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25 March 2019

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

,
,
and
1
Department of Computer Science and Engineering, Korea University, Seoul 02841, Korea
2
Department of Computer Engineering, Dongguk University, Gyeongju 38066, Korea
3
Smith Liberal Arts College, Sahmyook University, Seoul 01795, Korea
*
Author to whom correspondence should be addressed.
Sustainability2019, 11(6), 1801;https://doi.org/10.3390/su11061801
This article belongs to the Collection Advanced IT based Future Sustainable Computing
Version Notes
Order Reprints
The authors would like to make the following corrections to this paper [1]:
(1) We corrected the subscript error in Definition 2.
Replacing:
Definition 2.
The minimum distance between a cell gi and a cluster center cj for numeric attributes, denoted dmin(gi, cj), is:
d m i n ( g i ,   c j ) = ∑ i = 1 m r | o i − r i | 2 ,
where r i = { s i i f   o i < s i t i i f   o i > t i o i o t h e r w i s e .
with:
Definition 2.
The minimum distance between a cell gi and a cluster center cj for numeric attributes, denoted dmin(gi, cj), is:
d m i n ( g i ,   c j ) = ∑ i = 1 m r | c j i − r i | 2 ,
where r i = { s i i f   c j i < s i t i i f   c j i > t i o i o t h e r w i s e .
(2) We corrected the subscript error in Definition 3.
Replacing:
Definition 3.
The maximum distance between a cell gi and a cluster center cj for numeric attributes, denoted dmax(gi, cj), is:
d m a x ( g i ,   c j ) = ∑ i = 1 m r | p i − r i | 2 ,
where r i = { t i , p i ≤ s i + t i 2 s i , o t h e r w i s e
with:
d m a x ( g i ,   c j ) = ∑ i = 1 m r | c j i − r i | 2 ,
where r i = { t i , c j i ≤ s i + t i 2 s i , o t h e r w i s e
(3) We corrected the order of some commands in Algorithm 1.
Replacing:
1:
C[ ]← Ø // k cluster centers
2:
Randomly choosing k object, and assigning it to C.
3:
whileIsConverged() 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:
whileIsConverged() do
4:
 for each cell g in G
5:
  dmin[ ], dmax[ ] ← Calc(g, C)
6:
  dminmax ← min(dmax[ ])
(4) We corrected the order of some commands in Algorithm 2.
Replacing:
1:
C[k]← Ø // k cluster center
2:
Randomly choosing k object, and assigning it to C.
3:
whileIsConverged() 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:
whileIsConverged() do
4:
 for each cell g in G
5:
  dmin[ ], dmax[ ] ← Calc(g, C)
6:
  dminmax ← min(dmax[ ])
(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(kmc) to store the frequency of categorical data in each cluster and O(kd) to store cluster centers, where mc the number of categorical attributes. Additionally, KBP requires O(gmc) 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 + kmc + kd) and O(nd + gd + kmc + kd + gmc), 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(gmr), where g is the number of cells, to store the start point vector S and the end point vector T of each cell, O(ktmc), 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(gtmc) 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 + gmr + ktmc + kd) and O(nd + gmr + ktmc + kd + gtmc), respectively.
(6) We corrected the description of Figure 12.
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).
The authors would like to apologize for any inconvenience caused to the readers by these changes. The manuscript will be updated and the original will remain available from the article webpage.

Reference

  1. 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] [Scilit]

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