Online Circle Packing

TL;DR

This paper introduces an online algorithm for packing circles into a square with an improved area utilization of approximately 35%, surpassing previous bounds, and also addresses packing into rectangles with optimality conditions.

Contribution

The paper presents a novel online circle packing algorithm that achieves a higher area utilization than previous methods and extends to packing into rectangles with proven optimality.

Findings

Achieves an area utilization of 0.350389 of the square's area.

Improves upon the previous best of 0.31416 for online circle packing.

Provides an online algorithm for packing into rectangles with optimality for certain aspect ratios.

Abstract

We consider the online problem of packing circles into a square container. A sequence of circles has to be packed one at a time, without knowledge of the following incoming circles and without moving previously packed circles. We present an algorithm that packs any online sequence of circles with a combined area not larger than 0.350389 0.350389 of the square's area, improving the previous best value of {\pi}/10 \approx 0.31416; even in an offline setting, there is an upper bound of {\pi}/(3 + 2 \sqrt{2}) \approx 0.5390. If only circles with radii of at least 0.026622 are considered, our algorithm achieves the higher value 0.375898. As a byproduct, we give an online algorithm for packing circles into a 1\times b rectangle with b \geq 1. This algorithm is worst case-optimal for b \geq 2.36.

Tables

Table 1. Table 1: Circles are classified into the listed classes. Note that the lower bounds to the circles’ radii is relative to the lane width, e.g., the absolute lower bound for circles inside a small lane is q S ​ w 2 = 0.168261 ​ w 2 subscript 𝑞 𝑆 subscript 𝑤 2 0.168261 subscript 𝑤 2 q_{S}w_{2}=0.168261w_{2} .

Class $i$	(Relative) lower bound $q_i$	Lane width $w_i$
1 (Medium)	$q_1≔q_M≔0.25$	$w_1≔w$
\hdashline2 (Small)	$q_2≔q_S≔0.168261$	$w_2≔2q_{M}w=0.5\cdot w$
\hdashline3 (Tiny)	$q_3≔0.371446$	$w_3≔4q_M{q}_{S}w=0.168261\cdot w$
\hdashline4 (Tiny)	$q_4≔0.190657$	$w_4≔8q_M{q}_S{q}_{3}w\approx 0.125\cdot w$
\hdashline5 (Very tiny)	$q_5≔0.175592$	$w_5≔16q_M{q}_S{q}_3{q}_{4}w\approx 0.047664\cdot w$
\hdashline6 (Very tiny)	$q_6≔0.170699$	$w_6≔32q_M{q}_S{q}_3{q}_4{q}_{5}w\approx 0.016739\cdot w$
\hdashline7 (Very tiny)	$q_7≔0.169078$	$w_7\approx 0.005715\cdot w$
\hdashline8 (Very tiny)	$q_8≔0.168354$	$w_8\approx 0.001932\cdot w$
\hdashline9 (Very tiny)	$q_9≔0.168293$	$w_9\approx 0.000651\cdot w$
\hdashline10 (Very tiny)	$q_{10}≔0.168272$	$w_{10}\approx 0.000219\cdot w$
\hdashline11 (Very tiny)	$q_{11}≔0.168265$	$w_{11}\approx 0.000074\cdot w$
\hdashline12 (Very tiny)	$q_{12}≔0.168263$	$w_{12}\approx 0.000025\cdot w$
\hdashline13 (Very tiny)	$q_{13}≔0.168262$	$w_{13}\approx 0.000008\cdot w$
\hdashline…	$\dots$	$\dots$
\hdashlinek (Very tiny)	$q_k≔0.168262$	$w_k≔2^{k-1}{q}_M{q}_S{q}_3{q}_4\cdot \dots \cdot q_{k-1}w$

Equations

$$\delta :\left(0,\frac{1}{2}\right]\to \mathbb{R}\text{ with }q\mapsto \{\begin{array}{ll}\pi q& 0<q<\frac{1}{3\sqrt{3}}\\ \frac{\pi }{3\sqrt{3}}\approx 0.6046& \frac{1}{3\sqrt{3}}\le q\le \frac{1}{3}\\ \frac{\pi q^2}{\sqrt{4q-1}}& \frac{1}{3}<q\le \frac{1}{2}.\end{array}$$

$$mi{n}_{DSLP}(p_t(L),p_b(L),w,z,{\delta }_{min}):=$$

Full text

As a byproduct, we give an online algorithm for packing circles into a $1\times b$ rectangle with $b\geq 1$. This algorithm is worst case-optimal for $b\geq 2.36$.

1 Introduction

Packing a set of circles into a given container is a natural geometric optimization problem that has attracted considerable attention, both in theory and practice. Some of the many real-world applications are loading a shipping container with pipes of varying diameter [9], packing paper products like paper rolls into one or several containers [8], machine construction of electric wires [18], designing control panels [2], placing radio towers with a maximal coverage while minimizing interference [19], industrial cutting [19], and the study of macro-molecules or crystals [20]. See the survey paper of Castillo, Kampas, and Pintér [2] for an overview of other industrial problems. In many of these scenarios, the circles have to be packed online, i.e., one at a time, without the knowledge of further objects, e.g., when punching out a sequence of shapes from the raw material.

Even in an offline setting, deciding whether a given set of circles fits into a square container is known to be NP-hard [4], which is also known for packing squares into a square [10]. Furthermore, dealing with circles requires dealing with possibly complicated irrational numbers, incurring very serious additional geometric difficulties. This is underlined by the slow development of provably optimal packings of $n$ identical circles into the smallest possible square. In 1965, Schaer [16] gave the optimal solution for $n=7$ and $n=8$ and Schaer and Meir [17] gave the optimal solution for $n=9$. A quarter of a century later, Würtz et al. [3] provided optimal solutions for $10$,$11$,$12$, and $13$ equally sized circles. In 1998, Nurmela and Ostergård [14] provided optimal solutions for $n\leq 27$ circles by making use of computer-aided optimality proofs. Markót and Csendes [12] gave optimal solutions for $n=28,29,30$ also by using computer-assisted proofs within tight tolerance values. Finally, in 2002 optimal solutions were provided for $n\leq 35$ by Locatelli and Raber [11]; at this point, this is still the largest $n$ for which optimal packings are known. The extraordinary challenges of finding densest circle packings are also underlined by a long-standing open conjecture by Erdős and Oler from 1961 [15] regarding optimal packings of $n$ unit circles into an equilateral triangle, which has only been proven up to $n=15$.

These difficulties make it desirable to develop relatively simple criteria for the packability of circles. A natural bound arises from considering the packing density, i.e., the total area of objects compared to the size of the container; the critical packing density $\delta$ is the largest value for which any set of objects of total area at most $\delta$ can be packed into a unit square; see Fig. 1.

In an offline setting, two equally sized circles that fit exactly into the unit square show that $\delta\leq\delta^{*}=\pi/(3+2\sqrt{2})\approx 0.5390$. This is indeed tight: Fekete, Morr and Scheffer [7] gave a worst-case optimal algorithm that packs any instance with combined area at most $\delta^{*}$; see Fig. 2 (left). More recently, Fekete, Keldenich and Scheffer [6] established $0.5$ as the critical packing density of circles in a circular container.

The difficulties of offline circle packing are compounded in an online setting. This is highlighted by the situation for packing squares into a square, which does not encounter the mentioned issues with irrational coordinates. It was shown in 1967 by Moon and Moser [13] that the critical offline density is $0.5$: Refining an approach by Fekete and Hoffmann [5], Brubach [1] established the currently best lower bound for online packing density of $0.4$. This yields the previous best bound for the online packing density of circles into a square: Inscribing circles into bounding boxes yields a value of $\pi/10\approx 0.3142$.

1.1 Our Results

In this paper, we establish new lower bounds for the online packing density of circles into a square and into a rectangle. Note that in the online setting, a packing algorithm has to stop as soon as it cannot pack a circle. We provide three online circle packing results for which we provide constructive proofs, i.e., corresponding algorithms guaranteeing the claimed packing densities.

Theorem 1.1 ()

Let $b\geq 1$. Any online sequence of circles with a total area no larger than \min\Big{(}0.528607\cdot b-0.457876,\frac{\pi}{4}\Big{)} can be packed into the $1\times b$-rectangle $R$. This is worst-case optimal for $b\geq 2.36$.

We use the approach of Theorem 1.1 as a subroutine and obtain the following:

Theorem 1.2 ()

Any online sequence of circles with a total area no larger than $0.350389$ can be packed into the unit square.

If the incoming circles’ radii are lower bounded by $0.026623$, the density guaranteed by the algorithm of Theorem 1.2 improves to $0.375898$.

Theorem 1.3 ()

Any online sequence of circles with radii not smaller than $0.026623$ and with a total area no larger than $0.375898$ can be packed into the unit square.

We describe the algorithm of Theorem 1.1 in Section 2 and the algorithm of the Theorems 1.2 and 1.3 in Section 3.

2 Packing into a Rectangle

In this section, we describe the algorithm, Double-Sided Structured Lane Packing (DSLP), of Theorem 1.1. In particular, DSLP uses a packing strategy called Structured Lane Packing (SLP) and an extended version of SLP as subroutines.

2.1 Preliminaries for the Algorithms

A lane $L\subset\mathbb{R}^{2}$ is an $x$- and $y$-axis-aligned rectangle. The length $\ell(L)$ and the width $w(L)$ of $L$ are the dimensions of $L$ such that $w(L)\leq\ell(L)$. $L$ is horizontal if the length of $L$ is given via the extension of $L$ w.r.t. the $x$-axis. Otherwise, $L$ is vertical. The distance between two circles packed into $L$ is the distance between the orthogonal projections of the circles’ midpoints onto the longer side of $L$. A lane is either open or closed. Initially, each lane is open.

Packing a circle $C$ into a lane $L$ means placing $C$ inside $L$ such that $C$ does not intersect another circle that is already packed into $L$ or into another lane. A (packing) strategy for a lane $L$ is a set of rules that describe how a circle has to be packed into $L$. The (packing) orientation of a strategy for a horizontal lane is either rightwards or leftwards and the (packing) orientation of a strategy for a vertical lane is either downwards or updwards.

Let $w$ be the width of $L$. Depending on the radius $r$ of the current circle $C$, we say: $C$ is medium (Class 1) if $w>r\geq\frac{w}{4}$, $C$ is small if $\frac{w}{4}>r\geq 0.0841305w$ (Class 2), $C$ is tiny (Class 3 or 4) if $0.0841305w>r\geq 0.023832125w$, and $C$ is very tiny if $0.023832125w>r$ (Classes 5,6, …). For a more refined classification of $r$, we refer to Section 2.3. The general idea is to reach a certain density within a lane by packing only relatively equally sized circles into a lane with SLP.

For the rest of Section 2, for $0<w\leq b$, let $L$ be a horizontal $w\times b$ lane.

2.2 Structured Lane Packing (SLP) – The Standard Version

Rightwards Structured Lane Packing (SLP) packs circles into $L$ alternating touching the bottom and the top side of $L$ from left to right, see Fig. 3 (right).

In particular, we pack a circle $C$ into $L$ as far as possible to the left while guaranteeing: (1) $C$ does not overlap a vertical lane packed into $L$, see Section 2.3 for details111Requiring that $C$ does not overlap a vertical lane placed inside $L$ is only important for the extension of SLP (see Section 2.3), because the standard version of SLP does not place vertical lanes inside $L$.. (2) The distance between $C$ and the circle $C^{\prime}$ packed last into $L$ is at least $\min\{r,r^{\prime}\}$ where $r,r^{\prime}$ are the radii of $C,C^{\prime}$, see Fig. 3 (right).

Leftwards Structured Lane Packing packs circles by alternatingly touching the bottom and the top side of $L$ from right to left. Correspondingly, upwards and downwards Structured Lane Packing packs circles alternatingly touching the left and the right side of $L$ from bottom to top and from top to bottom.

2.3 Extension of SLP – Filling Gaps by (Very) Tiny Circles

Now consider packing medium circles with SLP. We extend SLP for placing tiny and very tiny circles within the packing strategy, see Fig. 4. Note that small circles are not considered for the moment, such that (very) tiny circles are relatevily small compared to the medium ones. In particular, if the current circle $C$ is medium, we apply the standard version of SLP, as described in Section 2.2. If $C$ is (very) tiny, we pack $C$ into a vertical lane inside $L$, as described next.

We pack (very) tiny circles into vertical lanes inside $L$, see Fig. 4. The vertical lanes are placed inside blocks that are the rectangles induced by vertical lines touching medium circles already packed into $L$, see Fig. 3 (right).

Blocks that include two halves of medium circles are called dense blocks, while blocks that include one half of a medium circle are called sparse blocks. The area of $L$ that is neither covered by a dense block or a sparse block is called a free block. Packing a vertical lane $L^{\prime}$ into a sparse block $B$ means placing $L^{\prime}$ inside $B$ as far as possible to the left, such that $L^{\prime}$ does not overlap another vertical lane already packed into $B$. Packing a vertical lane $L^{\prime}$ into $L$ means placing $L^{\prime}$ inside $L$ as far as possible to the left, such that $L^{\prime}$ does neither overlap another vertical lane packed into $L$, a dense block of $L$, or a sparse block of $L$.

We extend our classification of circles by defining classes $i$ of lane widths $w_{i}$ and relative lower bounds $q_{i}$ for the circles’ radii as described in Table 1. This means a circle with radius $r$ belongs to class 1 if $0.5w\geq r>q_{1}w_{1}$ and to class $i$ if $q_{i-1}w_{i-1}\geq r>q_{i}w_{i}$, for $i\geq 2$. Only circles of class $i$ are allowed to be packed into lanes of class $i$.

A sparse block is either free, reserved for class $3$, reserved for class $4$, reserved for all classes $i\geq 5$, or closed. Initially, each sparse block is free.

We use SLP in order to pack a circle $C$ of class $i\geq 3$, into a vertical lane $L_{i}\subset L$ of class $i$ and width $w_{i}$ by applying the Steps 1-5 in increasing order as described below. When one of the five steps achieves that $C$ is packed into a vertical lane $L_{i}$, the approach stops and returns successful.

• •

Step (1): If there is no open vertical lane $L_{i}\subset L$ of class $i$ go to Step 2. Assume there is an open vertical lane $L_{i}$ of class $i$. If $C$ can be packed into $L_{i}$, we pack $C$ into $L_{i}$. Else, we declare $L_{i}$ to be closed.

• •

Step (2): We close all sparse blocks $B$ that are free or reserved for class $i$ in which a vertical lane of class $i$ cannot be packed into $B$.

• •

Step (3): If there is an open sparse block $B$ that is free or reserved for class $i$ and a vertical lane of class $i$ can be packed into $B$:

• –

(3.1): We pack a vertical lane $L_{i}\subset L$ of class $i$ into $B$. If the circle half that is included in $B$ touches the bottom of $L$, we apply downwards SLP to $L_{i}$. Otherwise, we apply upwards SLP to $L_{i}$.

• –

(3.2): If $B$ is free and $i\in\{3,4\}$, we reserve $B$ for class $i$. If $B$ is free and $i\geq 5$, we reserve $B$ for all classes $i\geq 5$.

• –

(3.3): We pack $C$ into $L_{i}$.

• •

Step (4): If a vertical lane of class $i$ can be packed into $L$:

• –

(4.1): We pack a vertical lane $L_{i}$ of class $i$ into $L$ and apply upwards SLP to $L_{i}$.

• –

(4.2): We pack $C$ into $L_{i}$

• •

Step (5): We declare $L$ to be closed and return failed.

2.4 Double-Sided Structured Lane Packing (DSLP)

We use SLP as a subroutine in order to define our packing strategy Double-Sided Structured Lane Packing (DSLP) of Theorem 1.1. In particular, additionally to $L$, we consider two small lanes $L^{1},L^{2}$ that partition $L$, see Fig. 5.

Rightwards Double-Sided Structured Lane Packing (DSLP) applies the extended version of rightwards SLP to $L$ and leftwards SLP to $L^{1},L^{2}$. If the current circle $C$ is medium or (very) tiny, we pack $C$ into $L$. If $C$ is small, we pack $C$ into that lane of $L^{1},L^{2}$, resulting in a smaller packing length.

Leftwards DSLP is defined analogously, such that the extended version of leftwards SLP is applied to $L$ and rightwards SLP to $L^{1},L^{2}$. Correspondingly, upwards and downwards DSLP are defined for vertical lanes.

3 Packing into the Unit Square

We extend our circle classification by the class [math] of large circles and define a relative lower bound $q_{0}:=\frac{w}{2}$ and the lane width of corresponding large lanes as $w_{0}:=1-w$.

We set $w$ to $0.288480$ and $0.277927$ for Theorem 1.2 respectively Theorem 1.3. In order to pack large circles, we use another packing strategy called Tight Lane Packing (TLP) defined as SLP, but without restrictions (1) and (2), see Fig. 3.

We cover the unit square by the union of one large lane $L_{0}$ and four medium lanes $L_{1},\dots,L_{k}$ for $k=4$, see Fig. 6. We apply TLP to $L_{0}$ and DSLP to $L_{1},\dots,L_{4}$. The applied orientations for $L_{0},L_{1},L_{2}$ are leftwards, rightwards, and downwards. For $i=3,4$, the orientation for $L_{i}$ is chosen such that the first circle packed into $L_{i}$ is placed adjacent to the bottom side of $L_{i}$.

If the current circle to be packed is large, we pack $C$ into the large lane $L_{0}$ and stop if $C$ does not fit in $L_{0}$. Otherwise, in increasing order we try to pack $C$ into $L_{1},\dots,L_{4}$.

4 Analysis of the Algorithms

In this section we sketch the analysis of our approaches and refer to the appendix for full details. First, we analyze the packing density guaranteed by DSLP. Based on that, we prove our main results Theorems 1.1, 1.2, and 1.3.

4.1 Analysis of SLP

In this section, we provide a framework for analyzing the packing density guaranteed by DSLP for a horizontal lane $L$ of width $w$. It is important to note that this framework and its analysis in this subsection deals with the packing of only one class into a lane.

We introduce some definitions. The packing length $p(L)$ is the maximal difference of $x$-coordinates of points from circles packed into $L$. The circle-free length $f(L)$ of $L$ is defined as $\ell(L)-p(L)$. We denote the total area of a region $R\subset\mathbb{R}^{2}$ by $area(R)$ and the area of an $a\times b$-rectangle by $\mathcal{R}(a,b)$. Furthermore, we denote the area of a semicircle for a given radius $r$ by $\mathcal{\mathcal{H}}(r)\coloneqq\frac{\pi}{2}r^{2}$. The total area of the circles packed into $R$ is called occupied area denoted by $occ(R)$. Finally, the density $den(R)$ is defined by $occ(R)/area(R)$.

In order to apply our analysis for different classes of lanes, i.e., different lower bounds, we consider a general (relative) lower bound $q$ for the radii of circles allowed to be packed into $L$ with $0<q\leq 1/2$. The following lemma deduces a lower bound for the density of dense blocks depending on $q$.

Lemma 1 ()

Consider a dense block $D$ containing two semicircles of radii $r_{1}$ and $r_{2}$ such that $0<qw\leq r_{1},r_{2}\leq 1/2w$. Then $den(D)$ is lower-bounded by

[TABLE]

We continue with the analysis of sparse blocks. Sparse blocks have a minimum length $qw$. Lemma 2 states a lower bound for the occupied area of sparse blocks. This lower bound consists of a constant summand and a summand that is linear with respect to the actual length.

Lemma 2 ()

Given a density bound $\delta_{min}\leq\delta(q)$ for dense blocks. Let $S$ be a sparse block and $z$ be the lower bound for $\ell(S)$ with $\ell(S)\geq z\geq qw$. Then $occ(S)\geq\mathcal{R}(\ell(S)-z,w)\cdot\delta_{min}+\mathcal{H}(z)$.

The occupied area of a sparse block is at least a semicircle of a smallest possible circle plus the remaining length multiplied by the lane width and by the minimal density $\delta_{min}$ of dense blocks. This composition is shown in Fig. 8.

Next, we combine the results of Lemma 1 and Lemma 2. We define the term {min_{SLP}}\big{(}p,w,z,\delta_{min}\big{)}\coloneqq\mathcal{R}(p-2z,w)\cdot\delta_{min}+2\cdot\mathcal{H}(z) for some $p,w,z,\delta_{min}>0$ and state the following (see also Fig. 8 (4)).

Lemma 3 ()

Given a lane $L$ packed by SLP, a lower bound $q$, and a density bound $\delta_{min}\leq\delta(q)$ for dense blocks. Let $w$ be the width of $L$. The occupied area in $L$ is lower-bounded by {min_{SLP}}\big{(}p(L),w,qw,\delta_{min}\big{)}.

4.2 Analysis of DSLP

Let $L$ be a horizontal lane packed by DSLP. We define $p_{t}(L)$ ($p_{b}(L)$) as the sum of the packing lengths of the packing inside $L$ and the length of the packing inside the top (bottom) small lane inside $L$. Furthermore, we define $f_{t}(L):=\ell(L)-p_{t}(L)$ and $f_{b}(L):=\ell(L)-p_{b}(L)$, see Fig. 5.

By construction, vertical dense blocks packed into $L$ have a density of at least $\delta(q_{2})$. In fact, the definitions of circle sizes for all classes $i\geq 2$ ensure the common density bound $\hat{\delta}:=\delta(q_{2})$ for dense blocks.

We consider mixed dense blocks, that were defined as sparse blocks of $L$ in which vertical lanes are packed, also as dense blocks and extend the lower bound $den(D)\geq\hat{\delta}$ to all kinds of dense blocks by the following Lemma.

Lemma 4 ()

Let $D$ be a dense block of $L$. Assume all vertical lanes packed into $D$ to be closed. Then $den(D)\geq\hat{\delta}$.

As some vertical lanes may not be closed, we upper bound the error $\mathcal{O}(L)$ that we make by assuming that all vertical lanes $L_{1},\dots,L_{n}\subset L$ are closed.

Lemma 5 ()

$\mathcal{O}(L_{1}\cup\ldots\cup L_{n})<0.213297\cdot w^{2}$.

We lower bound the occupied area inside $L$ by using the following term:

[TABLE]

where $z$ denotes the minimal radius for the circles, i.e., $z:=q_{2}w_{2}$.

Applying Lemma 4 and Lemma 2 separately to $L$, $L^{1}$, and $L^{2}$, analogous to the combination of Lemma 1 and Lemma 2 in the last subsection, and estimating the error $\mathcal{O}(L)$ with Lemma 5, yields lower bounds for the occupied areas of $L$, $L^{1}$, and $L^{2}$. Fig. 9 separately illustrates the lower bounds for the occupied areas of $L$, $L^{1}$, and $L^{2}$ for two example packings and Lemma 6 states the result.

Lemma 6 ()

min_{DSLP}\big{(}p_{t}(L),p_{b}(L),w,z,\hat{\delta}\big{)}-\mathcal{O}(L)\geq occ(L).

4.3 Analysis of Packing Circles into a Rectangle

Given a $1\times b$ rectangle $R$, we apply DSLP for packing the input circles into $R$.

See 1.1

The lower bound for the occupied area implied by Lemma 6 is equal to \big{(}b-\frac{3}{4}-q_{2})\cdot\hat{\delta}+\frac{\pi}{16}+\frac{\pi}{2}(q_{2})^{2}-0.213297. This is lower bounded by $\frac{\pi}{4}$ for $b\geq 2.36$. Hence, the online sequence consisting of one circle with a radius of $\frac{1}{2}+\epsilon$ and resulting total area of $\frac{\pi}{4}+\epsilon$ is a worst case online sequence for $b\geq 2.36$, see Fig. 10. This concludes the proof of Theorem 1.1.

4.4 Analysis of Packing Circles into the Unit Square

In this section, we analyze the packing density of our overall approach for online packing circles into the unit square. In order to prove Theorem 1.3, i.e., a lower bound for the achieved packing density, we show that if there is an overlap or if there is no space in the last lane, then the occupied area must be at least this lower bound. Our analysis distinguishes six different higher-level cases of where and how the overlap can happen, see Fig. 11 left.

For each of the six cases and its subcases, we explicitly give a density function, providing the guaranteed packing density depending on the choice of $w$, see Fig. 11 right. The shown functions are constructed for the case of no (very) tiny circles with an alternative $q_{2}=0.191578$, which was chosen numerically in order to find a high provable density. The $w$ with the highest guaranteed packing density of $0.375898$ is $w=0.277927$. This concludes the proof of Theorem 1.3.

See 1.3

With the same idea but with $w=0.288480$ and all circles classes, especially with classes $i\geq 2$ as defined in Table 1, we prove Theorem 1.2.

See 1.2

5 Conclusion

We provided online algorithms for packing circles into a square and a rectangle. For the case of a rectangular container, we guarantee a packing density which is worst-case optimal for rectangles with a skew of at least $2.36$. For the case of a square container, we provide a packing density of $0.350389$ which we improved to $0.375898$ if the radii of incoming circles are lower-bounded by $0.026622$.