What are Maximal Frequent Itemsets?

A maximal frequent itemset is represented as a frequent itemset for which none of its direct supersets are frequent. The itemsets in the lattice are broken into two groups such as those that are frequent and those that are infrequent. A frequent itemset border, which is defined by a dashed line.

Each item set situated above the border is frequent, while those located under the border (the shaded nodes) are infrequent. Between the itemsets residing near the border, {a, d}, {a, c, e}, and {b, c, d, e} are treated to be maximal frequent itemsets because their direct supersets are infrequent.

An itemset including {a, d} is maximal frequent because some direct supersets, {a, b, d}, {a, c, d}, and {a, d, e}, are infrequent. In contrast, {a, c} is non-maximal because the direct supersets, {a, c, e}, is frequent.

Maximal frequent itemsets adequately support a compact description of frequent itemsets. In other terms, they form the smallest set of itemsets from which some frequent itemsets can be derived. For instance, the frequent itemsets can be broken into two groups such as follows −

  • Frequent itemsets that start with item a and that can include items c, d, or e. This group contains itemsets including {a), {a, c), {a, d}, {a, e}, and {a, c, e}.

  • Frequent itemsets that start with items b, c, d, or e. This group contains itemsets including {b}, {b, c}, {c, d}, {b, c, d, e}, etc.

Frequent itemsets that apply in the first group are subsets of either {a, c, e} or {a, d}, while those that apply in the second group are subsets of {b, c, d, e}. Therefore, the maximal frequent itemsets {a, c, e}, {a, d}, and {b, c, d, e} support a compact description of the frequent itemsets.

Maximal frequent itemsets support a valuable description for data sets that can make very high, frequent itemsets, as there are exponentially several frequent itemsets in such data. This method is practical only if an effective algorithm occurs to explicitly discover the maximal frequent itemsets without having to enumerate some subsets.

Despite supporting a compact description, maximal frequent itemsets do not include the support data of their subsets. For instance, the support of the maximal frequent itemsets {a,c,e}, {a,d}, and {b,c,d,e} do not give any idea about the support of their subsets.

An additional pass over the data set is required to decide the support counts of the non-maximal frequent itemsets. In some cases, it can be desirable to have a minimal description of frequent itemsets that preserves the support data.