# Multi-Objective Optimization for High-Dimensional Maximal Frequent Itemset Mining
## 4 Case Studies
### Accident Transaction Set Test
A total of 216 maximal frequent itemsets were gained in one of our papers on accurate algorithms, which aims to search for the maximal frequent itemsets (right-hand side expanding algorithm for maximal frequent itemsets mining). Two curves are seen in Figure 2e. The upper curve shows the average number of items in each frequent itemset, which is searched in each generation and fluctuates around 10. The lower curve is relatively complicated and was set to \(d\) if the average number of items  
Figure 2: Results of transactions set accidents: (**a**) support of the corresponding item used as the 1-itemsets; (**b**) support of frequent 2-itemsets; (**c**) number of frequent itemsets against generation; (**d**) sum of frequent itemsets against generation; (**e**) average item number curve and addable item number curve against generation; and (**f**) number of infrequent itemsets curve and number of identical itemsets curve against generation. in the frequent itemsets which are searched in each generation was \(a\). For each frequent itemset, the items which are absent in many itemsets can independently be added into the itemsets and these itemsets are still frequent (\(b\)). If the mean of \(b\) in each generation was \(c\), therefore, \(d=1+a-c\). In other words, all frequent itemsets which were searched from the current generation when \(d=1\) can be added with one item and the number of the frequent itemset was still zero. There is no item which can be added in. All frequent itemsets are maximal frequent itemset. It can be seen from the lower curve that the \(d\) value fluctuates around 1, which is a normal phenomenon. There are two curves in Figure 2f. The upper one is the number of non-frequent itemsets which are encountered in each generation of searching tasks. Since the population is 168, the number of the non-frequent itemset fluctuates around 150. The sum of the number of non-frequent itemsets and the number of frequent itemset in Figure 2c is basically consistent with population size. The lower curve represents the number of frequent itemsets that are searched from each generation and which are the same with previously identified frequent itemsets. The algorithm deletes these overlapping itemsets. In the many traditional algorithms, the two most classical ones are the Apriori and FP-growth algorithms. We know that FP-growth is faster than Apriori, which is also widely recognized. In order to compare with our algorithm, we also tested the FP-growth algorithm on the Accidents transaction set. Similarly, when the support threshold was set to 0.5, there was no result for FP-growth to mine the Accidents within 24 h.  
In order to estimate the running time of the FP-growth algorithm on Accidents, the number of transactions and the length of each transaction were reduced. Some transaction were taken out of the Accidents set, such as 100, 200, etc. Only 10%, 20% and 30% of the beginning of each transaction was taken, respectively. The running time of the algorithm is shown in Table 1. The first row represents the different numbers of transactions. The first column represents the different transaction lengths. Table 1 shows that the running time of FP-growth is insensitive to the number of transactions but sensitive to the transaction length. Since the item ID of each transaction is from small to large, the transaction length roughly corresponds to the dimension of the transaction set. That is, the running time of FP-growth is sensitive to the dimension of the transaction set. With the growth momentum shown in Table 1, if the dimension of the extracted accidents is 468, the algorithm time will exceed 100 h at least. In contrast, our proposed algorithm is self-evident for the high-dimensional transaction sets.