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

Hong-Jun Jang 1, Byoungwook Kim 2 , Jongwan Kim 3 and Soon-Young Jung 1,* 1 Department of Computer Science and Engineering, Korea University, Seoul 02841, Korea; hongjunjang@korea.ac.kr 2 Department of Computer Engineering, Dongguk University, Gyeongju 38066, Korea; bwkim@dongguk.ac.kr 3 Smith Liberal Arts College, Sahmyook University, Seoul 01795, Korea; kimj@syu.ac.kr * Correspondence: jsy@korea.ac.kr; Tel.: +82-2-3290-2394

where with: Definition 2. The minimum distance between a cell g i and a cluster center c j for numeric attributes, denoted d min (g i , c j ), is: where (2) We corrected the subscript error in Definition 3.

Replacing:
Definition 3. The maximum distance between a cell g i and a cluster center c j for numeric attributes, denoted d max (g i , c j ), is: where r i = t i , p i ≤ s i +t i with: where r i = t i , c ji ≤ s i +t i 2 s i , otherwise (3) We corrected the order of some commands in Algorithm 1. Replacing: Randomly choosing k object, and assigning it to C.

3:
while IsConverged() do 4: for each cell g in G with: Randomly choosing k object, and assigning it to C.

3:
while IsConverged() do 4: for each cell g in G 5: (4) We corrected the order of some commands in Algorithm 2. Replacing: Randomly choosing k object, and assigning it to C.

3:
while IsConverged() do 4: for each cell g in G with: Randomly choosing k object, and assigning it to C.

3:
while IsConverged() do 4: for each cell g in G 5: (5) We corrected the analysis of complexity.

Replacing:
However, our proposed algorithms based on heuristic techniques (KCP and KBP) can reduce the number of objects to be computed, n' ≤ n, and the number of dimensions to be computed, d' ≤ d, respectively.Therefore, the time complexities of our proposed algorithms are O(n'kd'i), n' ≤ n and d' ≤ d.
For space complexity, KCP requires O(nd) to store the entire dataset, O(gd) to store the start point vector S and the end point vector of each cell, O(km 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:
However, our proposed algorithms based on heuristic techniques (KCP and KBP) can reduce the number of cluster centers to be computed, k' ≤ k, and the number of dimensions to be computed, d' ≤ d.Therefore, the time complexities of our proposed algorithms are O(nk'd'i), k' ≤ k, and d' ≤ d.
For space complexity, KCP requires O(nd) to store the entire dataset, O(gm 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.

( 6 )Figure 12 .Figure 12 .
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).