Suppose you have a relation r with n_r tuples on which a secondary B⁺-tree is to be constructed.

a. Give a formula for the cost of building the B+-tree index by inserting on record at a time. Assume each block will hold an average of f entries and that all levels of the tree above the leaf are in memory.

b. Assuming random disk access takes 10 milliseconds, what is the cost of index construction on a relation with 10 million records?

c. Write pseudocode for bottom-up construction of a B⁺-tree, which was outlined. You can assume that a function to efficiently sort a large file is available.