Resizable Array and Amortized Runtime

Question

\"Therefore, inserting N elements takes O(N) work total. Each insertion is O(1) on average, each though some insertions take O(N) time in the worst case.\" This quote is fou

Answer 1

Let's split up all the insertions into "heavy" insertions that take time proportional to the number of elements and "light" insertions that only take a constant amount of time to complete. Then if you start with an empty list and keep appending and appending, you're going to have mostly light insertions, but every now and then you'll have a heavy insertion.

Let's say, for simplicity, that you double the size of the array every time you run out of space and that you start off with an array of size 4. Then the first resize will have to move four elements, the second will move eight, then sixteen, then thirty-two, then sixty-four, then 128, then 256, etc.

Notice that it's not just one single append that takes a long time - roughly speaking, if you have n total insertions, then roughly log n of them will be heavy (the time between them keeps growing) and the other roughly n - log n of them will be light.