【洛谷3112】【USACO14DEC】后卫马克Guard Mark

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#状压dp

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本题涉及一群牛通过堆叠来达到足够高度以接住飞盘的目标。每头牛都有特定的高度、重量和耐力值，需通过算法找出是否能成功堆叠并计算堆叠稳定性。

题目描述

Farmer John and his herd are playing frisbee. Bessie throws the

frisbee down the field, but it's going straight to Mark the field hand

on the other team! Mark has height H (1 <= H <= 1,000,000,000), but

there are N cows on Bessie's team gathered around Mark (2 <= N <= 20).

They can only catch the frisbee if they can stack up to be at least as

high as Mark. Each of the N cows has a height, weight, and strength.

A cow's strength indicates the maximum amount of total weight of the

cows that can be stacked above her.

Given these constraints, Bessie wants to know if it is possible for

her team to build a tall enough stack to catch the frisbee, and if so,

what is the maximum safety factor of such a stack. The safety factor

of a stack is the amount of weight that can be added to the top of the

stack without exceeding any cow's strength.

FJ将飞盘抛向身高为H(1 <= H <= 1,000,000,000)的Mark，但是Mark

被N(2 <= N <= 20)头牛包围。牛们可以叠成一个牛塔，如果叠好后的高度大于或者等于Mark的高度，那牛们将抢到飞盘。

每头牛都一个身高，体重和耐力值三个指标。耐力指的是一头牛最大能承受的叠在他上方的牛的重量和。请计算牛们是否能够抢到飞盘。若是可以，请计算牛塔的最大稳定强度，稳定强度是指，在每头牛的耐力都可以承受的前提下，还能够在牛塔最上方添加的最大重量。

输入输出格式

输入格式：

INPUT: (file guard.in)

The first line of input contains N and H.

The next N lines of input each describe a cow, giving its height,

weight, and strength. All are positive integers at most 1 billion.

输出格式：

OUTPUT: (file guard.out)

If Bessie's team can build a stack tall enough to catch the frisbee, please output the maximum achievable safety factor for such a stack.

Otherwise output "Mark is too tall" (without the quotes).

输入输出样例

输入样例#1：

输出样例#1：

题解

由于数据较小，所以想到状压dp，在当前状态加一头牛看最优解。

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#define ll long long
using namespace std;
ll H,dp[1<<21],f[1<<21];//f[i]表示i这种状态的最大稳定强度 dp[i]表示最大高度 
ll h[30],w[30],s[30],ans=-1;
int main(){ 
	int n;
	scanf("%d%lld",&n,&H);
	for(int i=0;i<n;i++){
		scanf("%lld%lld%lld",&h[i],&w[i],&s[i]);
	}
	memset(f,-1,sizeof(f));
	f[0]=2147483647;
	int end=(1<<n)-1;
	for(int i=0;i<=end;i++){
		for(int j=0;j<n;j++){
			if(f[i]>=w[j]){
				if((i>>j)&1)
			    continue;
			    int k=i^(1<<j);
			    f[k]=max(f[k],min(s[j],f[i]-w[j]));
				dp[k]=dp[i]+h[j];
				if(dp[k]>=H){
					ans=max(ans,f[k]); 
				}
			}
		}
	}
	if(ans<0) printf("Mark is too tall\n");
	else printf("%lld\n",ans);
	system("pause");
	return 0;
}