# Maximum Coverage by k Lines

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Problem**

**1**

**.**Given a set of n disks in the plane and a positive integer k, find k lines that together intersect the maximum number of input disks.

**Problem**

**2**

**.**Given a set of n disks in the plane and a positive integer k, find k parallel lines that together intersect the maximum number of input disks.

**Problem**

**3**

**.**Given a set of n disks in the plane and a positive integer k, find k lines that together intersect the maximum number of input disks and pass through a common point.

#### Related Work

## 2. Preliminaries and Notations

#### 2.1. Partial Interval Hitting Set

#### 2.2. Notations

## 3. Maximum Coverage by k Lines

#### 3.1. A First Algorithm

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 3.2. Maximum Coverage by One Line

**Theorem**

**1.**

**Proof.**

#### 3.3. Maximum Coverage by $k\ge 2$ Lines

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

## 4. Maximum Coverage by k Parallel Lines

**Observation**

**1.**

#### Improvements for $k=2$

**Lemma**

**4.**

- Let $T=\lfloor logn\rfloor $. Partition the disks into $\lceil n/T\rceil $ disjoint groups, each consisting of T disks, except possibly the last group containing less than T disks. Let $\mathcal{G}=\{{G}_{1},\dots ,{G}_{\lceil n/T\rceil}\}$ be the set of groups.
- For every subset of $\mathcal{S}\subseteq \mathcal{G}$ of size at most two,
- (a)
- Let ${\mathcal{D}}_{\mathcal{S}}=\bigcup \mathcal{S}$. For each disk $D\in {\mathcal{D}}_{\mathcal{S}}$, compute the sorted list of the other disks in $\mathcal{D}$ intersected by the tangent line rotating around D in $O(nlogn)$ time. It takes $O(Tnlogn)$ time for all disks in ${\mathcal{D}}_{\mathcal{S}}$.
- (b)
- For a fixed pair ${D}_{1},{D}_{2}\in {\mathcal{D}}_{\mathcal{S}}$, we can find the maximum number of disks intersected by ${\ell}_{1}$, ${\ell}_{2}$ in $O\left(n\right)$ time as the ordering of the events has been precomputed. So over all the pairs ${D}_{1},{D}_{2}\in {\mathcal{D}}_{\mathcal{S}}$, it takes $O\left(n{T}^{2}\right)$ time to find optimal lines ${\ell}_{1}$, ${\ell}_{2}$.

**Theorem**

**3.**

## 5. Maximum Coverage by k Lines through a Point

**Lemma**

**5.**

**Proof.**

**Theorem**

**4.**

## 6. On the Partial Interval Hitting Set Problem

**Lemma**

**6.**

**Proof.**

- ($\mathsf{LL}$)
- Two points p and q, which are both left endpoints of intervals, collide or separate. (See Figure 3a).
- ($\mathsf{LR}$)
- Two points p and q, where p is the left endpoint of an interval and q is the right endpoint of another interval, satisfy $p\left({t}_{0}^{-}\right)<q\left({t}_{0}^{-}\right)$ and collide at ${t}_{0}$, or p and q separate at ${t}_{0}$ and $p\left({t}_{0}^{+}\right)<q\left({t}_{0}^{+}\right)$. (See Figure 3b).
- ($\mathsf{RL}$)
- Two points p and q, where p is the right endpoint of an interval and q is the left endpoint of another interval, satisfy $p\left({t}_{0}^{-}\right)<q\left({t}_{0}^{-}\right)$ and collide at ${t}_{0}$, or p and q separate at ${t}_{0}$ and $p\left({t}_{0}^{+}\right)<q\left({t}_{0}^{+}\right)$. (See Figure 3c).
- ($\mathsf{R}$)
- Two points p and q, which are both right endpoints of intervals, collide or separate. (See Figure 3d).

**Lemma**

**7.**

**Proof.**

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**Left**) a set of seven disks in the plane, where and $k=2$, and $\overline{\mathcal{D}}=\{{D}_{1},{D}_{2}\}$. (

**Right**) The union of slabs induced by $({D}_{1},{D}^{\prime})$ and $({D}_{2},{D}^{\prime})$.

**Figure 3.**Types of events where two interval endpoints p and q collide or separate. (

**a**) Type $\mathsf{LL}$ (

**b**) Type $\mathsf{LR}$ (

**c**) Type $\mathsf{RL}$ (

**d**) Type $\mathsf{RR}$.

**Table 1.**The running times of our algorithms for Problem 1, 2 and 3 for a constant k. Space usage is $O\left(n\right)$ if not explicitly stated.

Problem 1 | Problem 2 | Problem 3 | |
---|---|---|---|

$k=1$ | $O\left({n}^{2}\right)$ time, $O\left({n}^{2}\right)$ space | ||

$k=2$ | $O({n}^{3}logn)$ time | $O({n}^{3}logn)$ time | $O({n}^{3}logn)$ time |

$O\left({n}^{3}\right)$ time, $O(nlogn)$ space | |||

$k=3$ | $O\left({n}^{3k/2}\right)$ time | $O\left({n}^{4}\right)$ time | $O({n}^{5}logn)$ time |

$O\left({n}^{5}\right)$ time, $O\left({n}^{2}\right)$ space | |||

$k\ge 4$ | $O\left({n}^{3k/2}\right)$ time | $O\left({n}^{4}\right)$ time | $O\left({n}^{6}\right)$ time |

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**MDPI and ACS Style**

Chung, C.; Vigneron, A.; Ahn, H.-K.
Maximum Coverage by *k* Lines. *Symmetry* **2024**, *16*, 206.
https://doi.org/10.3390/sym16020206

**AMA Style**

Chung C, Vigneron A, Ahn H-K.
Maximum Coverage by *k* Lines. *Symmetry*. 2024; 16(2):206.
https://doi.org/10.3390/sym16020206

**Chicago/Turabian Style**

Chung, Chaeyoon, Antoine Vigneron, and Hee-Kap Ahn.
2024. "Maximum Coverage by *k* Lines" *Symmetry* 16, no. 2: 206.
https://doi.org/10.3390/sym16020206