Given an unsorted array of distinct integers numbers and is given a non-negative integer number, k, k < n. you need to find an element from a such that its rank is k, i. e., there are exactly (k-1) numbers less than or equal to that element. example: input: a = [1, -3, 4, 3, 12, 20, 30, 7, 14, -1, 0] and k =8. output: 12, since 7, -3, -1, 0, 4, 3, 1 (8-1=7 numbers) are all less than or equal to 12 suggest a sub-quadratic running time complexity algorithm (only pseudo-code) to solve this problem. justify. (b) given an array a of n + m elements. it is know that the first n elements in a are sorted and the last m elements in a are unsorted. suggest an algorithm (only pseudo code) to sort a in o(mlogm +n) worst case running time complexity. justify. (c) the processing time of an algorithm is described by the following recurrence equation (c is a positive constant): t(n) = 3t(n/3) + 2cn; t(1) = 0 what is the running time complexity of this algorithm? justify. (d) you decided to improve insertion sort by using binary search to find the position p where the new insertion should take place. (d.1) what is the worst-case complexity of your improved insertion sort if you take account of only the comparisons made by the binary search? (d.2) what is the worst-case complexity of your improved insertion sort if only swaps/inversions of the data values are taken into account?