Fastest algorithm to hop through an array

Question

Start with an array A of positive numbers. Start at index 0. From index i, you can move to index i+x for any x <= A[i]. The goal is to find the minimum number of moves needed

Answer 1

Use your basic idea, but start from the beginning instead and you can get O(n).

The goal is to make a sequence (A_i1, A_i2, ..., A_ik, ...) such that

1. positions 0,1,2,...,ik can be reached in k or fewer steps

2. positions i(k-1)+1, i(k-1)+2, ..., ik cannot be reach in fewer than k steps

The base case is easy:

i0 = 0
i1 = A[0]

and the inductive part isn't too complicated:

i(k+2) = max { A_(ik+1) + ik , A_(ik+1) + ik+1, ..., A_(i(k+1)) + i(k+1) }