[英]Solving a Recurrence Relation: T(n) = T(n-1) + n-1
I have been asked to solve that recurrence relation. 我被要求解决这种复发关系。 I got the next solution: https://imgur.com/a/xWoTI40
我得到了下一个解决方案: https : //imgur.com/a/xWoTI40
So I decided to ask my teacher if it was right. 所以我决定问我的老师这是不对的。 He told me that it wasn't;
他告诉我,事实并非如此; and that this is the right solution: https://imgur.com/a/CGD0ta8
这是正确的解决方案: https : //imgur.com/a/CGD0ta8
I'm totally clueless right now. 我现在完全无能为力。 The most I try to understand why mine is wrong;
我最努力理解为什么我的错了; the most I think it's actually right.
我认为它是最正确的。
Can somebody explain? 有人可以解释一下吗?
Your solution is correct. 你的解决方案是对的。 Here's a different approach with the same result:
这是一个不同的方法,结果相同:
t(1) = 0 (given)
t(2) = t(1) + 1 = 1
t(3) = t(2) + 2 = 1 + 2 = 3
t(4) = t(3) + 3 = 1 + 2 + 3 = 6
...
t(n) = 1 + 2 + ... + (n-1) = n * (n - 1) / 2 = Theta(n^2).
The teacher's solution is wrong after the second =
sign. 在第二个
=
符号后,教师的解决方案是错误的。 Here's what the teacher wrote: 这是老师写的:
t(n-1) + n - 1 = t(n-2) + n - 1 - 2
But actually the following is correct: 但实际上以下是正确的:
t(n-1) + n - 1 = ( t(n-2) + n - 2 ) + n - 1
which is actually exactly what you got. 这实际上就是你得到的。 It appears that the teacher dropped an
n
term. 看来,老师下降的
n
项。
In fact, the teacher's solution ends with a dominant term of -n^2
which is clearly wrong, as t(n) >= 0
for all n >= 0
. 实际上,教师的解决方案以
-n^2
的显性项结束,这显然是错误的,因为对于所有n >= 0
, t(n) >= 0
n >= 0
。
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