If one has a set S$,$ of n distinct integer items, stored in a binary tree based min$-$heap T of height h$,$ where each node v in T is augmented with a count nv$,$ which indicates the number of items stored in the subtree of T rooted at v including v $($obviously every node of T has a parent pointer, a left child pointer, a right child pointer and the data field that stores an appropriate element of S to satisfy the min$-$heap property$).$ Design and analyze an efficient algorithm to compute the rank $3$ element of S$.$ Recall that the rank of an element e of S is defined to be the number of elements of S smaller than or equal to e$,$ hence, for this problem you are finding the $3$rd smallest element of S$.$ What are the time and space complexities of your solution? $($Note: Full credit for the most efficient $($in the big$-$oh sense$)$ and an elegant solution.$)$