From f62898961aa7d8db395843f497e429bac17e344a Mon Sep 17 00:00:00 2001 From: SunnyQjm Date: Wed, 24 Jun 2020 22:48:37 +0800 Subject: [PATCH] add: alogorithm-analysis --- .../image-20200623230918389.png | Bin 0 -> 13711 bytes .../image-20200623231215489.png | Bin 0 -> 14596 bytes .../image-20200624204630269.png | Bin 0 -> 97384 bytes .../image-20200624213050265.png | Bin 0 -> 30274 bytes .../image-20200624213228553.png | Bin 0 -> 26596 bytes .../image-20200624213240115.png | Bin 0 -> 26596 bytes .../image-20200624215757473.png | Bin 0 -> 26596 bytes .../image-20200624215759947.png | Bin 0 -> 26596 bytes algorithm-analysis.md | 394 ++++++++++++++++++ 9 files changed, 394 insertions(+) create mode 100644 algorithm-analysis.assets/image-20200623230918389.png create mode 100644 algorithm-analysis.assets/image-20200623231215489.png create mode 100644 algorithm-analysis.assets/image-20200624204630269.png create mode 100644 algorithm-analysis.assets/image-20200624213050265.png create mode 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qAeV!qx5c53rnLTld&BW#<{F*o1QxZkaiA@UPEXfJr{uO>)c*sqqsA}* literal 0 HcmV?d00001 diff --git a/algorithm-analysis.md b/algorithm-analysis.md new file mode 100644 index 0000000..28d5cf9 --- /dev/null +++ b/algorithm-analysis.md @@ -0,0 +1,394 @@ +# 算法分析与复杂性理论 + +## 1. 函数渐进的界 + +### 1.1 大 $O$ 符号 + +- **定义:** + + 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若存在正数 $c$ 和 $n_0$ ,使得对于一切 $n > n_0$ 有 $0 \le f(n) \le cg(n)$** 成立,则称 **$f(n)$ 的渐进上界是 $g(n)$**,记作: + + $f(n) = O(g(n))$ + + > [自然数]([https://baike.baidu.com/item/%E8%87%AA%E7%84%B6%E6%95%B0/385394?fr=aladdin](https://baike.baidu.com/item/自然数/385394?fr=aladdin)):自然数是指用以计量事物的件数或表示事物次序的数。即用数码0,1,2,[3](https://baike.baidu.com/item/3/5833),4……所表示的数。自然数由0开始,一个接一个,组成一个无穷的集体。自然数有有序性,无限性。分为偶数和奇数,合数和质数等。 + +- **说明:** + + - $f(n) = O(g(n))$,$f(n)$ 的阶不高于 $g(n)$ 的阶; + - 可能存在多个正数 $c$ ,只要指出一个即可; + - 对前面有限多个值可以不满足不等式; + - 常数函数可以写作 $O(1)$ + +- **栗子:** + 设 $f(n) = n^2 + n$,则: + + - $f(n) = O(n^2)$,取 $c = 2, n_0 = 1$ 即可; + - $f(n) = O(n^3)$,取 $c = 1, n_0 = 2$ 即可; + +### 1.2 大 $\Omega$ 符号 + +- **定义:** + + 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若存在正数 $c$ 和 $n_0$ ,使得对于一切 $n > n_0$ 有 $0 \le cg(n) \le f(n)$** 成立,则称 **$f(n)$ 的渐进下界是 $g(n)$**,记作: + + $f(n) = \Omega(g(n))$ + +- **说明:** + + - $f(n) = \Omega(g(n))$,$f(n)$ 的阶不低于 $g(n)$ 的阶; + - 可能存在多个正数 $c$ ,只要指出一个即可; + - 对前面有限多个值可以不满足不等式; + +- **栗子:** + + 设 $f(n) = n^2 + n$,则: + + - $f(n) = \Omega(n^2)$,取 $c = 1, n_0 = 1$ 即可; + - $f(n) = \Omega(100n)$,取 $c = \frac{1}{100}, n_0 = 1$ 即可; + +### 1.3 小 $o$ 符号 + +- **定义:** + + 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若对于任意正数 $c$ 都存在 $n_0$**,**使得对于一切 $n > n_0$ 有 $0 \le f(n) \le cg(n)$** 成立,则记作: + + $f(n) = o(g(n))$ + +- **说明:** + + - $f(n) = o(g(n))$,$f(n)$ 的阶低于 $g(n)$ 的阶; + - 对不同的正数 $c$,$n_0$ 不一样,$c$ 越小 $n_0$ 越大; + - 对前面有限多个值可以不满足不等式; + +- **栗子:** + + 设 $f(n) = n^2 + n$,则:$f(n) = o(n^3)$ + + 证: + + - 当 $c \ge 1$ 时显然成立,只要取 $n_0 = 2$,$n^2 + n < cn^3$; + + - 当 $0 < c < 1$ 时,取 $n_0 = \left \lceil \frac{2}{c} \right \rceil$ 即可,因为当 $n \ge n_0$: + + $cn \ge cn_0 > 2$ + + $n^2 + n < 2n^2 < cn^3$ + +### 1.4 小 $\omega$ 符号 + +- **定义:** + + 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若对于任意正数 $c$ 都存在 $n_0$**,**使得对于一切 $n > n_0$ 有 $0 \le cg(n) \le f(n) $** 成立,则记作: + + $f(n) = \omega(g(n))$ + +- **说明:** + + - $f(n) = o(g(n))$,$f(n)$ 的阶高于 $g(n)$ 的阶; + - 对不同的正数 $c$,$n_0$ 不一样,$c$ 越小 $n_0$ 越大; + - 对前面有限多个值可以不满足不等式; + +### 1.5 $\Theta$ 符号 + +- **定义:** + + 若 $f(n) = O(g(n))$ 且 $f(n) = \Omega(g(n))$,则记作: + + $f(n) = \Theta(g(n))$ + +- **说明:** + + - $f(n)$ 的阶与 $g(n)$ 的阶相同 + - 对前面有限多个值可以不满足条件; + +| | 定义 | 说明 | +| :------: | :----------------------------------------------------------: | :----------------------------------------------------------: | +| $O$ | 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若存在正数 $c$ 和 $n_0$ ,使得对于一切 $n > n_0$ 有 $0 \le f(n) \le cg(n)$** 成立,则称 **$f(n)$ 的渐进上界是 $g(n)$**,记作:$f(n) = O(g(n))$ | $f(n) = O(g(n))$,$f(n)$ 的阶不高于 $g(n)$ 的阶; 可能存在多个正数 $c$ ,只要指出一个即可; 对前面有限多个值可以不满足不等式; 常数函数可以写作 $O(1)$ | +| $\Omega$ | 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若存在正数 $c$ 和 $n_0$ ,使得对于一切 $n > n_0$ 有 $0 \le cg(n) \le f(n)$** 成立,则称 **$f(n)$ 的渐进下界是 $g(n)$**,记作:$f(n) = \Omega(g(n))$ | $f(n) = \Omega(g(n))$,$f(n)$ 的阶不低于 $g(n)$ 的阶; 可能存在多个正数 $c$ ,只要指出一个即可; 对前面有限多个值可以不满足不等式; | +| $o$ | 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若对于任意正数 $c$ 都存在 $n_0$**,**使得对于一切 $n > n_0$ 有 $0 \le f(n) \le cg(n)$** 成立,则记作:$f(n) = o(g(n))$ | $f(n) = o(g(n))$,$f(n)$ 的阶低于 $g(n)$ 的阶; 对不同的正数 $c$,$n_0$ 不一样,$c$ 越小 $n_0$ 越大; 对前面有限多个值可以不满足不等式; | +| $\omega$ | 设 $f$ 和 $g$ 是定义域为自然数集N上的函数。**若对于任意正数 $c$ 都存在 $n_0$**,**使得对于一切 $n > n_0$ 有 $0 \le cg(n) \le f(n) $** 成立,则记作:$f(n) = \omega(g(n))$ | $f(n) = o(g(n))$,$f(n)$ 的阶高于 $g(n)$ 的阶; 对不同的正数 $c$,$n_0$ 不一样,$c$ 越小 $n_0$ 越大; 对前面有限多个值可以不满足不等式; | +| $\Theta$ | 若 $f(n) = O(g(n))$ 且 $f(n) = \Omega(g(n))$,则记作:$f(n) = \Theta(g(n))$ | $f(n)$ 的阶与 $g(n)$ 的阶相同 对前面有限多个值可以不满足条件; | + +## 2. 函数渐进界的定理 + +### 2.1 定理1:`Knowledge` + +- **定理内容:** + + 设 $f$ 和 $g$ 是定义域为自然数集合的函数。 + + - 如果 $\lim_{n \to \infty}\frac{f(n)}{g(n)}$ 存在,并且等于某个常数 $c > 0$ ,那么: + + $f(n) = \Theta(g(n))$ + + - 如果 $\lim_{n \to \infty}\frac{f(n)}{g(n)} = 0$ ,那么: + + $f(n) = o(g(n))$ + + - 如果 $\lim_{n \to \infty}\frac{f(n)}{g(n)} = +\infty$ ,那么: + + $f(n) = \omega(g(n))$ + +- **栗子:** + + 设 $f(n) = \frac{1}{2}n^2 - 3n$,证明 $f(n) = \Theta(n^2)$ + + 证:因为 + + ​ $\lim_{n \to \infty}\frac{f(n)}{n^2} = \lim_{n \to \infty}\frac{\frac{1}{2}n^2 - 3n}{n^2} = \frac{1}{2}$ + + 根据定理1,有 $f(n) = \Theta(n^2)$ + +- **根据定理1,得到的一些重要的结论:** + + - $n^d = o(r^n), r > 1, d > 0$ => **多项式函数的阶低于指数函数的阶** + - $\ln n = o(n^d), d > 0$ => **对数函数的阶低于幂函数的阶** + +### 2.2 定理2: + +- **定理内容:** + + 设 $f, g, h$ 的定义域为自然数集合:(**函数阶之间的关系具有可传递性**) + + - 如果 $f = O(g)$,且 $f = O(h)$,那么 $f = O(h)$; + - 如果 $f = \Omega(g)$,且 $f = \Omega(h)$,那么 $f = \Omega(h)$; + - 如果 $f = \Theta(g)$,且 $f = \Theta(h)$,那么 $f = \Theta(h)$; + +### 2.3 定理3: + +- **定理内容:** + + 设 $f$ 和 $g$ 是定义域为自然数集合的函数,若对某个其它的函数 $h$ ,有 $f = O(h)$ 和 $g = O(h)$,那么: + + $f + g = O(h)$ + + => **该性质可以推广到有限个函数** + +- 算法的时间复杂度是各步操作时间之和,在常数步的情况下取最高阶的函数即可。 + +## 4. 基本函数 + +### 4.1 对数函数 + +- **符号:** + - $\log n = \log_2 n$ + - $\log^k n = (\log n)^k$ + - $\log\log n = \log(\log n)$ +- **性质:** + - $\log_2 n = \Theta(\log_l n)$ + - $log_b n = o(n^\alpha), \alpha > 0$ + - $\alpha ^{\log_b n} = n^{\log_b \alpha} $ + +### 4.2 指数函数与阶乘 + +- **斯特林公式(*Stirling*):** + + $n! = \sqrt{2\pi n}(\frac{n}{e})^n(1 + \Theta(\frac{1}{n}))$ + +- **由 *Stirling* 公式得到的结论:** + + - $n! = o(n^n)$ + - $n! = \omega(2^n)$ + - $\log(n!) = \Theta(n\log n)$ + +- **$\log(n!) = \Theta(n\log n)$ 证明:** **`Knowledge`** + + - $\log(n!) = \Omega(n\log n)$ 的证明: + + ![image-20200623230918389](algorithm-analysis.assets/image-20200623230918389.png) + + $\log(n!) = \sum_{k = 1}^n\log k \ge \int_1^n \log x dx = \log e(n\ln n - n + 1) = \Omega(n\log n)$ + + - $\log(n!) = O(n\log n)$ 的证明: + + ![image-20200623231215489](algorithm-analysis.assets/image-20200623231215489.png) + + $\log(n!) = \sum_{k = 1}^n\log k \le \int_{2}^{n + 1} \log x dx = \log e(n\ln n - n + 1) = O(n\log n)$ + +### 4.3 取整函数 + +- **定义:** + - $\left \lfloor x \right \rfloor$ :表示小于等于x的最大整数 + - $\left \lceil x \right \rceil$ :表示大于等于x的最大整数 +- **性质:** + - $x - 1 < \left \lfloor x \right \rfloor \le x \le \left \lceil x \right \rceil < x + 1$ + - $\left \lfloor x + n \right \rfloor = \left \lfloor x \right \rfloor + n, \left \lceil x + n \right \rceil = \left \lceil x \right \rceil + n$ + - $\left \lceil \frac{n}{2} \right \rceil + \left \lfloor \frac{n}{2} \right \rfloor = n$ + - $\left \lceil \frac{\left \lceil \frac{n}{a} \right \rceil}{b} \right \rceil = \left \lceil \frac{n}{ab} \right \rceil, \left \lfloor \frac{\left \lfloor \frac{n}{a} \right \rfloor}{b} \right \rfloor = \left \lfloor \frac{n}{ab} \right \rfloor$ + +### 4.4 按照阶排序 **`Knowledge`** + +$2^{2^n}, n!, n2^n, (\frac{3}{2})^n, (\log n)^{\log n} = n^{\log\log n}$ + +$n^3, \log{(n!)} = \Theta(n\log n), n = 2^{\log n}$ + +$\log^2n, \log n, \sqrt{\log n}, \log\log n$ + +$n^{\frac{1}{\log n}} = 1$ + +## 5. 序列求和的方法 + +### 5.1 引例 + + $$\begin{aligned}(1). \sum_{k = 1}^{n - 1}\frac{1}{k(k + 1)}&=\sum_{k = 1}^{n - 1}(\frac{1}{k} - \frac{1}{k + 1})\\ &=\sum_{k = 1}^{n - 1}\frac{1}{k} - \sum_{k = 1}^{n - 1}\frac{1}{k + 1}\\ &=\sum_{k = 1}^{n - 1}\frac{1}{k} - \sum_{k = 2}^{n}\frac{1}{k} \\ &= 1 - \frac{1}{n} \\ \end{aligned} $$ + + $$\begin{aligned}(2). \sum_{t = 1}^{k}t2^{t - 1}&=\sum_{t = 1}^{k}t(2^t - 2^{t - 1}) \\ &=\sum_{t = 1}^{k}t2^t - \sum_{t = 1}^{k}t2^{t - 1} \\ &=\sum_{t = 1}^{k}t2^t - \sum_{t = 0}^{k - 1}(t + 1)2^t \\ &=\sum_{t = 1}^{k}t2^t - \sum_{t = 0}^{k - 1}t2^t - \sum_{t = 0}^{k - 1}2^t \\ &=k2^t - (2^k - 1) = (k - 1)2^k + 1\\ \end{aligned} $$ + +### 5.2 二分检索的平均时间复杂度 **`knowledge`** + +![image-20200624204630269](algorithm-analysis.assets/image-20200624204630269.png) + +$A(n) = \left \lfloor \log n \right \rfloor + \frac{1}{2}$ + +### 5.3 估计和式上界的放大法 **`knowledge`** + +- **两个放大公式:** + + - $\sum_{k = 1}^n a_k \le na_{max}$ + + - 假设存在常数 $r < 1$ ,使得对一切 $k \ge 0$ 有 $\frac{a_{k + 1}}{a_k} \le r$ 成立,则有如下结论: + + $\sum_{k = 0}^n \le \sum_{k = 0}^\infty a_0r^k = a_0\sum_{k = 0}^\infty r^k = \frac{a_0}{1 - r}$ + +- **栗子:** + + 估计$\sum_{k = 1}^n \frac{k}{3^k}$ 的上界。 + + 解: + + ​ $\sum_{k = 1}^n \frac{k}{3^k} = \sum_{k = 0}^{n}\frac{k}{3^{k}}$ + + ​ 令 $a_k = \frac{k}{3^{k}}, a_{k + 1} = \frac{k + 1}{3^{k + 1}}$,则 $\frac{a_{k + 1}}{a_k} = \frac{(k + 1)3^{k}}{(k)3^{k + 1}} = \frac{k + 1}{3k} \le \frac{2}{3} (k >= 1)$ + + 所以,由上述第二个放大公式有: + + ​ $\sum_{k = 1}^n \frac{k}{3^k} \le \sum_{k = 1}^\infty \frac{1}{3}(\frac{2}{3})^{k - 1} = \frac{1}{3}\frac{1}{1 - \frac{2}{3}} = 1$ + +### 5.4 估计和式渐进的界 **`Knowledge`** + +估计$\sum_{k = 1}^n \frac{1}{k}$ 的渐进的界 + +- $\sum_{k = 1}^n \frac{1}{k} \ge \int_1^{n + 1}\frac{dx}{x} = \ln (n + 1)$ + + ![image-20200624213050265](algorithm-analysis.assets/image-20200624213050265.png) + +- $\sum_{k = 1}^n \frac{1}{k} = \frac{1}{1} + \sum_{k = 2}^n \frac{1}{k} \le 1 + \int_1^{n}\frac{dx}{x} = \ln n + 1$ + + ![image-20200624213240115](algorithm-analysis.assets/image-20200624213240115.png) + + 所以,$\sum_{k = 1}^n \frac{1}{k} = \Theta(\ln n) = \Theta(\log n)$ + +## 6. 递推方程与算法分析 **`Knowledge`** + +- **主定理的应用背景:** + + $T(n) = aT(\frac{n}{b}) + f(n)$ + + - $a$ :规约后的子问题个数 + - $\frac{n}{b} $ :规约后子问题的规模 + - $f(n)$ :规约过程以及组合子问题的解的工作量 + + 二分检索 => $T(n) = T(\frac{n}{2}) + 1$ + + 二分归并排序 => $T(n) = 2T(\frac{n}{2}) + n - 1$ + +- **主定理:** + + 设 $a > 1, b > 1$ 为常数,$f(n)$ 为函数,$T(n)$ 为非负整数,且 $T(n) = aT(\frac{n}{b}) + f(n)$,则: + + 1. 若 $f(n) = O(n^{log_b a - \epsilon})$,$\epsilon > 0$ ,那么: + + $T(n) = \Theta(n^{\log_b a})$ + + 2. 若 $f(n) = \Theta(n^{log_b a})$,那么: + + $T(n) = \Theta(n^{\log_b a}\log n)$ + + 3. 若 $f(n) = \Omega(n^{\log_b a + \epsilon})$,$\epsilon > 0$,且对于某个常数 $c < 1$ 和充分大的 $n$ 有 $af(\frac{n}{b}) \le cf(n)$,那么: + + $T(n) = \Theta(f(n))$ + +- **例1:** + + 求解递推方程:$T(n) = 9T(\frac{n}{3}) + n$ + + 解: + + ​ $a = 9, b = 3, f(n) = n$ + + ​ $n^{\log_b a} = n^{log_3 9} = n^2$,$f(n) = O(n^{log_3 9 - 1})$ + + ​ 根据主定理规则1,其中 $\epsilon = 1$: + + ​ $T(n) = \Theta(n^2)$ + +- **例2:** + + 求解递推方程:$T(n) = T(\frac{2n}{3}) + 1$ + + 解: + + ​ $a = 1, b = \frac{3}{2}, f(n) = 1$ + + ​ $n^{log_b a} = n^{log_{\frac{3}{2}} 1} = 1$ ,$f(n) = n^{log_{\frac{3}{2}}1}$ + + ​ 根据主定理规则2: + + ​ $T(n) = \Theta(n^{\log_{\frac{3}{2}} 1} \log n) = \Theta(\log n)$ + +- **例3:** + + 求解递推方程:$T(n) = 3T(\frac{n}{4}) + n\log n$ + + 解: + + ​ $a = 3, b = 4, f(n) = n\log n$ + + ​ $n^{\log_b a} = n^{\log_4 3} \approx 0.793$ + + ​ 取 $\epsilon = 0.2$,则 $f(n) = n\log n = \Omega(n^{\log_4 3 + 0.2})$ = $\Omega(n^{0.993})$ + + ​ 条件验证:要使 $af(\frac{n}{b}) \le cf(n)$ 成立,带入 $f(n) = n\log n$ 得到: + + ​ $3(\frac{n}{4})\log (\frac{n}{4}) \le cn\log n$ + + ​ 当 $c \ge \frac{3}{4}$ 时,上述不等式可以对充分打的n成立,根据主定理规则3: + + ​ $T(n) = \Theta(f(n)) = \Theta(n\log n)$ + +- **二分检索:** + + $W(n) = W(\frac{n}{2}) + 1, W(1) = 1$ + + 解: + + ​ $a = 1, b = 2, f(n) = 1, n^{\log_2 1} = 1$ + + ​ 根据主定理规则2: + + ​ $W(n) = \Theta(\log n)$ + +- **二分归并排序:** + + $W(n) = 2W(\frac{n}{2}) + n - 1, W(1) = 0$ + + 解: + $a = 2, b = 2, f(n) = n - 1, n^{\log_2 2} = n$ + + ​ 根据主定理规则2: + + ​ $W(n) = \Theta(n\log n)$ + +- **例4:** => 不能使用主定理的情形 + + 求解递推方程:$T(n) = 2T(\frac{n}{2}) + n\log n$ + + 解: + + ​ $a = 2, b = 2, f(n) = n\log n, n^{\log_b a} = n$ + + ​ 不存在 $\epsilon > 0$ 使得:$n\log n = \Omega(n^{1 + \epsilon})$ + + ​ 不存在 $c < 1$ 使 $af(\frac{n}{b}) \le cf(n)$ 对所有充分大的 $n$ 成立 + + ​ $2(\frac{n}{2})\log{\frac{n}{2}} = n(\log n - 1) \le cn\log n$ \ No newline at end of file