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Chapter 27: Problem 16

For the Partition algorithm, prove that any frequent itemset in the database must appear as a local frequent itemset in at least one partition.

Short Answer

Expert verified
According to Partition Algorithm principle, any frequent itemset in the whole database should appear as a local frequent itemset in at least one partition because the database is a superset of all partitions, and each transaction in the database belongs to exactly one partition. Therefore, if an itemset is frequent in the database, it must be frequent in at least one of the partitions.

Step by step solution

01

Understanding the Problem Terminology

An itemset is considered 'frequent' if its support (probability of occurrence) in the dataset is no less than a predetermined minimum support threshold. A 'local frequent itemset' is a frequent itemset in a partition of the data.
02

Partition Algorithm Basic Principle

The basic principle of the Partition algorithm is that it scans the entire dataset once and divides it into non-overlapping subsets or partitions. Each partition fits into main memory, allowing for local frequent itemsets to be generated. Given the constraint of the Partition algorithm, each transaction in the database belongs to exactly one partition.
03

Proving the Assertion

It is stated that every frequent itemset in the entire database must exist as a local frequent itemset in at least one partition. This holds true because if an itemset is 'frequent' in the entire database, it means its support is above the minimum support threshold in the full data. Since the database is a superset of all partitions, but each transaction can belong to only one partition, that itemset has to appear frequently in at least one partition. Thus, if an itemset does not appear as a local frequent itemset in any partition, it cannot be a frequent itemset in the entire database.
04

Counter-Argument for a Clearer Picture

To affirm the understanding, consider a counterargument. Assume that a frequent itemset does not appear as a local frequent itemset in any partition. But because the frequent itemset in the full set is constituted by transactions from some (or all) partitions, the absence of this itemset in any partition means it's not frequent in the database. This contradicts the initial assumption, hence the counterargument confirms our theory.

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Most popular questions from this chapter

The K-means algorithm uses a similarity metric of distance between a record and a cluster centroid. If the attributes of the records are not quantitative but categorical in nature, such as Income Level with values \\{low, medium, hight or Married with values \\{Yes, Nof or State of Residence with values \\{Alabama, Alaska, \(\ldots,\) Wyoming then the distance metric is not meaningful. Define a more suitable similarity metric that can be used for clustering data records that contain categorical data.

What is the downward closure property? How does it aid in developing an efficient algorithm for finding association rules, i.e., with regard to finding large itemsets?

What was the motivating factor for the development of the FP-tree algorithm for association rule mining?

Apply the Apriori algorithm to the following data set. $$\begin{array}{ll} \text { Trans ID } & \text { Items Purchased } \\ \hline 101 & \text { milk, bread, eggs } \\ 102 & \text { milk, juice } \\ 103 & \text { juice, butter } \\ 104 & \text { milk, bread, eggs } \\ 105 & \text { coffee, eggs } \\ 106 & \text { coffee } \\ 107 & \text { coffee, juice } \\ 108 & \text { milk, bread, cookies, eggs } \\ 109 & \text { cookies, butter } \\ 110 & \text { milk, bread } \end{array}$$ The set of items is \(\\{\text { milk, bread, cookies, eggs, butter, coffee, juice }\\}\). Use 0.2 for the minimum support value.

What are classification rules and how are decision trees related to them?
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