Question

Consider a binary heap with 0 elements in it. We have a sequence of N numbers, and we use the special "build" function to construct a heap directly from them. What is the worst-case time complexity of building the heap this way?

a. O(N)
b. O(N log N)
c. O(log N)
d. O(N²)

Answer 1

The worst-case time complexity of building a heap using the 'build' function from a sequence of N numbers is O(N), due to efficient down-heap operations that result in an overall linear time complexity.

Step-by-step explanation:

When constructing a binary heap from a sequence of N numbers using a special 'build' function, the worst-case time complexity is O(N).

This is because when using the bottom-up approach to construct a heap, we start from the lowest level non-leaf nodes and work our way up to the root, performing down-heap operations (bubble down) to maintain the heap property. While it might seem like each down-heap operation would take O(log N), due to the nature of a complete binary tree, most of the elements are in the bottom layers and do not require the maximum number of swaps. Therefore when analyzed rigorously, the overall complexity is revealed to be linear, not logarithmic or quadratic.