Question

With a C++ STL vector we are building a vector of N elements and for some
reason we chose to insert them at the front of the vector. Every element insertion at the front of a vector forces the shift of all existing elements by 1. This results in (1+2+3+…+N) overall shifts of vector elements, which is (N/2)(N+1) shifts.

My question is how the author came with (1+2+3+…N), I thought it should be 1+1+1..N as we are moving one element at one position to get empty at beginning?

Thanks!

Answer 1

From [vector.modifiers]/2 (which describes vector::insert):

Complexity: The complexity is linear in the number of elements inserted plus the distance to the end of the vector.

Each time that you add an element the distance to the end of the vector is increased by one.

The first time that you add an element, there is 1 to be inserted and the distance to the end is 0, so the complexity is 1 + 0 = 1. The second time, there is 1 to be inserted, and the distance to the end is 1, so the complexity is 1 + 1 = 2. The third time, the distance to the end is 2, so the complexity is 1 + 2 = 3. This is what creates the 1 + 2 + 3 + … + N pattern that the author is describing.