Question

Consider a binary heap with 0 elements in it. We have a sequence of N numbers that we insert into the heap one at a time. What is the (best) worst case time complexity of building the heap this way?

a) O(N log N)
b) O(N)
c) O(N²)
d) O(1)

Answer 1

The worst case time complexity of constructing a binary heap by sequentially inserting N elements is O(N log N), since each insertion has a worst case of O(log N) and there are N insertions.

Step-by-step explanation:

The question is about constructing a binary heap by inserting N numbers into an initially empty heap. The (best) worst case time complexity of building the heap in this way is a common question in data structures, which is a part of computer science and more specifically within the field of algorithms.

The worst-case time complexity for inserting an element into a binary heap is O(log N) because the newly added element may have to traverse from the bottom of the heap to its correct position, which in a complete binary tree represents a path of at most log N swaps.

Thus, if you insert N elements into the heap, one after the other, the worst case time complexity is the sum of a series of growth rates which approximates O(N log N). Therefore, the correct answer to the question is option (a) O(N log N).