We have a binary heap on n elements and wish to insert n more elements (not necessarily one after another) into this heap. The total time required for this is

Consider the process of inserting an element into a Max Heap, where the Max Heap is represented by an array. Suppose we perform a binary search on the path from the new leaf to the root to find the position for the newly inserted element, the number of comparisons performed is:

Given two arrays of numbers a₁,......,a_n and b₁,......, b_n where each number is 0 or 1, the fastest algorithm to find the largest span (i, j) such that a_i+a_i+1......a_j = b_i+b_i+1......b_j or report that there is not such span,

A 3-ary max heap is like a binary max heap, but instead of 2 children, nodes have 3 children. A 3-ary heap can be represented by an array as follows: The root is stored in the first location, a[0], nodes in the next level, from left to right, is stored from a[1] to a[3]. The nodes from the second level of the tree from left to right are stored from a[4] location onward. An item x can be inserted into a 3-ary heap containing n items by placing x in the location a[n] and pushing it up the tree to satisfy the heap property.

Which one of the following is a valid sequence of elements in an array representing 3-ary max heap?