在《暴露编程采访》一书中,该书说下面程序的复杂度为O(N),但我不知道这是怎么可能的。
In the book Programming Interviews Exposed it says that the complexity of the program below is O(N), but I don't understand how this is possible. Can someone explain why this is?
int var = 2; for (int i = 0; i < N; i++) { for (int j = i+1; j < N; j *= 2) { var += var; } }推荐答案
您需要一点数学才能看到。内循环迭代Θ(1 + log [N /(i + 1)])次( 1 + 是必需的,因为对于 i> = N / 2 , [N /(i + 1)] = 1 和对数为0,但循环迭代一次)。 j 取值(i + 1)* 2 ^ k 直到它至少等于 N ,并且
You need a bit of math to see that. The inner loop iterates Θ(1 + log [N/(i+1)]) times (the 1 + is necessary since for i >= N/2, [N/(i+1)] = 1 and the logarithm is 0, yet the loop iterates once). j takes the values (i+1)*2^k until it is at least as large as N, and
(i+1)*2^k >= N <=> 2^k >= N/(i+1) <=> k >= log_2 (N/(i+1))使用数学除法。因此,更新 j * = 2 称为 ceiling(log_2(N /(i + 1)))次,检查条件 1 +上限(log_2(N /(i + 1)))次。这样我们就可以写出总工作量
using mathematical division. So the update j *= 2 is called ceiling(log_2 (N/(i+1))) times and the condition is checked 1 + ceiling(log_2 (N/(i+1))) times. Thus we can write the total work
N-1 N ∑ (1 + log (N/(i+1)) = N + N*log N - ∑ log j i=0 j=1 = N + N*log N - log N!现在,斯特林公式告诉我们
log N! = N*log N - N + O(log N)所以我们发现完成的工作确实是 O(N)。
so we find the total work done is indeed O(N).
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