# Correction: An Efficient Grid-Based K-Prototypes Algorithm for Sustainable Decision Making Using Spatial Objects. Sustainability 2018, 10, 2614

^{1}

^{2}

^{3}

^{*}

**Replacing:**

**Definition**

**2.**

_{i}and a cluster center c

_{j}for numeric attributes, denoted d

_{min}(g

_{i}, c

_{j}), is:

**with:**

**Definition**

**2.**

_{i}and a cluster center c

_{j}for numeric attributes, denoted d

_{min}(g

_{i}, c

_{j}), is:

**Replacing:**

**Definition**

**3.**

_{i}and a cluster center c

_{j}for numeric attributes, denoted d

_{max}(g

_{i}, c

_{j}), is:

**with:**

**Replacing:**

- 1:
- C[ ]← Ø // k cluster centers
- 2:
- Randomly choosing k object, and assigning it to C.
- 3:
**while**IsConverged()**do**- 4:
- dmin[], dmax[] ← Calc(g, C)
- 5:
- dminmax ← min(dmax[])
- 6:
**for each**cell g**in**G

- 1:
- C[ ]← Ø // k cluster centers
- 2:
- Randomly choosing k object, and assigning it to C.
- 3:
**while**IsConverged()**do**- 4:
**for each**cell g**in**G- 5:
- dmin[ ], dmax[ ] ← Calc(g, C)
- 6:
- dminmax ← min(dmax[ ])

**Replacing:**

- 1:
- C[k]← Ø // k cluster center
- 2:
- Randomly choosing k object, and assigning it to C.
- 3:
**while**IsConverged()**do**- 4:
- dmin[], dmax[] ← Calc(g, C)
- 5:
- dminmax ← min(dmax[])
- 6:
**for each**cell g**in**G

**with:**

- 1:
- C[ ]← Ø // k cluster centers
- 2:
- Randomly choosing k object, and assigning it to C.
- 3:
**while**IsConverged()**do**- 4:
**for each**cell g**in**G- 5:
- dmin[ ], dmax[ ] ← Calc(g, C)
- 6:
- dminmax ← min(dmax[ ])

**Replacing:**

_{c}) to store the frequency of categorical data in each cluster and O(kd) to store cluster centers, where m

_{c}the number of categorical attributes. Additionally, KBP requires O(gm

_{c}) to store the frequency of categorical data in each cell, where g is the number of cells. Therefore, the space complexities of KCP and KBP are O(nd + gd + km

_{c}+ kd) and O(nd + gd + km

_{c}+ kd + gm

_{c}), respectively.

**with:**

_{r}), where g is the number of cells, to store the start point vector S and the end point vector T of each cell, O(ktm

_{c}), where t is the number of categorical data, to store the frequency of categorical data in each cluster and O(kd) to store cluster centers. KBP requires O(gtm

_{c}) to store the bitmap index in each cell, where g is the number of cells. Therefore, the space complexities of KCP and KBP are O(nd + gm

_{r}+ ktm

_{c}+ kd) and O(nd + gm

_{r}+ ktm

_{c}+ kd + gtm

_{c}), respectively.

**Replacing:**

**Figure 12.**Effect of the number of clusters (numeric data and categorical data are on uniform distribution).

**with:**

**Figure 12.**Effect of the number of clusters (numeric data are uniformly distributed and the distribution of categorical data is skewed).

## Reference

- Jang, H.-J.; Kim, B.; Kim, J.; Jung, S.-Y. An Efficient Grid-Based K-Prototypes Algorithm for Sustainable Decision-Making on Spatial Objects. Sustainability
**2018**, 10, 2614. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jang, H.-J.; Kim, B.; Kim, J.; Jung, S.-Y.
Correction: An Efficient Grid-Based K-Prototypes Algorithm for Sustainable Decision Making Using Spatial Objects. Sustainability 2018, *10*, 2614. *Sustainability* **2019**, *11*, 1801.
https://doi.org/10.3390/su11061801

**AMA Style**

Jang H-J, Kim B, Kim J, Jung S-Y.
Correction: An Efficient Grid-Based K-Prototypes Algorithm for Sustainable Decision Making Using Spatial Objects. Sustainability 2018, *10*, 2614. *Sustainability*. 2019; 11(6):1801.
https://doi.org/10.3390/su11061801

**Chicago/Turabian Style**

Jang, Hong-Jun, Byoungwook Kim, Jongwan Kim, and Soon-Young Jung.
2019. "Correction: An Efficient Grid-Based K-Prototypes Algorithm for Sustainable Decision Making Using Spatial Objects. Sustainability 2018, *10*, 2614" *Sustainability* 11, no. 6: 1801.
https://doi.org/10.3390/su11061801