exercises for help with exercisesplease visit the sitewww wiley com/college/goodrich reinforcement - give complete justification of proposition - in the merge-sort tree shown in figures through some edges are drawn as arrows what is the meaning of downward arrowhow about an upward arrowr- show that the running time of the merge-sort algorithm on an -element sequence is ( log )even when is not power of - is our array-based implementation of merge-sort given in section stableexplain why or why not - is our linked-list-based implementation of merge-sort given in code fragment stableexplain why or why not - an algorithm that sorts key-value entries by key is said to be straggling ifany time two entries ei and have equal keysbut ei appears before in the inputthen the algorithm places ei after in the output describe change to the merge-sort algorithm in section to make it straggling - suppose we are given two -element sorted sequences and each with distinct elementsbut potentially some elements that are in both sequences describe an ( )-time method for computing sequence representing the union (with no duplicatesas sorted sequence - suppose we modify the deterministic version of the quick-sort algorithm so thatinstead of selecting the last element in an -element sequence as the pivotwe choose the element at index  / what is the running time of this version of quick-sort on sequence that is already sortedr- consider modification of the deterministic version of the quick-sort algorithm where we choose the element at index  / as our pivot describe the kind of sequence that would cause this version of quick-sort to run in ( time - show that the best-case running time of quick-sort on sequence of size with distinct elements is ( log nr- suppose function inplace quick sort is executed on sequence with duplicate elements prove that the algorithm still correctly sorts the input sequence what happens in the partition step when there are elements equal to the pivotwhat is the running time of the algorithm if all the input elements are equal