[Algorithm] Asymptotic Growth Rate

f(n) 的形式 vs 判定形势

但，此题型过于简单，一般不出现在考题中。

Extended:

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Let's set n = 2^m, so m = log(n)

T(n) = 2*T(n^(1/2)) + 1 =>

T(2^m) = 2*T(2^(m/2)) + 1 =>

S(M) = 2*S(M/2)) + 1

通过变量替换，变为了熟悉的、主定理能解决的 形式 => S(m) ~ O(m)

故，S(m) = T(log(n)) ~ O(log(n))

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关键点：match 主定理（3）的条件。

a*f(n/b) <= c*f(n)  ->

3*f(n/4) <= c*f(n)  ->

3*(n/4)*log(n/4) <= c*n*log(n)  ->

3/4 * n*log(n/4) <= c*n*log(n)

可见，有c<1时，是满足的。

答案就是O(nlogn)

When f(x) = x*sqrt(x+1)

这里的常数1是个tricky.

去掉它，x^(3/2)

变为x，sqrt(2)*[x^(3/2)]

可见，f(x)~Θ(x^(3/2))

T(n)=T(⌊n/2⌋)+T(⌊n/4⌋)+T(⌊n/8⌋)+n.

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n+(7/8)*n+(7/8)^2 *n+(7/8)^3 *n+⋯ 等比数列！

so we have a geometric series and thus，

for our upper bound.

In a similar way, we could count just the contribution of the leftmost branches and conclude that T(n)≥2n.

Putting these together gives us the desired bound, T(n)=Θ(n)

Extended:

Recurrence Relation T(n)=T(n/8)+T(n/4)+lg(n)

太难：Solution.

(a) T(n) = T(n-1) + cn         link

(b) T(n) = T(n-1) + log(n)    link

(c) T(n) = T(n-1) + n^2      link

(d) T(n) = T(n-1) + 1/n       link

(a) 递归展开，探寻规律：

T(n)=T(n−3)+(n−2)c+(n−1)c+nc

At the end we use T(2)=T(1)+2c=1+2cT(2)=T(1)+2c=1+2c. We conclude that

T(n)=1+(2+3+⋯+n)c.

可见是O(n^2)

(b) 递归展开，探寻规律：

T(n) = T(n - 1) + log n

= T(n - 2) + log (n - 1) + log n

= T(n - 3) + log (n - 2) + log (n - 1) + log n

= ...

= T(0) + log 1 + log 2 + ... + log (n - 1) + log n

= T(0) + log n!

According to Stirling's formula, 可见是O(n*log(n))

(c) 递归展开，探寻规律：

T(n)=T(0)+1^2+2^2+3^2+⋯+n^2

Or simply,

O(n^3)

(d) 这里是个调和级数

This is the nth harmonic number, H_n = 1 + 1/2 + 1/3 + ... + 1/n.

"所有调和级数都是发散于无穷的。但是其拉马努金和存在，且为欧拉常数。"

In fact, H_n = ln(n) + γ + O(1/n)

Hint：比较难，需换两次元。