no4.htm

常用经典算法及讲解

style="mso-spacerun: yes">&nbsp; </span></span><span style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>由已知及定义，易知</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x xor y&lt;2<sup>t</sup></span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>，</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'> </span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>由性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>8</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>，</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'> (x xor y)+2<sup>t
</sup>= x xor y xor 2<sup>t<span style="mso-spacerun: yes">&nbsp; </span></sup>=
x xor (y xor 2<sup>t</sup>) = x xor (y +2<sup>t</sup>), </span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>故性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>10</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>成立<span lang=EN-US>.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>11</span></b><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>:
x与偶数个y 的异或仍为x ,即：x xor y xor y xor</span><span lang=EN-US style='font-size:
12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:宋体'>…</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'> xor y=x（共偶数个y）.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>这是性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>3</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>、</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>4</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>的直接推论<span lang=EN-US>.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'><span
style="mso-spacerun: yes">&nbsp; </span>定义3：（T） </span></b><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>设<span lang=EN-US>x<sub>1</sub>,x<sub>2</sub>,</span></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,x<sub>n</sub>∈N<sup>*</sup>,<span
style="mso-spacerun: yes">&nbsp; </span>则：T=x<sub>1</sub> xor x<sub>2</sub> xor</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>xor x<sub>n</sub>.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span><b style='mso-bidi-font-weight:normal'>定义4：（T<sub>i</sub>）</b>设x<sub>1</sub>,x<sub>2</sub>,</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,x<sub>n</sub>∈N<sup>*</sup>,
<span style="mso-spacerun: yes">&nbsp;</span>则T<sub>i</sub>=T xor x<sub>i</sub>,
i=1,2,</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
mso-ascii-font-family:宋体'>…</span><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,n.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>由性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>3</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>、</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>4</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>，<span lang=EN-US> T<sub>i</sub>实质上就是对{ x<sub>1</sub>,x<sub>2</sub>,</span></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,x<sub>i-1</sub>,x<sub>i+1</sub>，</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,x<sub>n</sub>}中的n-1个数求异或.</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>定义</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>5</span></b><b style='mso-bidi-font-weight:normal'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>：（取数操作）</span></b><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'> </span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>在</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>n</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>个非负整数序列</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub></span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>中，选取一个</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x<sub>i</sub></span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>及正整数</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>b, </span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>满足</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x<sub>i</sub></span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>≥</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt'>b</span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>＞</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>0</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>，再用</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>y<sub>i</sub>=x<sub>i</sub></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>-</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>b</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>取代序列中的</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x<sub>i</sub></span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>，称为对序列</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>x<sub>1</sub>,x<sub>2</sub>,</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-hansi-font-family:"Times New Roman"'>，<span lang=EN-US>x<sub>n</sub>的一个取数操作.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span><b style='mso-bidi-font-weight:normal'>定理1：</b>给定N<sup>*</sup>中的序列x<sub>1</sub>,x<sub>2</sub>,</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-hansi-font-family:"Times New Roman"'>，<span lang=EN-US>x<sub>n</sub>（n≥2），若T≠0，必存在一个取数操作，使y<sub>i</sub>取代x<sub>i</sub>后，</span></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>T ’=x<sub>1</sub>
xor…x<sub>i-1</sub><span style="mso-spacerun: yes">&nbsp; </span>xor<span
style="mso-spacerun: yes">&nbsp; </span>y<sub>i</sub><span style="mso-spacerun:
yes">&nbsp; </span>xor<span style="mso-spacerun: yes">&nbsp; </span>x<sub>i+1 </sub><span
style="mso-spacerun: yes">&nbsp;</span>xor …x<sub>n</sub>=0.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp;</span><span style="mso-spacerun:
yes">&nbsp;</span></span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>证明：取</span><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>T<sub>i</sub>=T xor x<sub>i</sub> , </span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>如果存在某个</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x<sub>i</sub>&gt;T<sub>i</sub>
</span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>令</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>b=x<sub>i</sub></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>-</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>T<sub>i</sub>&gt;0,<span
style="mso-spacerun: yes">&nbsp; </span>y<sub>i</sub>=x<sub>i</sub></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>-</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>b=T<sub>i</sub>. </span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>由性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>3</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>、</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>4</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>,知，用y<sub>i</sub>代替序列x<sub>1</sub></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>x<sub>n</sub>中的x<sub>i,
</sub>即T</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;mso-ascii-font-family:宋体'>’</span><span lang=EN-US style='font-size:
12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>=(T
xor x<sub>i</sub> )xor y<sub>i </sub>= T<sub>i</sub> xor T<sub>i</sub> =0.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span>下面证明必存在某个i ∈</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt'>{1,2,…,n}, </span><span style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>使</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt'>x<sub>i</sub>&gt;T<sub>i </sub></span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span>设序列x<sub>1</sub></span><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;mso-ascii-font-family:宋体'>…</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>x<sub>n</sub> 对应二进制数序列中，最大数的最高非零位为t+1位，最高位的值对应于2<sup>t</sup>，设这样的数有r个(n≥r≥1)，