# Outdoors Evacuation Routes Algorithm Using Cellular Automata and Graph Theory for Uphills and Downhills

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Geographical Information

- Latitude $\left(lat\right)$, Longitude $\left(long\right)$, and Elevation $\left(ele\right)$: The geographical coordinates used refer to Google Maps with the cellular automata. A precision of eight digits is used so that the evaluation points are not so distant in the cellular automata.
- $({X}_{i},{Y}_{i})$: Positions in the matrix are used for the user’s movement in the cellular automata, i.e., positions are used to create evacuation routes.
- States S: This is a finite non-empty set of states used in cellular automata. Each state changes depending on the user’s location and if it has been evaluated in the route calculation.

#### 2.2. Cellular Automata

- Network Infrastructure: The cells in a CA are usually rectangular or square, although they can have other topologies, but always with a finite-dimensional. This CA case is 256 × 256.
- State S: There is a finite set of states in a CA; each cell must be in a defined state. The set of states on the different cells will help us to identify the possibility that there is a path through the cells.
- ${S}_{N}$, ${S}_{NW}$, ${S}_{NE}$, ${S}_{S}$, ${S}_{SW}$, ${S}_{SE}$, ${S}_{W}$, and ${S}_{E}$: They are labels of the cellular automaton states to show the user the direction of their evacuation route.
- Cell neighborhood: Each cell will have a neighborhood, usually defined by the developer; for two-dimensional structures, Moore and von Neumann schemes are the most used. Each cell’s status and those of neighboring cells within the neighborhood will depend on the transition rules that will be applied in each case [30]. A general scheme is shown in Figure 2.
- $({X}_{i},{Y}_{i})$: Cell position within the cellular automata. The position $\left(i\right)$ is considered as the initial position, and the surrounding positions as its neighborhood.
- Transition rules: Each cell in the CA has applied a rule, which determines its state based on its state in time $(T-1)$ and on the state of its neighbors in time T.

- Empty Zone (0): Initial state of the cell.
- Evaluated Zone (1): If the cell is previously evaluated but is not considered acceptable in the final route.
- Accepted Zone (2): When the cells around a cell (${X}_{i},{Y}_{i}$) are analyzed and it presents a lower slope with respect to the initial cell (considering if the person is walking downhill).

#### 2.3. Evacuation Routes

- ${P}_{start}=({X}_{start},{Y}_{start})$: Initial starting point (when the user starts walking).
- ${P}_{save}=({X}_{save},{Y}_{save})$: These points are the final position where the user is going; they are used to know the graph’s optimal evacuation route.
- $N,NW,$$NE,S,$$SW,$$SE,W\mathrm{and}E$: Labels on the edges to know the direction of evacuation routes.

Algorithm 1: Evacuation Routes Algorithm |

## 3. Case Study

## 4. Results and Discussion

**PATH:**- ${P}_{start}=(150,180)$ -> (149,179) -> (148,178) -> (147,179) -> (146,180) -> (145,181) -> (144,182) -> (143,183) -> (142,184) -> (141,185) -> (140,186) -> (138,187) -> (138,188) -> (137,189) -> (136,190) -> (135,191) -> ${P}_{save\_1}=(134,192)$.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Evacuation Routes Flowchart

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Point | Latitude | Longitude | Elevation |
---|---|---|---|

A | 40.13897826 | $-8.69440953$ | 7.77376699 |

B | 40.13802564 | $-8.69434342$ | 16.68270111 |

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**MDPI and ACS Style**

Velasquez, W.; Alvarez-Alvarado, M.S. Outdoors Evacuation Routes Algorithm Using Cellular Automata and Graph Theory for Uphills and Downhills. *Sustainability* **2021**, *13*, 4731.
https://doi.org/10.3390/su13094731

**AMA Style**

Velasquez W, Alvarez-Alvarado MS. Outdoors Evacuation Routes Algorithm Using Cellular Automata and Graph Theory for Uphills and Downhills. *Sustainability*. 2021; 13(9):4731.
https://doi.org/10.3390/su13094731

**Chicago/Turabian Style**

Velasquez, Washington, and Manuel S. Alvarez-Alvarado. 2021. "Outdoors Evacuation Routes Algorithm Using Cellular Automata and Graph Theory for Uphills and Downhills" *Sustainability* 13, no. 9: 4731.
https://doi.org/10.3390/su13094731