(1)
y[i] = A*y[i-1] + x[i]
可以写成
y[z] = A^z * y[0] + Sum(A^(z-j) * x[j])
,where j E [z,1].
A^z * y[0]
可以计算在O(log(z))
Sum(A^(z-j) * x[j])
中可以计算O(z)
。
如果事先知道序列的最大大小(例如max
),那么您可以预先计算修改后的前缀和数组x
为
prefix_x[i] = A*prefix_x[i-1] + x[i]
then Sum(A^(z-j) * x[j]) is simply prefix_x[z]
and the query becomes O(1) with O(max) precomputation.
(2)
y[i] = y[i-1] + y[i-2] + x[i]
可以写成
y[z] = (F[z] * y[1] + F[z-1] * y[0]) + Sum(F[z-j+1] * x[j])
,where j E [z,2] and F[x] = xth fibonaci number
(F[z] * y[1] + F[z-1] * y[0])
可以计算在O(log(z))
Sum(F[z-j+1] * x[j])
中可以计算O(z)
。
如果事先知道序列的最大大小(例如max
),那么您可以预先计算修改后的 x 前缀和数组为
prefix_x[i] = prefix_x[i-1] + prefix_x[i-2] + x[i]
then Sum(F[z-j+1] * x[j]) is simply prefix_x[z]
and the query becomes O(1) with O(max) precomputation.