logn subproblems in size group ifor randomized quick-sortis at most / - suppose algorithm inplacequicksort (code fragment is executed on sequence with duplicate elements show that the algorithm still correctly sorts the input sequencebut the result of the divide step may differ from the highlevel description given in section and may result in inefficiencies in particularwhat happens in the partition step when there are elements equal to the pivotis the sequence (storing the elements equal to the pivotactually computeddoes the algorithm recur on the subsequences and gor on some other subsequenceswhat is the running time of the algorithm if all the input elements are equalr- of the npossible inputs to given comparison-based sorting algorithmwhat is the absolute maximum number of inputs that could be sorted with just comparisonsr- jonathan has comparison-based sorting algorithm that sorts the first elements in sequence of size in (ntime give big-oh characterization of the biggest that can ber- is the merge-sort algorithm in section stablewhy or why notr- an algorithm that sorts key-value entries by key is said to be straggling ifany time two entries ei and ej have equal keysbut ei appears before ej in the inputthen the algorithm places ei after ej in the output describe change to the merge-sort algorithm in section to make it straggling - describe radix-sort method for lexicographically sorting sequence of triplets ( , , )where kland are integers in the range [ , ]for some >= how could this scheme be extended to sequences of -tuples ( , ,kd)where each ki is an integer in the range [ - is the bucket-sort algorithm in-placewhy or why not