算法设计与分析——回溯法——01背包问题

短命女 2022-10-08 00:56 132阅读 0赞

//0-1背包问题 回溯法求解
    #include<bits/stdc++.h>
    #include <iostream>
    
    using namespace std; 
    
    template<class Typew,class Typep>
    class Knap
    {
        
    	public:
    		Typep Bound(int i);
    		void Backtrack(int i);
    	
    		Typew c;	//背包容量
    		int n;		//物品数
    		Typew *w;	//物品重量数组
    		Typep *p;	//物品价值数组
    		Typew cw;	//当前重量
    		Typep cp;	//当前价值
    		Typep bestp;//当前最后价值
    
    };
    
    template<class Typew,class Typep>
    
    Typep Knapsack(Typep p[],Typew w[],Typew c,int n);
    
    template <class Type>
    
    inline void Swap(Type &a,Type &b);
    
    template<class Type>
    
    void BubbleSort(Type a[],int n);
    int main()
    {
        
    	int n;
    
    	int c = 7;//背包容量	
    	cout<<"请输入物品数目:";	//物品数
    	cin>>n;
    	int p[n+1];// p[] = {0,9,10,7,4};//物品价值 下标从1开始
    		p[0]=0;
    	int w[n+1];//int w[] = {0,3,5,2,1};//物品重量 下标从1开始
    		w[0]=0;
     	cout<<"输入物品的价值序列：";
    	for(int i=1;i<=n;i++)
    	{
        
    		cin>>p[i];
    	} 
    	cout<<"输入物品的重量序列：";
    	for(int i=1;i<=n;i++)
    	{
        
    		cin>>w[i];
    	}
    
    	cout<<"背包容量为："<<c<<endl;
    	cout<<"物品重量和价值分别为："<<endl;
    	for(int i=1; i<=n; i++)
    	{
        
    		cout<<"("<<w[i]<<","<<p[i]<<") ";
    	}
    	cout<<endl;
    	cout<<"背包能装下的最大价值为："<<Knapsack(p,w,c,n)<<endl;
    	return 0;
    }
    template<class Typew,class Typep>
    void Knap<Typew,Typep>::Backtrack(int i)
    {
        
    	if(i>n)//到达叶子节点
    	{
        
    		bestp = cp;
    		return;
    	}
    	if(cw + w[i] <= c)//进入左子树
    	{
        
    		cw += w[i];
    		cp += p[i];
    		Backtrack(i+1);
    		cw -= w[i];
    		cp -= p[i];
    	}
    	if(Bound(i+1)>bestp)//进入右子树
    	{
        
    		Backtrack(i+1);
    	}
    }
    template<class Typew, class Typep>
    Typep Knap<Typew, Typep>::Bound(int i)// 计算上界
    {
        
    	Typew cleft = c - cw;  // 剩余容量
    	Typep b = cp;
    	// 以物品单位重量价值递减序装入物品
    	while (i <= n && w[i] <= cleft) 
    	{
        
    		cleft -= w[i];
    		b += p[i];
    		i++;
    	}
       // 装满背包
       if (i <= n)
       {
        
    	   b += p[i]/w[i] * cleft;
       }
       return b;
    }
    class Object
    {
        
    	template<class Typew,class Typep>
    	friend Typep Knapsack(Typep[],Typew [],Typew,int);
    	public:
    		int operator <= (Object a)const
    		{
        
    			return (d>=a.d);
    		}
    	private:
    		int ID;
    		float d;	
    };
    template<class Typew,class Typep>
    Typep Knapsack(Typep p[],Typew w[],Typew c,int n)
    {
        
    	//为Knap::Backtrack初始化
    	Typew W = 0;
    	Typep P = 0;
    	Object *Q = new Object[n];
    	for(int i=1; i<=n; i++) 
    	{
        
    		Q[i-1].ID = i;
    		Q[i-1].d = 1.0 * p[i]/w[i];
    		P += p[i];
    		W += w[i];
    	}
    	if(W <= c)//装入所有物品
    	{
        
    		return P;
    	}
    
    	//依物品单位重量价值排序
    
    	BubbleSort(Q,n);
    	Knap<Typew,Typep> K;
    	K.p = new Typep[n+1];
    	K.w = new Typew[n+1];
    	for(int i=1; i<=n; i++)
    	{
        
    		K.p[i] = p[Q[i-1].ID];
    		K.w[i] = w[Q[i-1].ID];
    	}
    	K.cp = 0;
    	K.cw = 0;
    	K.c = c;
    	K.n = n;
    	K.bestp = 0;
    	//回溯搜索
    	K.Backtrack(1);
    	delete []Q;
    	delete []K.w;
    	delete []K.p;
    	return K.bestp;
    }
    
    template<class Type>
    void BubbleSort(Type a[],int n)
    {
        
    	 //记录一次遍历中是否有元素的交换   
         bool exchange;  
    	 for(int i=0; i<n-1;i++)  
    	 {
          
    		exchange = false ;  
    		for(int j=i+1; j<=n-1; j++)  
    		{
          
    			if(a[j]<=a[j-1])  
    			{
          
    				Swap(a[j],a[j-1]); 
    				exchange = true;  
    			}   
    		}   
    		//如果这次遍历没有元素的交换,那么排序结束   
    		if(false == exchange)  
    		{
        
    			 break ;  
    		}
    	 }
    }
    template <class Type>
    inline void Swap(Type &a,Type &b)
    {
        
    	Type temp = a;
    	a = b;
    	b = temp;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzUwNjc1ODEz_size_16_color_FFFFFF_t_70]  
贪心算法解决背包问题的主要时间用在了将其各种物品按其单位重量的价值从小到大排序 O(n\*logn)

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzUwNjc1ODEz_size_16_color_FFFFFF_t_70]: /images/20221005/0f89c72e505f42aba6ba9aa28c150ac2.png