no22.htm

常用经典算法及讲解

mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>B </span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>和</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>L</span></i><i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>B</span></i><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>。通过比较</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>H</span></i><i><span lang=EN-US style='font-size:5.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>A </span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>和</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>H</span></i><i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>B</span></i><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>（</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>L</span></i><i><span lang=EN-US style='font-size:5.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>A </span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>和</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>L</span></i><i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>B</span></i><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>），就能找出所有金块中最重和最轻的。因此，当</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n</span></i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>= 8</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>时，这种分而治之的方法需要</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>1 0</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>次比较。如果使用程序</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>1 - 3 1</span><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>，则需要</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>1 3</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>次比较。如果使用程序</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>2 - 2 6</span><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>和</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>2 - 2 7</span><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>，则最多需要</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>1 4</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>次比较。设</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>c</span></i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>(<i>n</i>)</span><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>为使用分而治之方法所需要的比较次数。为了简便，假设</span><i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>n</span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>是</span><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>2</span><span style='font-size:
10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>的幂。当</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n</span></i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>= 2</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>时，</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>c</span></i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>(<i>n</i>)
= 1</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>。对于较大的</span><i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>n</span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>，</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>c</span></i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>(<i>n</i>) = 2<i>c</i>(<i>n</i>/ 2 ) + 2</span><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>。当</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>n</span></i><span style='font-size:
10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>是</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>2</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>的幂时，使用迭代方法（见例</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>2 - 2 0</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>）可知<span lang=EN-US><o:p></o:p></span></span></p>

<p class=MsoNormal align=left style='text-align:left;mso-layout-grid-align:
none;text-autospace:none'><i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>c</span></i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>(<i>n</i>) = 3<i>n</i>/ 2 - 2</span><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>。在本例中，使用分而治之方法比逐个比较的方法少用了</span><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>2 5</span><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>％的比较次数。<span lang=EN-US><o:p></o:p></span></span></p>

<p class=MsoNormal align=left style='text-align:left;text-indent:20.1pt;
mso-char-indent-count:2.0;mso-char-indent-size:10.05pt;mso-layout-grid-align:
none;text-autospace:none'><b><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>例<span lang=EN-US>2</span></span></b><b><span
lang=EN-US style='font-size:10.0pt;font-family:Arial;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>-3 [</span></b><b><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>矩阵乘法</span></b><b><span lang=EN-US style='font-size:10.0pt;font-family:
Arial;mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>] </span></b><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>两个</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>n</span></i><span style='font-size:
10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>×</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n </span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>阶的矩阵</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>A</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>与</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>B</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>的乘积是另一个</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>×</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n </span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>阶矩阵</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>C</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>，</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>C</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>可表示为假如每一个</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>C</span></i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>(<i>i</i>, <i>j</i>) </span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>都用此公式计算，则计算</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>C</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>所需要的操作次数为</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n</span></i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>3 </span><i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>m</span></i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>+<i>n</i></span><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>2 </span><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>(<i>n</i>-
1) <i>a</i></span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>，其中</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>m</span></i><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>表示一次乘法，</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>a </span></i><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>表示一次加法或减法。<span lang=EN-US><o:p></o:p></span></span></p>

<p class=MsoNormal align=left style='text-align:left;text-indent:20.0pt;
mso-char-indent-count:2.0;mso-char-indent-size:10.0pt;mso-layout-grid-align:
none;text-autospace:none'><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>为了得到两个矩阵相乘的分而治之算法，需要： </span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>1) </span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>定义一个小问题，并指明小问题是如何进行乘法运算的；
</span><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>2) </span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>确定如何把一个大的问题划分成较小的问题，并指明如何对这些较小的问题进行乘法运算；
</span><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>3) </span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>最后指出如何根据小问题的结果得到大问题的结果。为了使讨论简便，假设</span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n </span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>是</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>2</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>的幂（也就是说， </span><i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>n</span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>是</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>1</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>，</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>2</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>，</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>4</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>，</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>8</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>，</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>1 6</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>，<span lang=EN-US>.）。<o:p></o:p></span></span></p>

<p class=MsoNormal align=left style='text-align:left;text-indent:20.0pt;
mso-char-indent-count:2.0