# A Hybrid of Differential Evolution and Genetic Algorithm for the Multiple Geographical Feature Label Placement Problem

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## Abstract

**:**

## 1. Introduction

## 2. Candidate-Position Generation and Quality Evaluation Model

#### 2.1. Candidate-Position Generation

#### 2.2. Quality Metric

## 3. Label Placement Model of DDEGA

#### 3.1. A Hybrid Algorithm of Discrete Differential Evolution and Genetic Algorithm

#### 3.2. DDEGA Process

#### 3.2.1. Initial Population Generation and Coding

#### 3.2.2. Fitness Function

#### 3.2.3. Selection Operation

#### 3.2.4. Variation and Crossover

## 4. Experimental Results

#### 4.1. Label Placement Experiment Using DDEGA

#### 4.2. Experimental Analysis

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A candidate-position model of a point feature; (

**b**) a candidate-position model of a line feature; and (

**c**) a candidate-position model of an area feature.

**Figure 4.**The progression of label placement with the DDEGA algorithm: (

**a**) the initial random labeling of a generated map; (

**b**) the map with label placement after 50 iterations of the algorithm; and (

**c**) the final labeling of the map.

**Figure 5.**The map of Washington State with labels generated by four approaches: (

**a**) GA; (

**b**) DDE; (

**c**) DDEGA; and (

**d**) Maplex.

**Figure 6.**Map of near Buena Vista Park in San Francisco with labels generated by four approaches: (

**a**) GA; (

**b**) DDE; (

**c**) DDEGA; and (

**d**) Maplex.

**Figure 7.**(

**a**) Convergence of each algorithm for the first case study; and (

**b**) convergence of each algorithm for the second case study.

Notation | Description |
---|---|

$N$ | Total number of labels of the map |

${S}_{i1}$ | Score of label conflict factor for the ith label, $i\in \left[1,N\right]$ |

${S}_{1}$ | Score of label conflict factor of the map |

${C}_{i1}$ | Number of overlaps between the $i$th label and the other labels |

${S}_{i2}$ | Score of label-feature conflict factor for the $i$th label |

${S}_{2}$ | Score of label-feature conflict factor of the map |

${C}_{i2}$ | Number of overlaps between the $i$th label and other features |

$C$ | Number of the label rectangles on the map |

${S}_{i3}$ | Score of label non-ambiguity factor for the $i$th label |

${S}_{3}$ | Score of label non-ambiguity factor of the map |

${D}_{i}$ | Minimum Euclidean distance between the $i$th label rectangle and the center of the line feature or the center of the skeleton line of the area feature |

${D}_{max}$ | The maximum Euclidean distance between the label and the center of the line feature or the center of the skeleton line of the area feature |

${S}_{i4}$ | Score of label priority factor for the $i$th label |

${S}_{4}$ | Score of label priority factor on the map |

$\alpha $ | The angle between the line of the center of label rectangle and point feature with the horizontal line |

${S}_{i}$ | Score of quality evaluation for the $i$th label |

${S}_{j}$ | Score of the $j$th factor, $j\in \left[1,2,3,4\right]$ |

${W}_{j}$ | Weight of each impact factor |

$S$ | Score of quality evaluation for the cartographic label placement |

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**MDPI and ACS Style**

Lu, F.; Deng, J.; Li, S.; Deng, H.
A Hybrid of Differential Evolution and Genetic Algorithm for the Multiple Geographical Feature Label Placement Problem. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 237.
https://doi.org/10.3390/ijgi8050237

**AMA Style**

Lu F, Deng J, Li S, Deng H.
A Hybrid of Differential Evolution and Genetic Algorithm for the Multiple Geographical Feature Label Placement Problem. *ISPRS International Journal of Geo-Information*. 2019; 8(5):237.
https://doi.org/10.3390/ijgi8050237

**Chicago/Turabian Style**

Lu, Fuyu, Jiqiu Deng, Shiyu Li, and Hao Deng.
2019. "A Hybrid of Differential Evolution and Genetic Algorithm for the Multiple Geographical Feature Label Placement Problem" *ISPRS International Journal of Geo-Information* 8, no. 5: 237.
https://doi.org/10.3390/ijgi8050237