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Consider a comparison-based sorting algorithm for sorting an input sequence of n numbers (x1 x2 . . .

xn).

Screen Shot 2019-09-30 at 11.40.57 PM.png

Screen Shot 2019-09-30 at 11.40.57 PM.png

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).

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