Let * be an associative binary operator, and let a be a field maintained in each node of a red-black tree. Suppose that we want to include in each node x an additional field f such that f[x] = a[x₁] * a[x₂] * ··· * a[x_m], where x₁, x₂,..., x_m is the inorder listing of nodes in the subtree rooted at x. Show that the f fields can be properly updated in O(1) time after a rotation. Modify your argument slightly to show that the size fields in order-statistic trees can be maintained in O(1) time per rotation.

1)容易得到不变式 f(x) = f(left[x])*a[x]*f(right[x])]

2)在一次反转过程中,无非是a,b,r三个子树的调整,对于三个子树的f域不需要调整,因为不影响他们内部的顺序,

3)对于求B的f域, f(B) = f(b)*a[B]*f(r)

4)求A的f域f(A) = f(a)*a[A]*f(B)

以上以left-rotate为例,right-roate类似

B A

/ / -->/ /

A ra B

a b b r