Question

Input is a boolean array a_0,i with at most 1000,000 elements.

each time the new array is made by xor of adjacent(cyclic) elements in previous array:

a_t,i = a_t-1,i ^ a_t-1,(i+1)%n     // n is size of input

The p-th array(a_p,i) is wanted.(p <= 1000,000,000).

According to high bound on p I think maybe there is a structure of arrays or maybe the array can be calculated in O(lg(p) * n).

Answer 1

If t is a power of two (t=2^k),

a_t,i = a_0,i ^ a_0,(i+t)%n

Also, if t is a sum of two components, and one of them is a power of two (t = v + w, w=2^m),

a_t,i = a_v,i ^ a_v,(i+w)%n

This allows using binary representation of p to recursively compute the resulting array. The complexity is as requested: O(lg(p) * n):

shift = 1;
while (p != 0)
{
  if (p&1)
    a ^= a.rotate(shift);
  shift *= 2
  p /= 2
}