Question

The running time should be O(n), this is what I came up with, is it correct? THANKS

for (i = 1 ; i < A[n]; i++)
  A[i] = 0

B[i] = 0;
for i in A[1..n]
  B[i] = B[i] + A[i]

Answer 1

Yes, this is correct. Your solution is the most trivial case of dynamic programming, a technique to greatly speed up algorithms for solutions that can be constructed from solutions from smaller sub-problems.

Your problem at hand has precisely this property: a solution to B[n] can be constructed in O(1) if you are given a solution to B[n-1], and that is what your algorithm did, for a running time of O(N).

Solutions like that are very helpful in practice: I once used an algorithm like yours to speed up a piece of start-up code in my program from several minutes to several seconds (it was adding vectors, so it went from O(3) to O(2)).