Advanced Algorithm 听课笔记（Introduction & Complexity Class）

Posted 2022-12-01 糖果天王

0x00 前言

作为学术生涯的最后一门课，选了一门据说是最难的，上下来的感觉也确实是难得不行，不太懂……
决定照着ppt和上课的笔记整理一下，以此争取达到复习的目的。
（意思是有些虽然写出来了，但自己都不见得明白，有的部分存疑后续去询问之后再做修改）

Introduction of Randomized Algorithm

0x01 Big O notation

• $f(n)=O(g(n)):\le$
  • which means $\exists c,{\overline{lim}}_{n\to \infty }\frac{f(n)}{g(n)}\le c$, c is constant
• $f(n)=o(g(n)):<$
  • which means ${\overline{lim}}_{n\to \infty }\frac{f(n)}{g(n)}=0$
• $f(n)=\Omega (g(n)):\ge$
  • which means $g(n)=O(f(n))$
• $f(n)=\omega (g(n)):>$
  • which means $g(n)=o(f(n))$
• $f(n)=\Theta (g(n)):=$
  • which means $f(n)=O(g(n))$ and $f(n)=\Omega (g(n))$

0x02 QuickSort

Sorting Problem: Given a set S of n numbers, sort them into ascending order.

• Find random $y$ of set $S$; (O(1))
• Partition $S\setminus y$ into two sets $S_1$ and $S_2$; (n)
• Recursively sort $S_1$ and $S_2$
• Time complexity: $O(nlogn)$ does not depend on input. It holds for every input.
  $$\begin{array}{rl}E(T(n))& =\frac{1}{n}\sum _{k=1}^n\left(E(T(n))|X_n=k\right)\\ & =\frac{1}{n}\sum _{k=1}^n\left(E(T(k-1))+E(T(n-k))+n-1|X_n=k\right)\\ & =\frac{1}{n}\sum _{k=1}^n\left(E(T(k-1))+E(T(n-k))+n-1\right)\\ \therefore E(T(n))& =\frac{1}{n}\sum _{k=1}^n\left(E(T(k-1))+E(T(n-k))+(n-1)\right)\end{array}$$

0x03 Comparison Probability

判断序列中的两个数在QuickSort中被比较过的概率

当第$i$个数字和第$j$个数字曾经被比较过，设 $X_{ij}=1$，反之为0。
有 $E(X_{ij})=P_{ij}$，从而有 $T(n)={\sum }_{1\le i<j\le n}{X}_{ij}$