# Publishing Anonymized Set-Valued Data via Disassociation towards Analysis
## 2 Disassociation and Utility Awareness for Associations
### Disassociation of a Set-valued Data  
This section is dedicated to show how disassociation works in terms of privacy and utility. We use Figure 2 to illustrate an example of disassociation, applied with \(k=3\), \(m=2\) and \(\delta=6\).  
**Horizontal partitioning**: is the process of clustering the records of a datasets following a naive similarity function. Iteratively and until all records are clustered, from non clustered records, horizontal partitioning groups at most \(\delta\) records containing the most frequent item at the current iteration. With that, horizontal partitioning fails to take into consideration the associations in the dataset, relying only on one common term to cluster the records. Figure 2b reflects the process of horizontal partitioning, where records containing the most frequent item, \(a,s(a,\mathcal{T})=6\), are grouped together within cluster \(C_{1}\), and all the other records within \(C_{2}\). Both clusters have a size less than \(\delta\).  
**Vertical partitioning:** is the process that verifies the \(k^{m}\)-anonymity privacy constraint by vertically cutting every cluster into record chunks \(RC\). A record chunk represents sub-itemsets of the cluster's records verifying \(k^{m}\)-anonymity between their items. A term chunk \(TC\), is added to the vertical cut for items with a support of less than \(k\). This process is known as vertical partitioning and is the core of privacy preservation in disassociation.  
In our example, vertical partitioning is applied over \(C_{1}\) and \(C_{2}\). Associations in the clusters are split into different record chunks when \(k^{m}\)-anonymity is not verified.  
Figure 2c represents the final result of horizontal and vertical partitioning.  
To illustrate the effect of disassociation on the utility of associations, let us consider that the frequency of the association \(\{b,c\}\), is valuable for future analysis. From the result of disassociation, Figure 2c, we can note that the bond between items \(b\) and \(c\) is ambiguous. In \(C_{1}^{*}\), items \(b\) and \(c\) are dropped in the term chunk \(TC\) having a support less than \(k=3\), therefore, camouflaging their exact support and the association between them, if it exists. Similarly, the association between \(b\) and \(c\) is unclear in \(C_{2}^{*}\), with only one advantage over \(C_{1}^{*}\); knowing the support of item \(b\).  
Let us suppose that we can guide the horizontal partitioning process for the favor of the association \(\{b,c\}\) by keeping together all records that its supersets, as in Figure 3a. Now, association \(\{b,c\}\) verifies \(k^{m}-anonymity\) and is totally preserved associated after vertical partitioning, Figure 3b.  
From this example, we deduce that preserving associations depends significantly on horizontal partitioning. When a set of associations is important for future analysis, a special treatment of horizontal partitioning must be considered. In what follows, we present a solution to preserve high utility for a set of predefined associations, we call utility rules, within the disassociated dataset.  
Figure 2: Example of disassociation.