no11.htm

常用经典算法及讲解：贪婪

color:black;mso-font-kerning:0pt'>i</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'>。相应的优化函数是</span><i><span
lang=EN-US style='font-size:5.0pt;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;
font-family:Symbol;mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>&aring;</span><i><span lang=EN-US style='font-size:5.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>i</span></i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>= 1</span><i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>x</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'>i </span></i><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;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'>满足限制条件的每一组</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>x</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'>i </span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>都是一个可行解，能使</span><i><span lang=EN-US style='font-size:5.0pt;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;font-family:Symbol;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>&aring;</span><i><span lang=EN-US style='font-size:
5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>i</span></i><span
lang=EN-US style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>= 1</span><i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>x</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'>i </span></i><span style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>取得最大值的方案是最优解。<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></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'>1-3 [</span></b><b><span style='font-size:
10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>最小代价通讯网络</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'>城市及城市之间所有可能的通信连接可被视作一个无向图，图的每条边都被赋予一个权值，权值表示建成由这条边所表示的通信连接所要付出的代价。包含图中所有顶点（城市）的连通子图都是一个可行解。设所有的权值都非负，则所有可能的可行解都可表示成无向图的一组生成树，而最优解是其中具有最小代价的生成树。<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'>在这个问题中，需要选择一个无向图中的边集合的子集，这个子集必须满足如下限制条件：所有的边构成一个生成树。而优化函数是子集中所有边的权值之和。<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'><span lang=EN-US style='font-size:12.0pt;font-family:
Arial;mso-fareast-font-family:仿宋_GB2312;color:blue;mso-font-kerning:0pt'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p class=MsoNormal align=left style='text-align:left;mso-layout-grid-align:
none;text-autospace:none'><span lang=EN-US style='font-size:12.0pt;font-family:
Arial;mso-fareast-font-family:仿宋_GB2312;color:blue;mso-font-kerning:0pt'>1.2 </span><span
style='font-size:12.0pt;font-family:仿宋_GB2312;color:blue;mso-font-kerning:0pt'>算法思想<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'><span lang=EN-US style='font-size:10.0pt;font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></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'>在贪婪算法（</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>greedy method</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>）中采用逐步构造最优解的方法。在每个阶段，都作出一个看上去最优的决策（在一定的标准下）。决策一旦作出，就不可再更改。作出贪婪决策的依据称为贪婪准则（</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>greedy criterion</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:20.1pt;
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></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'>1-4 [</span></b><b><span style='font-size:
10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>找零钱</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'>一个小孩买了价值少于</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'>美元的糖，并将</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'>美元的钱交给售货员。售货员希望用数目最少的硬币找给小孩。假设提供了数目不限的面值为</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'>美分、</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'>美分、</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>5</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>美分、及</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'>美分的硬币。售货员分步骤组成要找的零钱数，每次加入一个硬币。选择硬币时所采用的贪婪准则如下：每一次选择应使零钱数尽量增大。为保证解法的可行性（即：所给的零钱等于要找的零钱数），所选择的硬币不应使零钱总数超过最终所需的数目。<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'>假设需要找给小孩</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>6 7</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>美分，首先入选的是两枚</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'>美分的硬币，第三枚入选的不能是</span><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>2 5</span><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>美分的硬币，否则硬币的选择将不可行（零钱总数超过</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>6 7</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>美分），第三枚应选择</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'>美分的硬币，然后是</span><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>5</span><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>美分的，最后加入两个</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'>美分的硬币。<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'>贪婪算法有种直觉的倾向，在找零钱时，直觉告诉我们应使找出的硬币数目最少（至少是接近最少的数目）。可以证明采用上述贪婪算法找零钱时所用的硬币数目的确最少（见练习</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 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'><b><span style='font-size:10.0pt;font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>例</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'>1-5 [</span></b><b><span style='font-size:
10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>机器调度</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'>现有</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'>件任务和无限多台的机器，任务可以在机器上得到处理。每件任务的开始时间为</span><i><span lang=EN-US
style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>s</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'>i</span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>，完成时间为</span><i><span lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:
仿宋_GB2312;color:black;mso-font-kerning:0pt'>f</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'>i </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'>s</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'>i </span></i><span
lang=EN-US style='font-size:10.0pt;mso-fareast-font-family:仿宋_GB2312;
color:black;mso-font-kerning:0pt'>&lt; <i>f</i></span><i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>i </span></i><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'>[<i>s</i></span><i><span lang=EN-US style='font-size:
5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>i </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>f</i></span><i><span lang=EN-US
style='font-size:5.0pt;mso-fareast-font-family:仿宋_GB2312;color:black;
mso-font-kerning:0pt'>i </span></i><span lang=EN-US style='font-size:10.0pt;
mso-fareast-font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>] </span><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>为处理任务</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 </span></i><span
style='font-size:10.0pt;font-family:仿宋_GB2312;color:black;mso-font-kerning:
0pt'>的时间范围。两个任务</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</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'>j </span></i><span style='font-size:10.0pt;
font-family:仿宋_GB2312;color:black;mso-font-kerning:0pt'>重指两个任务的时间范围区间有重叠，而并非是指</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</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'>j </span></i><span style='font