Below is the data from problem 6:

Suppose we run the Multistage Algorithm on the data of Problem 6, with the same support threshold of 4. The first pass is the same as in that problem, and for the second pass, we hash pairs to nine buckets, using the hash function that hashes (i, j} to bucket i+j mod 9. (a) Determine the counts of the buckets on the second pass. (b) Does the second pass reduce the set of candidate pairs? Note that all items are frequent, so the only reason a pair would not be hashed on the second pass is if it hashed to an infrequent bucket on the first pass. Here is a collection of twelve baskets. Each contains three of the six items 1 through 6. {1,2,3} {2,3,4} {3,4,5} {4,5,6} {1,3,5} {2,4,6} {2,4,6} {1,3,4} {2,4,5} {3,5,6} {1, 2,4} {2,3,5} {3,4,6} Suppose the support threshold is 4. On the first pass of the PCY Algorithm we use a hash table with 11 buckets, and the set {i,j} is hashed to bucket ixj mod 11. (a) By any method, compute the support for each item and each pair of items. (b) Which pairs hash to which buckets? (c) Which buckets are frequent? (c) Which pairs are counted on the second pass of the PCY Algorithm? Suppose we run the Multistage Algorithm on the data of Problem 6, with the same support threshold of 4. The first pass is the same as in that problem, and for the second pass, we hash pairs to nine buckets, using the hash function that hashes (i, j} to bucket i+j mod 9. (a) Determine the counts of the buckets on the second pass. (b) Does the second pass reduce the set of candidate pairs? Note that all items are frequent, so the only reason a pair would not be hashed on the second pass is if it hashed to an infrequent bucket on the first pass. Here is a collection of twelve baskets. Each contains three of the six items 1 through 6. {1,2,3} {2,3,4} {3,4,5} {4,5,6} {1,3,5} {2,4,6} {2,4,6} {1,3,4} {2,4,5} {3,5,6} {1, 2,4} {2,3,5} {3,4,6} Suppose the support threshold is 4. On the first pass of the PCY Algorithm we use a hash table with 11 buckets, and the set {i,j} is hashed to bucket ixj mod 11. (a) By any method, compute the support for each item and each pair of items. (b) Which pairs hash to which buckets? (c) Which buckets are frequent? (c) Which pairs are counted on the second pass of the PCY Algorithm