Lecture2

Recurrences¶

Bentley, Haken and Saxe

General method

Example

\[T(n) = 9 T(\lfloor n / 3 \rfloor) + n\]

Example

A tighter upper bound?

Assume $T(k) \leq ck^2$ for $k < n$

Wrong!

但是这个失败的证明可以用来证明 $T(n) = \Omega(n^2)$

Example

Strengthen the inductive hypothesis

Assume $T(k) \leq c_1 k^2 - c_2 k$ for $k < n$

hence $T(n) = \Theta(n^2)$

The master method applies to recurrences of the form

\[ T(n) = aT(n/b) + f(n)\]

where $a\geq 1$, $b \geq 1$, and $f$ is asymptotically positive.

The master theorem

if $f(n) = O(n^{\log_b a-\epsilon})$ for some constant $\epsilon > 0$, then $T(n) = \Theta(n^{\log_b a})$
if $f(n) = \Theta(n^{\log_b a)$, then $T(n) = \Theta(n^{\log_b a} \lg n$
if $f(n) = \Omega(n^{\log_b a+\epsilon})$ for some constant $\epsilon > 0$, and if $af(n/b) \leq cf(n)$ for some constant $c < 1$ and all sufficiently large $n$, then $T(n) = \Theta(f(n))$

When $f(n)$ is smaller than $n^{\log_b a}$ but not polynomially smaller. This is a gap between case 1 and 2
smaller -> larger
When the regularity condition in case 3 fails to hold

Example

\[T(n) = 2T(n/2) + n\lg n\]

The master method fails

\[ T(n) = \sum_{i = 1}^k a_i T(\lfloor n/b_i \rfloor) + f(n) \]

We first find $p$ s.t. $$ \sum_{i = 1}^{p} a_i b_i ^{-p} = 1 $$

The solution is

\[ T(n) = \Theta(n^p) + \Theta(n^p \int_{n'}^{n} \frac{f(x)}{x^{p + 1}} \mathrm{d}x) \]

Note

\[ a^h = a^{\log_b n} = n^{\log_b a} \]

Case1: The weight increases geometrically from the root to the leaves. The leaves hold a constant fraction of the total weight.
Case2: The weight is approximately the same on each of the $\log_b n$ levels
Case3: The weight decreases geometrically from the root to the leaves.

Example

\[ T(n) = 2T(\lfloor \sqrt{n} \rfloor) + \lg n \]