01背包问题总结一
2013-11-05 23:57
211 查看
01背包问题总结(一)
一 问题描述:
有N件物品和一个容量为V的背包。第i件物品的费用是c[i],价值是w[i]。求解将哪些物品装入背包可使价值总和最大。
所谓01背包,表示每一个物品只有一个,要么装入,要么不装入。
二 解决方案:
考虑使用dp问题 求解,定义一个递归式 opt[i][v] 表示前i个物品,在背包容量大小为v的情况下,最大的装载量。
opt[i][v] = max(opt[i-1][v] , opt[i-1][v-c[i]] + w[i])
解释如下:
opt[i-1][v] 表示第i件物品不装入背包中,而opt[i-1][v-c[i]] + w[i] 表示第i件物品装入背包中。
花费如下:
时间复杂度为o(V * T) ,空间复杂度为o(V * T) 。 时间复杂度已经无法优化,但是空间复杂度则可以进行优化。
但必须将V 递减的方式进行遍历,即V.......0 的方式进行。
三 初始化:
(1)若要求背包必须放满,则初始如下:
f[0] = 0 , f[1...V]表示-INF。表示当容积为0时,只接受一个容积为0的物品入包。
(2)若要求背包可以空下,则初始化如下:
f[0...V] = 0 ,表示任意容积的背包都有一个有效解即为0。
具体解释如下:
初始化的f数组事实上就是在没有任何物品可以放入背包时的合法状态。
如果要求背包恰好装满,那么此时只有容量为0的背包可能被价值为0的nothing“恰好装满”,
其它容量的背包均没有合法的解,属于未定义的状态,它们的值就都应该是-∞了。
如果背包并非必须被装满,那么任何容量的背包都有一个合法解“什么都不装”,
这个解的价值为0,所以初始时状态的值也就全部为0了。
四 代码如下:
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
/**//*
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
01背包,使用了优化后的存储空间
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
建立数组
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[i][v] = max(f[i-1][v] , f[i-1][v-c[i]] + w[i])
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
将前i件物品,放入容量为v的背包中的最大值。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
下面介绍一个优化,使用一维数组,来表示
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
(1) f[v]表示每一种类型的物品,在容量为v的情况下,最大值。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
但是体积循环的时候,需要从v----1循环递减。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
初始化问题:
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
(1)若要求背包中不允许有剩余空间,则可以将f[0]均初始化为0,其余的f[1
![](http://www.cppblog.com/Images/dot.gif)
..n]均初始化为-INF 。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
表示只有当容积为0 的时候,允许放入质量为0的物品。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
而当容积不为0的情况下,不允许放入质量为0的物品,并且把状态置为未知状态。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
(2)若要求背包中允许有剩余空间 ,则可以将f[1
![](http://www.cppblog.com/Images/dot.gif)
![](http://www.cppblog.com/Images/dot.gif)
n],均初始化为0。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
这样,当放不下去的时候,可以空着。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockEnd.gif)
*/
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
#include <iostream>
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
using namespace std ;
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
const int V = 1000 ; //总的体积
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
const int T = 5 ; //物品的种类
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
int f[V+1] ;
#define max(a,b) a>b?a:b
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
//#define EMPTY //可以不装满
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
int w[T] =
![](http://www.cppblog.com/Images/dot.gif)
{8 , 10 , 4 , 5 , 5}; //价值
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
int c[T] =
![](http://www.cppblog.com/Images/dot.gif)
{600 , 400 , 200 , 200 , 300}; //每一个的体积
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
const int INF = -66536 ;
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
int package()
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
#ifdef EMPTY
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int i = 0 ; i <= V ;i++) //条件编译,表示背包可以不存储满
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[i] = 0 ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
#else
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[0] = 0 ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int i = 1 ; i <= V ;i++)//条件编译,表示背包必须全部存储满
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[i] = INF ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
#endif
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int i = 0 ; i < T ; i++)
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedSubBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int v = V ; v >= c[i] ;v--) //必须全部从V递减到0
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedSubBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[v] = max(f[v-c[i]] + w[i] , f[v]) ; //此f[v]实质上是表示的是i-1次之前的值。
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockEnd.gif)
}
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockEnd.gif)
}
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
return f[V] ;
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockEnd.gif)
}
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
int main()
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
int temp = package() ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
cout<<temp<<endl ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
system("pause") ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
return 0 ;
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockEnd.gif)
}
一 问题描述:
有N件物品和一个容量为V的背包。第i件物品的费用是c[i],价值是w[i]。求解将哪些物品装入背包可使价值总和最大。
所谓01背包,表示每一个物品只有一个,要么装入,要么不装入。
二 解决方案:
考虑使用dp问题 求解,定义一个递归式 opt[i][v] 表示前i个物品,在背包容量大小为v的情况下,最大的装载量。
opt[i][v] = max(opt[i-1][v] , opt[i-1][v-c[i]] + w[i])
解释如下:
opt[i-1][v] 表示第i件物品不装入背包中,而opt[i-1][v-c[i]] + w[i] 表示第i件物品装入背包中。
花费如下:
时间复杂度为o(V * T) ,空间复杂度为o(V * T) 。 时间复杂度已经无法优化,但是空间复杂度则可以进行优化。
但必须将V 递减的方式进行遍历,即V.......0 的方式进行。
三 初始化:
(1)若要求背包必须放满,则初始如下:
f[0] = 0 , f[1...V]表示-INF。表示当容积为0时,只接受一个容积为0的物品入包。
(2)若要求背包可以空下,则初始化如下:
f[0...V] = 0 ,表示任意容积的背包都有一个有效解即为0。
具体解释如下:
初始化的f数组事实上就是在没有任何物品可以放入背包时的合法状态。
如果要求背包恰好装满,那么此时只有容量为0的背包可能被价值为0的nothing“恰好装满”,
其它容量的背包均没有合法的解,属于未定义的状态,它们的值就都应该是-∞了。
如果背包并非必须被装满,那么任何容量的背包都有一个合法解“什么都不装”,
这个解的价值为0,所以初始时状态的值也就全部为0了。
四 代码如下:
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
/**//*
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
01背包,使用了优化后的存储空间
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
建立数组
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[i][v] = max(f[i-1][v] , f[i-1][v-c[i]] + w[i])
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
将前i件物品,放入容量为v的背包中的最大值。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
下面介绍一个优化,使用一维数组,来表示
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
(1) f[v]表示每一种类型的物品,在容量为v的情况下,最大值。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
但是体积循环的时候,需要从v----1循环递减。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
初始化问题:
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
(1)若要求背包中不允许有剩余空间,则可以将f[0]均初始化为0,其余的f[1
![](http://www.cppblog.com/Images/dot.gif)
..n]均初始化为-INF 。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
表示只有当容积为0 的时候,允许放入质量为0的物品。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
而当容积不为0的情况下,不允许放入质量为0的物品,并且把状态置为未知状态。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
(2)若要求背包中允许有剩余空间 ,则可以将f[1
![](http://www.cppblog.com/Images/dot.gif)
![](http://www.cppblog.com/Images/dot.gif)
n],均初始化为0。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
这样,当放不下去的时候,可以空着。
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockEnd.gif)
*/
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
#include <iostream>
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
using namespace std ;
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
const int V = 1000 ; //总的体积
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
const int T = 5 ; //物品的种类
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
int f[V+1] ;
#define max(a,b) a>b?a:b
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
//#define EMPTY //可以不装满
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
int w[T] =
![](http://www.cppblog.com/Images/dot.gif)
{8 , 10 , 4 , 5 , 5}; //价值
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
int c[T] =
![](http://www.cppblog.com/Images/dot.gif)
{600 , 400 , 200 , 200 , 300}; //每一个的体积
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
const int INF = -66536 ;
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
int package()
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
#ifdef EMPTY
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int i = 0 ; i <= V ;i++) //条件编译,表示背包可以不存储满
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[i] = 0 ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
#else
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[0] = 0 ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int i = 1 ; i <= V ;i++)//条件编译,表示背包必须全部存储满
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[i] = INF ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
#endif
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int i = 0 ; i < T ; i++)
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedSubBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
for(int v = V ; v >= c[i] ;v--) //必须全部从V递减到0
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedSubBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
f[v] = max(f[v-c[i]] + w[i] , f[v]) ; //此f[v]实质上是表示的是i-1次之前的值。
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockEnd.gif)
}
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedSubBlockEnd.gif)
}
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
return f[V] ;
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockEnd.gif)
}
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/None.gif)
int main()
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockStart.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/ContractedBlock.gif)
![](http://www.cppblog.com/Images/dot.gif)
{
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
int temp = package() ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
cout<<temp<<endl ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
system("pause") ;
![](http://www.cppblog.com/Images/OutliningIndicators/InBlock.gif)
return 0 ;
![](http://www.cppblog.com/Images/OutliningIndicators/ExpandedBlockEnd.gif)
}
相关文章推荐
- linux kernel奔溃时候的检测点
- lib和dll文件的区别和联系
- 四种语言完美实现组合算法
- 计算机组成原理-存储器的分级
- 市场调研报告:手机刷机用户市场大摸底
- 迷茫的羔羊-失落了,伤心了,难过了
- BackBone 学习 (一)
- linux信号的默认处理方式
- 实时开发框架Meteor基础入门系列<零>--简介
- jdbc批处理优缺点
- 预处理
- Win32位图、坐标系,做出自己喜欢钟表
- python 中闭包理解,及函数string.maketrans(),string.translate()
- 硬盘坏块检测和隔离 推荐
- kill命令实现分析
- WindowsMobile配置英文ROM显示中文问题(解决程序中文乱码)
- Python基础:元类
- 设计模式学习系列4 简单工厂模式
- IIS 设置
- Java File和byte数据之间的转换