt{z_{t}^{2}-z_{1}^{2}}}{\sqrt{1+v^{2}}}\quad{\rm
and}\quad z_{t}=\frac{\sqrt{(1+v^{2})\,\Delta\eta^{2}+\Delta z^{2}}\
\sqrt{(1+v^{2})\,\Delta\eta^{2}+(z_{1}+z_{2})^{2}}}{2\sqrt{1+v^{2}}\,\Delta\eta},$
(3.16)

where $\Delta z=|z_{1}-z_{2}|$.

After plugging these solutions into the geodesic formula, the minimal geodesic
length connecting two operators at $\left\\{\eta_{1},x_{1}\right\\}$ and
$\left\\{\eta_{2},x_{2}\right\\}$ is given by

$\displaystyle L(\eta_{1},x_{1};\eta_{2},x_{2})=\
\left(\int_{z_{t}}^{z_{1}}dz-\int_{z_{2}}^{z_{t}}dz\right)dz\frac{R\,z_{t}}{z\sqrt{z_{t}^{2}-z^{2}}},$
(3.17)

where $z_{1}$ and $z_{2}$ are determined as function of
$\eta_{1},x_{1},\eta_{2}$ and $x_{2}$. After performing the integral and
applying the inverse Wick rotation ($\eta=i\tau$), we finally obtain the
following minimal length

$\displaystyle L(T_{1},x_{1};T_{2},x_{2})=R\log\left[\frac{\left(\sqrt{-\Delta
T^{2}+\Delta x^{2}}-\sqrt{-\Delta\tau^{2}+\Delta
x^{2}+(z_{1}+z_{2})^{2}}\right)^{2}}{4z_{1}z_{2}}\right],$ (3.18)

where $\Delta T^{2}=|T_{1}-T_{2}|^{2}=\Delta\tau^{2}-\Delta z^{2}$ due to
(3.9). If we specify the initial position of the braneworld at $t=t_{0}$ as

$\displaystyle
z_{0}=\frac{HR\left(x_{1}-x_{2}\right)e^{H(t_{2}-t_{0})}}{\sqrt{1-2\left(1-2H^{2}R^{2}\right)e^{H\left(t_{2}-t_{1}\right)}+\left(1-8H^{2}R^{2}\right)e^{2H(t_{2}-t_{1})}}},$
(3.19)

the resulting two-point function, according to the holographic proposal in
(1.1), results in

$\displaystyle\left\langle
O\left(T_{1},x_{1}\right)O\left(T_{2},x_{2}\right)\right\rangle=\left(\frac{4H^{2}R^{2}\
T_{1}T_{2}}{-|T_{1}-T_{2}|{}^{2}+|x_{1}-x_{2}|{}^{2}}\right)^{\Delta},$ (3.20)

This is consistent with the two-point function of the CFT (3.1) in a dS space
Kim  _et al._ (2023).

At a given time with $T_{1}=T_{2}$, the spatial two-point function in the
inflating universe becomes is reduced to

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle=\left(\frac{2Re^{Ht_{0}}}{|x_{1}-x_{2}|}\right)^{2\Delta}\
e^{-2\Delta Ht}.$ (3.21)

This shows that the spatial two-point function in the inflating universe
suppreses exponentially. In addition, the correlator of an operator having a
large conformal dimension suppresses more rapidly. In other words, the
operator in the inflationary universe has an effective mass $m_{eff}=2H\Delta$
in the time direction. However, it behaves as a massless one and gives rise to
a power-law two-point function due to the scale symmetry. As a result, the
scale symmetry is broken in the time direction but is still preserved in the
spatial directions.

##  4 Two-point functions in the FLRW cosmology

Now, we investigate two-point functions in the FLRW cosmology expanding by a
power-law. To do so, we take into account the case of $p\neq 4$ with
$\sigma=\sigma_{c}$. In the holographic dual geometry, the $p$-brane gas
geometry in terms of $z$ is rewritten as

$\displaystyle ds^{2}=\frac{R^{2}}{z^{2}}\left(-f(z)\
d\tau^{2}+\frac{1}{f(z)}dz^{2}+\delta_{ij}\,dx^{i}dx^{j}\right),$ (4.1)

with the following blackening factor

$\displaystyle f(z)=1-\frac{z^{4-p}}{z_{h}^{4-p}},$ (4.2)

where the black hole horizon $z_{h}$ is related to the density of $p$-branes,
$\rho_{p}=R^{2(4-p)}/z_{h}^{4-p}$.

In the braneworld model discussed before, it was shown that the braneworld’s
position $\bar{r}=R^{2}/\bar{z}$ in the bulk can be reinterpreted as the scale
factor. The relation in (2.12) is in terms of $\bar{z}$

$\displaystyle\bar{z}=\frac{R}{a(t)}\sim t^{-2/(4-p)}.$ (4.3)

In this background geometry, we calculate a geodesic length to evaluate the
boundary two-point function. To do so, we assume that the braneworld is
located at $\bar{z}$ at a given time $t$, and that two operators are located
at $\left\\{t,x_{1},\bar{z}\right\\}$ and $\left\\{t,x_{2},\bar{z}\right\\}$.
In the braneworld model, the time $t$ can be determined as a function of the
braneworld position $\bar{z}$ due to (4.3). Therefore, the geodesic length
connecting two operators at a given time $t$ is governed by

$\displaystyle
L\left(t,x_{1};t,x_{2}\right)=R\int_{x_{1}}^{x_{2}}dx\frac{\sqrt{f+z^{\prime
2}}}{z\sqrt{f}},$ (4.4)

Introducing a turning point $z_{t}$ where $z^{\prime}=0$, the geodesic length
extends to only the range of $\bar{z}\leq z\leq z_{t}$. Using this fact, we
determine the distance and geodesic length in terms of the turning point
$z_{t}$ and the braneworld’s position $\bar{z}$

$\displaystyle|x_{1}-x_{2}|$ $\displaystyle=$
$\displaystyle\int_{\bar{z}}^{z_{t}}dz\frac{2z}{\sqrt{f}\
\sqrt{z_{t}^{2}-z^{2}}},$ (4.5) $\displaystyle L\left(t,x_{1};t,x_{2}\right)$
$\displaystyle=$
$\displaystyle\int_{\bar{z}}^{z_{t}}dz\frac{2Rz_{t}}{z\sqrt{f}\
\sqrt{z_{t}^{2}-z^{2}}}$ (4.6)

It is worth to note that, since the background geometry (4.1) is time
independent, the equation of motion derived from (4.4) is also time
independent. However, the time-dependent braneworld position $\bar{z}$ should
be applied as a boundary condition in order to determine the geodesic curve.
Due to this time-dependent boundary condition, as a consequence, the distance
$|x_{1}-x_{2}|$ and geodesic length $L\left(t,x_{1};t,x_{2}\right)$ become
time-dependent.

In the short distance limit ($|x_{1}-x_{2}|\to 0$ and $z_{t}/z_{h}\to 0$),
since the geodesic extends only in the asymptotic region, the blackening
factor can be approximated by $f\approx 1$. In this case, leading terms of the
distance and geodesic length are given by

$\displaystyle|x_{1}-x_{2}|$ $\displaystyle\approx$ $\displaystyle
2z_{t}+\cdots,$ $\displaystyle L\left(t,x_{1};t,x_{2}\right)$
$\displaystyle\approx$ $\displaystyle 2R\log\frac{2z_{t}}{\bar{z}}+\cdots$
(4.7)

Applying the holographic formula (1.1), therefore, the leading contribution to
a short-distance two-point function is given by

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim\frac{\bar{z}^{2\Delta}}{|x_{1}-x_{2}|^{2\Delta}}\sim\frac{1}{|x_{1}-x_{2}|^{2\Delta}}\
t^{-4\Delta/(4-p)},$ (4.8)

where we use (4.3).

In the long distance limit ($|x_{1}-x_{2}|\to\infty$ and $z_{t}/z_{h}\to 1$),
on the other hand, the geodesic length is rewritten as the following form

$\displaystyle L\left(t,x_{1};t,x_{2}\right)=\lim_{z_{t}\to
z_{h}}\left(\frac{R|x_{1}-x_{2}|}{z_{t}}+2R\int_{\bar{z}}^{z_{t}}dz\frac{\sqrt{1-z^{2}/z_{t}^{2}}}{z\sqrt{f}}\right).$
(4.9)

In the late time era ($t\to\infty$ and $\bar{z}\to 0$) , the geodesic length
is approximately

$\displaystyle
L\left(t,x_{1};t,x_{2}\right)=\frac{R|x_{1}-x_{2}|}{z_{h}}-2R\log\bar{z}+{\cal
O}\left(1\right).$ (4.10)

Using this fact, the long-distance two-point function in the late time era is
reduced to

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim
e^{-\Delta|x_{1}-x_{2}|/z_{h}}\ t^{-4\Delta/(4-p)}.$ (4.11)

In sum, when one kind of matter with the equation of state parameter $w$ is
uniformly distributed, the holographic study shows that a short-distance two-
point function in the FLRW cosmology becomes in the late time era

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim\frac{1}{|x_{1}-x_{2}|^{2\Delta}}\
t^{-\frac{4\,\Delta}{3(1+w)}},$ (4.12)

whereas a long-distance two-point function in the late time era reads

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim
e^{-\Delta|x_{1}-x_{2}|/z_{h}}\ t^{-\frac{4\,\Delta}{3(1+w)}}.$ (4.13)

In the FLRW cosmology, the dimensionful density of matter breaks the conformal
symmetry. In the short distance limit, the finite density effect is
negligible. This is why the short-distance two-point function in the spatial
direction suppresses by a power-law. In the long distance limit, the spatial
two-point function decays exponentially. This indicates that the local
operator has the following effective mass in the FLRW universe

$\displaystyle m_{eff}=\frac{\Delta}{R^{2}}\ \rho_{p}^{\frac{1}{3(1+w)}}.$
(4.14)

In order to check the validity of the expectation in (4.12) and (4.13), we
take into account the case of $p=2$ because it allows an analytic solution.
$2$-branes in the bulk are observed as one-dimensional cosmic strings by an
observer living in the brane world, so the equation of state parameter of
cosmic strings is given by $w=-1/3$. When $2$-branes are uniformly distributed
in a five-dimensional AdS space, the $2$-brane gas geometry has the following
metric factor

$\displaystyle f(z)=1-\frac{z^{2}}{z_{h}^{2}},$ (4.15)

where $\rho_{2}=R^{4}/z_{h}^{2}$. If the braneworld has the critical tension
$\sigma_{c}$, the radial motion of the braneworld is governed by the following
junction equation

$\displaystyle\dot{z}=-\frac{z^{2}}{Rz_{h}},$ (4.16)

where the dot is a derivative with respect to the cosmological time $t$.
Solving this junction equation, the braneworld’s radial motion is described by

$\displaystyle z=c+\frac{Rz_{h}}{t}.$ (4.17)

Requiring that $z\to 0$ at $t\to\infty$, the integral constant is determined
as $c=0$. This solution implies that the braneworld expands linaearly in time,
$a(t)\sim 1/z\sim t$.

Now, we look into a spatial two-point function in this linearly expanding
universe. We assume that two local operators are located at
$\left\\{t,x_{1}\right\\}$ and $\left\\{t,x_{2}\right\\}$. Then, a geodesci
curve connecting these two operators is governed by

$\displaystyle
L\left(t,x_{1};t,x_{2}\right)=R\int_{x_{1}}^{x_{2}}dx\frac{\sqrt{f+z^{\prime
2}}}{z\sqrt{f}},$ (4.18)

where the prime means a derivative with respect to $x$. Using the translation
symmetry in the $x$-direction, $z$ is given by a function of $x$

$\displaystyle\frac{dz}{dx}=\frac{\sqrt{f}\,\sqrt{z_{t}^{2}-z^{2}}}{z},$
(4.19)

where $z_{t}$ is a turning point where $z^{\prime}=0$. A solution of this
differential equation becomes

$\displaystyle
x=x_{t}-z_{h}\tanh^{-1}\left(\frac{\sqrt{z_{t}^{2}-z^{2}}}{\sqrt{z_{h}^{2}-z^{2}}}\right),$
(4.20)

where $x_{t}$ is an integral constant. Since the geodesic formula in (4.18) is
invariant under $x\to x^{\prime}=(x_{1}+x_{2})-x$, a turning point has to
appear at $x=(x_{1}+x_{2})/2$ and $z=z_{t}$. This fact determines the integral
constant as

$\displaystyle x_{t}=\frac{x_{1}+x_{2}}{2}.$ (4.21)

Denoting the braneworld position as $\bar{z}$, the geodesic curve must be
anchored at the operator’s position, $\left\\{t,x_{1}\right\\}$ and
$\left\\{t,x_{2}\right\\}$ at $z=\bar{z}$. In this case, it should be noted
that, since we consider two operators at the same time $t$, they have to lie
in the same braneworld located at $\bar{z}=z(t)$. Imposing this constraint,
the turning point $z_{t}$ appears at

$\displaystyle
z_{t}=\frac{\sqrt{\bar{z}^{2}+z_{h}^{2}\sinh^{2}(|x_{1}-x_{2}|/2z_{h})}}{\cosh(|x_{1}-x_{2}|/2z_{h})}.$
(4.22)

Substituting these solutions into the geodesic formula (4.18), we finally
obtain a spatial two-point function in the linearly expanding universe results
in

$\displaystyle\left\langle O\left(t,x_{1}\right)\
O\left(t,x_{2}\right)\right\rangle=\left(\frac{\sqrt{R^{2}+t^{2}\sinh^{2}(|x_{1}-x_{2}|/2z_{h})}-t\sinh(|x_{1}-x_{2}|/2z_{h})}{\sqrt{R^{2}+t^{2}\sinh^{2}(|x_{1}-x_{2}|/2z_{h})}+t\sinh(|x_{1}-x_{2}|/2z_{h})}\right)^{\Delta}.$
(4.23)

In the late time era ($t\gg R/\sinh(|x_{1}-x_{2}|/2z_{h})$), a short-distance
two-point function for $|x_{1}-x_{2}|\ll z_{h}\sim 1/T_{H}$ is described by

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim\frac{1}{|x_{1}-x_{2}|^{2\Delta}}\
t^{-2\Delta}.$ (4.24)

On the other hand, a long-distance two-point function for $|x_{1}-x_{2}|\gg
1/T_{H}$ behaves as

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim
e^{-\Delta|x_{1}-x_{2}|/z_{h}}\ t^{-2\Delta}.$ (4.25)

These results are consistent with the previous expectation in (4.12) and
(4.13) for $p=2$, which leads to $w=-1/3$ .

(a) Two-point functions (b) Suppression power

Figure 1:  (a) Time evolution of a two-point function in FLRW cosmologies
where we take $\Delta=|x_{1}-x_{2}|=R=1$ and $z_{h}=10$. (b) In the late time
era, we numerically obtain $\gamma=1$ for $w=0$, $\gamma=4/3$ for $w=-1/3$,
and $\gamma=2$ for $w=-2/3$.

In the braneworld model, $0$\- and $1$-branes in the bulk map to radiation and
massive particles, respectively. In the expanding universe with uniformly
distributed radiation or massive particles, the time-dependent two-point
function can be understood by studying a geodesic length in the dual bulk
geometry in (2.4). The distance between operators and the geodesic length are
determined by (4.6). For $p=0$ and $1$ unlike the $p=2$ case, one cannot
perform the integrals analytically. Even in this case, however, it is possible
to figure out the relation between the distance and geodesic length by using
the numerical calculation. In Fig. 1, we depict (a) the time evolution of a
two-point function with a a fixed distance, $|x_{1}-x_{2}|=1$,

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim t^{-\gamma},$
(4.26)

and (b) the change of a suppression power with time. In the late time era, the
suppression powers obtained in the numerical simulation are perfectly mathced
with the analytic results in (4.13)

$\displaystyle\gamma=\left.-\frac{d\log\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle}{d\log
t}\right|_{|x_{1}-x_{2}|=1}=\frac{4\Delta}{3(1+w)}.$ (4.27)

##  5 Discussion

In this work, we have investigated the two-point function in expanding
universes. To do so, we took into account the braneworld model. When we
consider the junction equation in the $p$-brane gas geometry, its solution
resulted in two types of cosmologies, an eternal inflation and FLRW cosmology
expanding by a power, $2/(4-p)$. These cosmologies in the braneworld is
crucially related to the radial motion of the braneworld in the bulk.
$p$-branes in the bulk are detected as $(p-1)$-dimensional objects to an
observer living in the braneworld because the observer cannot see the bulk’s
radial direction. Therefore, the junction equation in the $p$-brane gas
geometry determines the braneworld cosmology involving uniformly distributed
$(p-1)$-dimensional objects. We showed that the braneworld cosmology is
consistent with the standard cosmology. Using the equivalence between the
braneworld and standard cosmologies, we investigate time-dependent two-point
functions of expanding universes in the braneworld model.

In the holographic setup, a two-point function of QFT is described by a
geodesic curve extending to the dual geometry. To verify the computation of
the two-point function in the braneworld model, we first calculated the two-
point function in inflating universe and successfully reproduced the known
two-point function of a dS space in the braneworld model. In the eternally
inflating universe, since the metric relies on time explicitly, the time
translation symmetry is broken. In the early time era, the spatial two-point
function in the inflating universe suppresses by a power-law. In the late time
era, however, it suppresses exponentially, due to the breaking of the
translation symmetry,

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim e^{-2\Delta Ht}.$
(5.1)

Next, we investigated the spatial two-point functions in the FLRW cosmology
expanding by a power-law. In the short distance limit, the spatial two-point
function decreases by a power-law because the finite density effect is
negligible in the UV region. We further showed that in the late time era the
two-point function in the FLRW cosmology suppresses by a power law, unlike the
previous inflating universe,

$\displaystyle\left\langle
O\left(t,x_{1}\right)O\left(t,x_{2}\right)\right\rangle\sim
t^{-4\Delta/3(1+w)},$ (5.2)

where the equation of state parameter $w$ is crucially related to $p$

$\displaystyle w=-\frac{p-1}{3}.$ (5.3)

To validate this result, we first took into account the case of $p=2$ where it
is possible to obtain an analytic solution. We successfully derive the above
late time behavior of the two-point function. For $p=0$ and $1$ cases, where
analytic solutions are unavailable, we performed numerical simulation about
the time evolution of a spatial two-point function. We found that the
suppression powers in the numerical simulation are perfectly matched to the
theoretical prediction in (5.2).

In this work, we investigated two-point functions in expanding universe in
terms of the distance between two local operators. It would be interesting to
look into further the Fourier transformation of the obtained two-point
functions because they give us more information about the power spectrum in
the FLRW cosmology. We hope to report more interesting results in future
works.

Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korean government(MSIT) (No. NRF-2019R1A2C1006639).

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