# How Many Lions Are Needed to Clear a Grid?

^{*}

## Abstract

**:**

## 1. Introduction

- belong to $W\left(t\right)$ and are not visited by a lion at time $t+1$, or
- are not occupied by a lion at time $t+1$, and have a direct neighbor q in $W\left(t\right)$ that is not visited at time $t+1$ by a lion coming directly from p.

## 2. Definitions

- belong to $W\left(t\right)$, or
- have a direct neighbor q in $W\left(t\right)$ that is not visited at time $t+1$ by a lion coming directly from p.

**Clearing question in ${G}_{n}^{d}$**. Given k lion paths over $\{0,\dots ,T\}$ in ${G}_{n}^{d}$, does there exist a time $t\in \{0,\dots ,T\}$ such that $W\left(t\right)=\varnothing $? We say that ${G}_{n}^{d}$ has clearing number ${k}_{d}\left(n\right)$, if

- for arbitrary large T and arbitrary ${k}_{d}\left(n\right)-1$ lion paths, we have $W\left(t\right)\ne \varnothing $ for all $t\in \{0,\dots ,T\}$,
**and** - there exist ${k}_{d}\left(n\right)$ lion paths and a time t such that $W\left(t\right)=\varnothing $.

**Figure 1.**$k=3$ lions try to decontaminate a 2-dimensional $n\times n$-grid, where $n=4$. (In this illustration the vertices are cells, and edges exist between neighbor cells.)

## 3. Results in Dimension 2

**Theorem 1.**

**Lemma 1.**

**Lemma 2.**

- are occupied by a lion at time $t+1$, or
- are belonging to $\partial \mathcal{C}\left(t\right)$ and are protected from recontamination in the following step by leaving lions.

**Lemma 3.**

**Lemma 4.**

**Lemma 5.**

**Figure 4.**(a) There exists a column satisfying ${n}_{2}\left({i}^{\prime}\right)<n-{i}^{\prime}$. (b) Column i contains at least ${n}_{2}\left(i\right)-{n}_{2}(i+1)$ boundary vertices.

## 4. Results on d-Dimensional Grids

#### 4.1. An upper bound

**Figure 5.**Eight lions (rather than ${n}^{d-1}=9$) suffice to clear the $3\times 3\times 3$-grid. If two lions occupy the same vertex at the same time, this is indicated by two crosses of different orientation.

**Lemma 6.**

**Lemma 7.**

**Lemma 8.**

#### 4.2. A lower bound

**Lemma 9.**

**Lemma 10.**

#### 4.3. An asymptotic estimate

**Lemma 11.**

**Theorem 2.**

## 5. Flying lions in the 2-dimensional grid

**Figure 6.**Flying lions gain one row. The vertical black line segments partition the grid with ratio 2:1. The vertices corresponding to the black segments are occupied by lions, hence no contamination crosses them. Red circles are contaminated vertices, green boxes are cleared vertices and blue crosses are lions.

**Lemma 12.**

**Lemma 13.**

**Theorem 3.**

## Acknowledgements

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## Share and Cite

**MDPI and ACS Style**

Berger, F.; Gilbers, A.; Grüne, A.; Klein, R.
How Many Lions Are Needed to Clear a Grid? *Algorithms* **2009**, *2*, 1069-1086.
https://doi.org/10.3390/a2031069

**AMA Style**

Berger F, Gilbers A, Grüne A, Klein R.
How Many Lions Are Needed to Clear a Grid? *Algorithms*. 2009; 2(3):1069-1086.
https://doi.org/10.3390/a2031069

**Chicago/Turabian Style**

Berger, Florian, Alexander Gilbers, Ansgar Grüne, and Rolf Klein.
2009. "How Many Lions Are Needed to Clear a Grid?" *Algorithms* 2, no. 3: 1069-1086.
https://doi.org/10.3390/a2031069