1: (top) The input transactional dataset D, represented in its horizontal form. (bottom) Lattice of all the frequent itemsets (σ = 1), with closed itemsets and equivalence classes highlighted.

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... 4.1 Figure 4.1 shows the lattice of frequent itemsets derived from the simple dataset reported in the same figure, mined with σ = 1. We can see that the itemsets with the same closure are grouped in the same equivalence class. ...

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... 4.2 The closed itemset {abcd} of Figure 4.1 would be mined twice, since it can be obtained as the closure of two minimal elements of its equivalence class, namely {ab} and {bc}. ...

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... similarly to the A-Close approach based on key patterns, it is possible to generate the same closed itemset multiple times. Example 4.3 As shown in Figure 4.1, the itemset {a} is a generator since it can be obtained as a superset of the closed itemsets ∅, and its closure is {acd}. But also {cd} is a generator obtained as a superset of {c}, and it has the same closure {acd}. ...

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... performance improvements resulting from each one of these optimizations are shown in Figures 4.4-4.3, which plots, for several datasets, the actual number of bit-wise AND operations performed DCI-Closed as a function of the support threshold. ...

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... plots, for several datasets, the actual number of bit-wise AND operations performed DCI-Closed as a function of the support threshold. In particular, Figure 4.3 refer to the tid-list intersections to perform task (1), while Figure 4.4 deal with the inclusion checks to complete tasks (3) and (4). ...

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... plots, for several datasets, the actual number of bit-wise AND operations performed DCI-Closed as a function of the support threshold. In particular, Figure 4.3 refer to the tid-list intersections to perform task (1), while Figure 4.4 deal with the inclusion checks to complete tasks (3) and (4). ...

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... interesting remark regards the comparison between the number of bit- wise intersection operations actually executed (Figure 4.3), and the number of op- erations carried out for the various inclusion checks (Figure 4.4). The two amounts, given a dataset and a support threshold, appear to be similar. ...

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... order to compare the efficiency of our duplication check technique with respect to that adopted by competitor algorithms, we instrumented the publicly available FP-Close code and our DCI-Closed code. Figure 4.8 shows the absolute times spent checking for duplicates while mining two dense datasets, Chess and Connect, as a function of the support threshold. ...

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... table also shows, for each support threshold, the number of closed frequent itemsets extracted (|C|). Other algorithms hash the already mined closed itemsets to detect duplicates, but since C grows exponentially, the hash performances may degrade suddenly as confirmed in Figure 4.8. Conversely, Tab. ...

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... issue related to the efficiency of a CFIM algorithm is the memory us- age. Figure 4.9(a) plots memory requirements of FP-Close, Closet+ and our algorithm DCI-Closed when mining the connect dataset as a function of the support threshold. ...

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... of the technique adopted, mining sparse datasets should not be a big issue from the point of view of memory occupation, because the number and length of frequent itemsets do not explode even for very low support thresholds. In Figure 4.9(b) we plotted the results of the tests conducted on T40I10D100K, a sparse dataset where Closet+ is supposed to use its upward checking technique. ...

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... results of the tests conducted are reported in Figure 4.10, which shows execution times (Figure 4.10(a)) and memory requirements (Figure 4.10(b)) of DCI- Closed and FP-Close for different support thresholds, as a function of the size of the sample extracted from the synthetic dataset. ...

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... results of the tests conducted are reported in Figure 4.10, which shows execution times (Figure 4.10(a)) and memory requirements (Figure 4.10(b)) of DCI- Closed and FP-Close for different support thresholds, as a function of the size of the sample extracted from the synthetic dataset. ...

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... 5.1 For example, consider Figure 4.1, which shows a dataset D and its frequent closed itemsets extracted with σ = 1. Consider now that from D, we can build two projected datasets D [a,b) ≡ D a (see Figure 5.1), and D [b,d] (see Figure 5.2), where D [b,d] is the projected dataset obtained by merging D b , D c , and D d . ...