Consider a comparison—based sorting algorithm for sorting an input sequence of 'n.

numbers ($1 :32 .. . :3”). Suppose that the sorting algorithm has the following ad—

ditional information about the sequence to be sorted: The sorting algorithm knows that the input sequence is a concatenation

of n/k subsequences, each of which contains 1: elements. The algorithm

also knows that all the elements in each subsequence occur before all the

elements in any later subsequence. That is, the overall input sequence

($1,322, . . . , mn) also can be written as ($14 311,2 $1,k (32,1 (132,2 mu: (3&1 31%,2 $30, where, for all 1 S i < j S g and for all a,b E {1,2,...,k}, we have

that raw < $335. Put another way, after each subsequence is sorted, the

concatenation of all the sorted subsequences produces a sorted sequence. Using a decision tree argument (in class we used this type of argument to prove the

runtime lower bound for comparison—based sorting algorithms), show that the lower

bound on the number of comparisons needed to sort this sequence is 901.10g k).