1. The Apriori algorithm uses a generate-and-count strategy for finding frequent itemsets.

Candidate generation: candidate itemsets of size k+1 are created by merging a pair of frequent itemsets of size k if their first k-1 items are the same. An itemset created in this way is discarded if any of its subsets is found to be infrequent during the candidate pruning step ( see Page 52 of the slides for an example of candidate generation).

Support counting: after candidate itemsets of size k+1 are determined, Apriori algorithm counts the support of each candidate to determine frequent itemsets of size k+1.

Suppose the Apriori algorithm is applied to the data set shown in the following table with minsup = 30%, that is any itemset occurring in less than 3 transactions is considered to be infrequent.

The itemset lattice representing the data set is given below.

(a) Mark out the pruned itemsets using Apriori algorithm. (10%)

(b) frequent itemsets found by the Apriori algorithm. (15%)

(c) Find all association rules involving 3 items, satisfying support >=30% and confidence >=70%. (25%)

2. Consider the following set of candidate 3-itemsets: {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {2, 3, 5}, {2, 4, 5}, {3, 4, 6}, {4, 5, 6}.

(a) Construct a hash tree for the above candidate 3-itemsets. (25%)

Assume the tree uses a hash function where all odd-numbered items are hashed to the left child of a node, while all even-numbered items are hashed to the right child. A candidate 3-itemset is inserted into the tree by hashing on each successive item in the candidate and then following the appropriate branch of the tree according to the hash value. Once a leaf node is reached, the candidate is inserted based on one of the following conditions:

Condition 1: If the depth of the leaf node is equal to 3 (the root is assumed to be at depth 0), then the candidate is inserted regardless of the number of itemsets already stored at the node.

Condition 2: If the depth of the leaf node is less than 3, then the candidate can be inserted as long as the number of itemsets stored at the node is less than maxsize. Assume maxsize = 2 for this question.

Condition 3: If the depth of the leaf node is less than 3 and the number of itemsets stored at the node is equal to maxsize, then the leaf node is converted into an internal node. New leaf nodes are created as children of the old leaf node. Candidate itemsets previously stored in the old leaf node are distributed to the children based on their hash values. The new candidate is also hashed to its appropriate leaf node.

(b) Consider a transaction that contains the following items: {1, 3, 4, 5, 6}. Using the hash tree constructed in Part (a), which leaf nodes will be checked against the transaction? What are the candidate 3-itemsets contained in the transaction? (Bonus Point) (Hint: See Page 59 of the slides.)

3. Given the following transactions:

(a) Build the FP Tree if minimum support is 3 out of 5 (20%).

(b) What is the confidence of {HDMI cable, camera} → {printer}? (5%)