Suppose we wish to implement LSH by Map Reduce. Specifically, assume chunks of the signature matrix consist of columns, and elements are key-value pairs where the key is the column number and the value is the signature itself (i.e., a vector of values).

(a) Show how to produce the buckets for all the bands as output of a single Map Reduce process. Hint: Remember that a Map function can produce several key-value pairs from a single element.

(b) Show how another Map Reduce process can convert the output of (a) to a list of pairs that need to be compared. Specifically, for each column i, there should be a list of those columns j > i with which i needs to be compared.