算法引论

快来打我* 2022-06-09 09:15 302阅读 0赞

**算法引论**

## 何谓算法（Algorithm） ##

通俗的讲，算法是指解决问题的一种方法或一个过程。严格的讲，算法是若干指令的有穷序列，满足性质：

(1) 输入：有零个或者多个外部量作为算法的输入。

(2) 输出：算法产生至少一个量作为输出。

(3) 确定性：组成算法的每条指令是清晰，无歧义的。

(4) 有限性：算法中每条指令的执行次数是有限的，执行每条指令的时间也是有限的。

## 何谓程序(Program) ##

程序是算法用某种程序设计语言的具体实现。程序可以不满足算法的性质(4)即有限性。例如操作系统，它是在无限循环中执行的程序，因而不是算法。然后可把操作系统的各种任务看成一些单独的问题，每一个问题由操作系统中的一个子程序通过特定的算法实现。该子程序得到的输出结果后便终止。

## 问题求解(Problem Solving) ##

![2010121110575078.jpg][]

## 算法设计过程 ##

1． 理解问题

2． 了解计算机设备的性能

3． 在精确解法和近似解法间做选择

4． 确定适当的数据结构

5． 算法设计技术

6． 详细表述算法的方法

7． 证明算法的正确性

8． 分析算法

9． 为算法写代码

## 算法复杂性分析 ##

算法复杂性 = 算法所需要的计算机资源;

算法的时间复杂性T(n);

算法的空间复杂性S(n);

其中n是问题的规模（输入大小）.

1、算法渐近复杂性

![2010121110584689.jpg][]

t(n)是T(n)的渐近性态，为算法的渐近复杂性。

在数学上， t(n)是T(n)的渐近表达式，是T(n)略去低阶项留下的主项。它比T(n) 简单。

2、渐近分析的记号

在下面的讨论中，对所有n，f(n) >= 0，g(n)>=0。

(1)渐近上界记号Ο

O(g(n)) = \{ f(n) | 存在正常数c和n0使得对所有n >=n0有：0 <= f(n)<= cg(n) \}

(2)渐近下界记号Ω

Ω(g(n)) = \{ f(n) | 存在正常数c和n0使得对所有n>=n0有：0 <=cg(n)<=f(n) \}

(3)非紧上界记号o

o(g(n)) = \{ f(n) | 对于任何正常数c>0，存在正数和n0 >0使得对所有n>=n0有：0 <=f(n)<cg(n) \}

等价于  f(n) / g(n) ->0 ，as  n->∞。

(4)非紧下界记号ω

ω(g(n)) = \{ f(n) | 对于任何正常数c>0，存在正数和n0 >0使得对所有n>=n0有：0 <=cg(n) < f(n) \}

等价于  f(n) / g(n)-> ∞，as  n->∞。 f(n) ∈ω(g(n))  等价于g(n) ∈o (f(n))

(5)紧渐近界记号Θ

Θ(g(n)) = \{ f(n) | 存在正常数c1,c2和n0使得对所有n>=n0有：c1g(n)<=f(n)<=c2g(n) \}

定理1： Θ(g(n)) = O (g(n)) ∩Ω(g(n))

3、渐近分析记号在等式和不等式中的意义

f(n)= Θ(g(n))的确切意义是：f(n)∈Θ(g(n))。

一般情况下，等式和不等式中的渐近记号Θ(g(n))表示Θ(g(n))中的某个函数。

例如：2n\*n + 3n + 1 = 2n\*n + Θ(n) 表示

2n\*n+3n +1=2n\*n + f(n)，其中f(n) 是Θ(n)中某个函数。

等式和不等式中渐近记号O,o,Ω和Θ的意义是类似的。

4、渐近分析中函数比较

f(n)= O(g(n)) ≈a <= b;

f(n)= Ω(g(n)) ≈ a >= b;

f(n)= Θ(g(n)) ≈ a = b;

f(n)= o(g(n)) ≈ a < b;

f(n)= ω(g(n)) ≈ a > b

5、渐近分析记号的若干性质

（1）传递性：

![2010121111121649.jpg][]

（2）反身性：

![2010121111130227.jpg][]

（3）对称性

![2010121111134359.jpg][]

（4）互对称性

![2010121111145385.jpg][]

几种复杂度比较

<table style="margin:5px 0px; padding:5px; border:1px solid silver; overflow:auto; color:rgb(0,0,0); border-collapse:collapse; word-break:break-word; font-family:'Courier New'!important; font-size:12px!important; background-color:rgb(245,245,245)"> 
 <tbody style="margin:0px; padding:0px"> 
  <tr style="margin:0px; padding:0px"> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">n</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">Logn</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">n</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">nlogn</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">n^2</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">n^3</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">2^n</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">n!</p> </td> 
  </tr> 
  <tr style="margin:0px; padding:0px"> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">3.3</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">3.3*10</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^2</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^3</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^3</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">3.6*10^6</p> </td> 
  </tr> 
  <tr style="margin:0px; padding:0px"> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^2</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">6.6</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^2</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">6.6*10^2</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^4</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^6</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">1.3*10^30</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">9.3*10^157</p> </td> 
  </tr> 
  <tr style="margin:0px; padding:0px"> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^3</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^3</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">1.0*10^4</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^6</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">10^9</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">&nbsp;</p> </td> 
   <td style="margin:0px; padding:3px; border:1px solid silver; border-collapse:collapse"> <p style="margin:10px auto; padding-top:0px; padding-bottom:0px">&nbsp;</p> </td> 
  </tr> 
 </tbody> 
</table>

## 算法的效率 ##

一个算法的最差效率：是指当输入规模为N时，算法在最坏情况下的效率。

一个算法的最优效率：是指当输入规模为N时，算法在最优情况下的效率。

一个算法的平均效率：在“典型”或者“随机”输入的情况下，算法会具有什么样的行为。

（1）最坏情况下的时间复杂性

Tmax(n) = max\{ T(I) | size(I)=n \}

（2）最好情况下的时间复杂性

Tmin(n) = min\{ T(I) | size(I)=n \}

（3）平均情况下的时间复杂性

Tavg(n) = ![2010121111170029.jpg][]

其中I是问题的规模为n的实例，p(I)是实例I出现的概率。

## 算法分析方法 ##

例：顺序搜索算法

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      <code style="white-space:pre-wrap; margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.8em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; min-height:auto!important; background:none!important">i=0;i&lt;n;i++)</code> 
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      <code style="white-space:pre-wrap; margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.8em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; min-height:auto!important; background:none!important">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</code> 
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      <code style="white-space:pre-wrap; margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.8em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; min-height:auto!important; color:rgb(0,0,255)!important; background:none!important">return</code>&nbsp; 
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     </div> 
    </div> </td> 
  </tr> 
 </tbody> 
</table>

（1）Tmax(n) = max\{ T(I) | size(I)=n \} = O(n)

（2）Tmin(n) = min\{ T(I) | size(I)=n \} = O(1)

（3）在平均情况下，假设：

(a) 搜索成功的概率为p ( 0 <= p<= 1 )；

(b) 在数组的每个位置i ( 0 <=i < n )搜索成功的概率相同，均为 p/n。

![2010121111181362.jpg][]

## **算法分析的基本法则** ##

1、非递归算法

（1）for / while 循环

循环体内计算时间 \* 循环次数；

（2）嵌套循环

循环体内计算时间 \* 所有循环次数；

（3）顺序语句

各语句计算时间相加；

（4）if-else语句

if语句计算时间和else语句计算时间的较大者。

2、递归算法复杂性分析

![2010121111184611.jpg][]

[2010121110575078.jpg]: /images/20220609/6f241b40d4784f45850bcb58904a6b5a.png
[2010121110584689.jpg]: /images/20220609/79146aa812d742dabb6a4aa936e91757.png
[2010121111121649.jpg]: /images/20220609/2b8f018bad6c4107b0d594bc049477fe.png
[2010121111130227.jpg]: /images/20220609/3c932251acf946f9bf9afd8fc30b34ea.png
[2010121111134359.jpg]: /images/20220609/8912080cdedb4150b2dfec893368addf.png
[2010121111145385.jpg]: /images/20220609/68aaa459864b4bd09a05c1cf39899bc5.png
[2010121111170029.jpg]: /images/20220609/539e77f4de784b63a13a86e6166221bf.png
[2010121111181362.jpg]: /images/20220609/e94bd7232f6246c38317b2bb02a7750e.png
[2010121111184611.jpg]: /images/20220609/9190c4336e504e2abfe752d4a207125e.png