bzoj1629 [Usaco2007 Demo]Cow Acrobats

Posted 2021-01-15 kafuuchino

Description

Farmer John‘s N (1 <= N <= 50,000) cows (numbered 1..N) are planning to run away and join the circus. Their hoofed feet prevent them from tightrope walking and swinging from the trapeze (and their last attempt at firing a cow out of a cannon met with a dismal failure). Thus, they have decided to practice performing acrobatic stunts. The cows aren‘t terribly creative and have only come up with one acrobatic stunt: standing on top of each other to form a vertical stack of some height. The cows are trying to figure out the order in which they should arrange themselves within this stack. Each of the N cows has an associated weight (1 <= W_i <= 10,000) and strength (1 <= S_i <= 1,000,000,000). The risk of a cow collapsing is equal to the combined weight of all cows on top of her (not including her own weight, of course) minus her strength (so that a stronger cow has a lower risk). Your task is to determine an ordering of the cows that minimizes the greatest risk of collapse for any of the cows. //有三个头牛，下面三行二个数分别代表其体重及力量 //它们玩叠罗汉的游戏，每个牛的危险值等于它上面的牛的体重总和减去它的力量值，因为它要扛起上面所有的牛嘛. //求所有方案中危险值最大的最小

Input

* Line 1: A single line with the integer N. * Lines 2..N+1: Line i+1 describes cow i with two space-separated integers, W_i and S_i.

Output

* Line 1: A single integer, giving the largest risk of all the cows in any optimal ordering that minimizes the risk.

Sample Input

3
10 3
2 5
3 3

Sample Output

2

OUTPUT DETAILS:

Put the cow with weight 10 on the bottom. She will carry the other
two cows, so the risk of her collapsing is 2+3-3=2. The other cows
have lower risk of collapsing.

 

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原来危险值可以为负数（大雾）

我们发现，交换两头牛的位置并不会改变其他牛的危险值。

考虑贪心

设两头牛$a,b$的体重为$w_{a},w_{b}$，力量为$h_{a}h_{b}$

分类讨论产生的代价：

1.牛$a$在牛$b$下面：$h_{a}-w_{b}$

2.牛$a$在牛$b$上面：$h_{b}-w_{a}$

显然，牛$a$在牛$b$下方更优时$h_{a}-w_{b}<=h_{b}-w_{a}$

移项得：$h_{a}+w_{a}<=h_{b}+w_{b}$

于是我们按$h_{i}+w_{i}$排序，贪心即可。

 1 #include<iostream>
 2 #include<cstdio>
 3 #include<cstring>
 4 #include<algorithm>
 5 #define re register
 6 using namespace std;
 7 struct data{
 8     int x,y;
 9     bool operator < (const data &tmp) const{return x+y<tmp.x+tmp.y;}
10 }a[50002];
11 int n; long long ans=-1e17,tot;
12 int main(){
13     scanf("%d",&n);
14     for(re int i=1;i<=n;++i) scanf("%d%d",&a[i].x,&a[i].y);
15     sort(a+1,a+n+1);
16     for(re int i=1;i<=n;++i){
17         ans=max(ans,tot-a[i].y);
18         tot+=a[i].x;
19     }printf("%lld",ans);
20     return 0;
21 }

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