貪心算法——找紙幣問題
<table border="1" style="font-family:Simsun; border-collapse:collapse; width:80%; padding:0pt 5.4pt"><tbody><tr><td width="882" valign="top" style="width:661.75pt; padding:0pt 5.4pt; border:0.5pt solid rgb(0,0,0)"><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:14pt; font-family:宋體"><strong>問題主題:</strong></span><span style="font-size:10.5pt; font-family:宋體">找錢</span><span style="font-size:14pt; font-family:宋體"/></p></td></tr><tr><td width="882" valign="top" style="width:661.75pt; padding:0pt 5.4pt; border-left-width:0.5pt; border-style:none solid solid; border-left-color:rgb(0,0,0); border-right-width:0.5pt; border-right-color:rgb(0,0,0); border-bottom-width:0.5pt; border-bottom-color:rgb(0,0,0)"><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:14pt; font-family:宋體"><strong>問題描述:</strong></span><span style="font-size:14pt; font-family:宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'; text-indent:28pt"><span style="font-size:14pt; font-family:宋體">假設有</span><span style="font-size:14pt; font-family:宋體">1<span style="font-family:宋體">元、</span><span style="font-family:Times New Roman">2</span><span style="font-family:宋體">元、</span><span style="font-family:Times New Roman">5</span><span style="font-family:宋體">元、</span><span style="font-family:Times New Roman">10</span><span style="font-family:宋體">元、</span><span style="font-family:Times New Roman">20</span><span style="font-family:宋體">元、</span><span style="font-family:Times New Roman">50</span><span style="font-family:宋體">元、</span><span style="font-family:Times New Roman">100</span><span style="font-family:宋體">的紙幣分別為</span><span style="font-family:Times New Roman">c0,?c1,?c2,?c3,?c4,?c5,?c6,</span><span style="font-family:宋體">張。現在要用這些錢來支付</span><span style="font-family:Times New Roman">K</span><span style="font-family:宋體">元,至少要用多少張紙幣?如果能找,則輸出紙幣的張數,不能找則輸出</span><span style="font-family:Times New Roman">No</span></span><span style="font-size:14pt; font-family:宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:14pt; font-family:宋體"><strong>限制條件:</strong></span><span style="font-size:14pt; font-family:宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'; text-indent:28pt"><span style="font-size:14pt; font-family:宋體">0</span><span style="font-size:14pt; font-family:宋體"><=</span><span style="font-size:14pt; font-family:宋體">c0,?c1,c2,c3,c4,c5,c6</span><span style="font-size:14pt; font-family:宋體"><=</span><span style="font-size:14pt; font-family:宋體">1</span><span style="font-size:14pt; font-family:宋體">0</span><span style="font-size:14pt; font-family:宋體; vertical-align:super">9</span><span style="font-size:14pt; font-family:宋體; vertical-align:super"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'; text-indent:28pt"><span style="font-size:14pt; font-family:宋體">0</span><span style="font-size:14pt; font-family:宋體"><=</span><span style="font-size:14pt; font-family:宋體">K</span><span style="font-size:14pt; font-family:宋體"><=10</span><span style="font-size:14pt; font-family:宋體; vertical-align:super">9</span><span style="font-size:14pt; font-family:宋體"/></p></td></tr><tr><td width="882" valign="top" style="width:661.75pt; padding:0pt 5.4pt; border-left-width:0.5pt; border-style:none solid solid; border-left-color:rgb(0,0,0); border-right-width:0.5pt; border-right-color:rgb(0,0,0); border-bottom-width:0.5pt; border-bottom-color:rgb(0,0,0)"><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:14pt; font-family:宋體"><strong>樣例:</strong></span><span style="font-size:14pt; font-family:宋體"><strong/></span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:14pt; font-family:宋體">輸入</span><span style="font-size:14pt; font-family:宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'; text-indent:27pt"><span style="font-size:14pt; font-family:宋體">C0=3,?c1=0,?c2=2,?c3=1,?c4=0,?c5=3,?c6</span><span style="font-size:14pt; font-family:宋體">=</span><span style="font-size:14pt; font-family:宋體">5</span><span style="font-size:14pt; font-family:宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:14pt; font-family:宋體">輸出</span><span style="font-size:14pt; font-family:宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'; text-indent:27pt"><span style="font-size:14pt; font-family:宋體">6</span><span style="font-size:14pt; font-family:宋體"/></p></td></tr></tbody></table>
### 【解法一】
### 解題分析:
???本題使用貪心算法,只需要考慮“盡可能多地使用面值大的紙幣”,然后根據面值的大小從大到小排序依次選擇。
### 程序實現:
**C++**
<table border="1" style="font-family:Simsun; border-collapse:collapse; width:80%; padding:0pt 5.4pt"><tbody><tr><td width="882" valign="top" style="width:661.8pt; padding:0pt 5.4pt; border:0.5pt solid rgb(0,0,0)"><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">#include</span><span style="font-size:9.5pt; font-family:新宋體">?</span><span style="color:rgb(163,21,21); font-size:9.5pt; font-family:新宋體">"iostream"</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">using</span><span style="font-size:9.5pt; font-family:新宋體">?</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">namespace</span><span style="font-size:9.5pt; font-family:新宋體">?std;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">const</span><span style="font-size:9.5pt; font-family:新宋體">?</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?N?=?7;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">static</span><span style="font-size:9.5pt; font-family:新宋體">?</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?K?=?6200;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?min(</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?num1,?</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?num2);</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?momeyCount[N]?=?{3,?0,?2,?1,?0,?3,?5};</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?value[N]?=?{1,?2,?5,?10,?20,?50,?100};</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?giveChange()?{</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?num?=?0;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">for</span><span style="font-size:9.5pt; font-family:新宋體">(</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?i?=?N-1;?i?>=?0;?i?--)?{</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?c?=?min(K/value[i],?momeyCount[i]);</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體">K?=?K?-?c?*?value[i];</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體">num?+=?c;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體">}</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">if</span><span style="font-size:9.5pt; font-family:新宋體">(K?>?0)?</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體">num?=?-1;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">return</span><span style="font-size:9.5pt; font-family:新宋體">?num;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">}</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?min(</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?num1,?</span><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?num2)?{</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">return</span><span style="font-size:9.5pt; font-family:新宋體">?num1?<?num2???num1?:?num2;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">}</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">?</span></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?main()?{</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">int</span><span style="font-size:9.5pt; font-family:新宋體">?result?=?giveChange();</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">if</span><span style="font-size:9.5pt; font-family:新宋體">(result?!=?-1)</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體">cout?<<?result?<<?endl;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">else</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體"/><span style="font-size:9.5pt; font-family:新宋體">cout?<<?</span><span style="color:rgb(163,21,21); font-size:9.5pt; font-family:新宋體">"NO"</span><span style="font-size:9.5pt; font-family:新宋體">?<<?endl;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體"/><span style="color:rgb(0,0,255); font-size:9.5pt; font-family:新宋體">return</span><span style="font-size:9.5pt; font-family:新宋體">?0;</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:9.5pt; font-family:新宋體">}</span><span style="font-size:9.5pt; font-family:新宋體"/></p><p class="p0" style="margin:0pt; text-align:justify; font-size:10.5pt; font-family:'Times New Roman'"><span style="font-size:10.5pt; font-family:宋體">?</span></p></td></tr></tbody></table>
**Java**
~~~
package greed;
import java.util.Arrays;
/**
* 找錢問題
* User: luoweifu
* Date: 14-1-14
* Time: 下午8:08
*/
public class GiveChange {
public static int gaiveChange(MoneyBox[] moneyBoxes, int target) {
Arrays.sort(moneyBoxes);
int count = 0;
for(int i = moneyBoxes.length-1; i >= 0; i--) {
int currentCount = Math.min(moneyBoxes[i].getCount(), target/moneyBoxes[i].getValue());
target -= currentCount*moneyBoxes[i].getValue();
count += currentCount;
}
if(target > 0) {
count = -1;
}
return count;
}
public static void main(String args[]) {
MoneyBox[] moneyBoxes = {
new MoneyBox(1, 3),
new MoneyBox(2, 0),
new MoneyBox(5, 2),
new MoneyBox(10, 1),
new MoneyBox(20, 0),
new MoneyBox(50, 3),
new MoneyBox(100, 5),
};
int result = gaiveChange(moneyBoxes, 620);
if(result > 0)
System.out.println(result);
else
System.out.println("No");
}
}
/**
* 錢盒子
*/
class MoneyBox implements Comparable{
private int value;
private int count;
MoneyBox(int value, int count) {
this.value = value;
this.count = count;
}
int getValue() {
return value;
}
void setValue(int value) {
this.value = value;
}
int getCount() {
return count;
}
void setCount(int count) {
this.count = count;
}
@Override
public int compareTo(Object o) {
MoneyBox moneyBox = (MoneyBox)o;
if(this.value < moneyBox.getValue())
return -1;
else if(this.value == moneyBox.getValue())
return 0;
else
return 1;
}
}
~~~
****
### 算法復雜度:
???時間復雜度:如果紙幣已經排序(如C++代碼),O(n);如果紙幣未經排序 (如Java代碼),O(nlogn);