更新时间:2023-02-10 13:30:24
感谢@Michael Foukarakis指出了丢失的K.
Thanks @Michael Foukarakis for pointing the missing K.
Base case: T(n, 0) = 1
Recurion : T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-3, k-1) + .. + T(0, k-1) + 1
= n * T(n-1, k-1) + 1
如下扩展
T(n, k) = n * T(n-1, k-1) + 1
= n * (n-1) * T(n-2, k-2) + 1 + 1
= n * (n-1) * T(n-2, k-2) + 2
= n * (n-1) * (n-2) * T(n-3, k-3) + 3
...
= n * (n-1) * (n-2) * ..(n-k) T(n-k, k-k) + k
= n * (n-1) * (n-2) * ..(n-k) (1) + k
= O(n^k) (As it is a k th order polynomial)
总体而言,我们可以说 O(n k )运行时.
Overall we can say O(nk) runtime.