公式求幂→二分求幂→快速求幂→快速求幂取模

来源:http://blog.csdn.net/qq_32680617/article/details/50640622

公式求幂→二分求幂→快速求幂→快速求幂取模 
等不急的可以直接下拉到最后看快速幂取模。

直接用C语言的库函数pow()（别忘了它的头文件#include<math.h>），似乎很简单，但是它的时间复杂度高达O(n)。 
显然，这很容易超时。 
于是有了下面的二分求幂（时间复杂度O(lgn)）

二分求幂的原理可以用下面这张图表示 

用递归来实现，虽然代码有点长，但是很好理解

<code class="hljs mel has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">pow</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> a,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> n)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//返回值是a的n次方</span>{    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(n==<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//递归终止条件</span>        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(n==<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> a;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> result=<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">pow</span>(a,n/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>);<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//二分递归</span>    result=result<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*result</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//这部分奇数偶数都一样</span>    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(n<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%2</span>==<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果n是奇数，就要多乘一次</span>        result=result<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*a</span>;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> result;}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul>

用非递归，更加简洁

<code class="hljs mel has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">pow</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> a,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> n)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//返回值是a的n次方</span>{    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> result=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(n!=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)    {        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(n<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%2</span>==<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果n是奇数</span>            result=result<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*a</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//就要多乘一次</span>        a=a<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*a</span>;        n=n/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//二分</span>    }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> result;}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul>

快速幂顾名思义比二分幂又快一些， 
快速幂借助了强大的位运算，时间复杂度达到O(log₂N)。 
位运算http://baike.baidu.com/view/379209.htm 
原理其实不难理解，用的还是小学生的公式。 
例如 求a的11次方 
11的二进制是1011 
11 = 2³×1 + 2²×0 + 2¹×1 + 2º×1 
因此，我们将a¹¹转化为算  
用非递归的代码实现

<code class="hljs glsl has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">pow</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> a,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> n)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//返回值是a的n次方</span>{    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> result=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,flag=a;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(n!=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)    {        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(n&<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果n是奇数，即n的二进制最末位为1时</span>            result=result*flag;        flag=flag*flag;        n=n>><span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//n的二进制右移一位，即n/2</span>    }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> result;}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul>

当然还能用递归来实现，但是太复杂，我没学会…

刷题中让直接求幂的不多，求幂后取模的却不少，毕竟求幂结果太大了。 
水平所限，只会用二分幂取模，时间复杂度与二分幂一样O(lgn)。 
基本可以在各种比赛中顺利通过，也是目前比较常用的方法

原理同样很简单，都是小学学过的：积的取余等于取余的积取余 
接下来用代码实现

<code class="hljs mel has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">pow</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> a,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> n,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> b)<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//返回值是a的n次方对b取余后的值</span>{    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> result=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;    a=a<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%b</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//积的取余等于取余的积取余</span>    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(n><span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)    {        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(n<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%2</span>==<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)            result=result<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*a</span><span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%b</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//n是奇数的话就要多乘一次，原理和前面的二分求幂一样</span>        n=n/<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//二分</span>        a=a<span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">*a</span><span class="hljs-variable" style="color: rgb(102, 0, 102); box-sizing: border-box;">%b</span>;<span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//积的取余等于取余的积取余</span>    }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> result;}</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li></ul><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li></ul>

影响计算机效率的是运算次数，而不是运算结果。 
所以前面几个算法都是通过增大运算结果，减少运算次数，提高计算机效率。