2020牛客暑期多校训练营（第八场）I Interesting Computer Game K Kabaleo Lite

K Kabaleo Lite

Tired of boring WFH (work from home), Apollo decided to open a fast food restaurant, called Kabaleo Lite extbf{Kabaleo Lite}Kabaleo Lite.
The restaurant serves n kinds of food, numbered from 1 to n. The profit for the i-th kind of food is aia_iai​. Profit may be negative because it uses expensive ingredients. On the first day, Apollo prepared bib_ibi​ dishes of the i-th kind of food.
The peculiarity of Apollo's restaurant is the procedure of ordering food. For each visitor Apollo himself chooses a set of dishes that this visitor will receive. When doing so, Apollo is guided by the following rules:

• every visitor should receive at least one dish.
• each visitor should receive continuous kinds of food started from the first food. And the visitor will receive exactly 1 dish for each kind of food. For example, a visitor may receive 1 dish of the 1st kind of food, 1 dish of the 2nd kind of food, 1 dish of the 3rd kind of food.

What's the maximum number of visitors Apollo can feed? And he wants to know the maximum possible profits he can earn to have the maximum of visitors.

输入描述:

The first line of the input gives the number of test case, Tmathbf{T}T (1≤T≤101 leq mathbf{T} leq 101≤T≤10). Tmathbf{T}T test cases follow.
Each test case begins with a line containing one integers n (1≤n≤1051 le n le 10^51≤n≤105), representing the number of different kinds of food.
The second line contains n space-separated numbers aia_iai​ (−109≤ai≤109-10^9 le a_i le 10^9−109≤ai​≤109), where aia_iai​ denotes the profit of one dish of the i-th kind.
The third line contains n space-separated numbers bib_ibi​ (1≤bi≤1051 le b_i le 10^51≤bi​≤105), where bib_ibi​ denotes the number of the i-th kind of dishes.

输出描述:

For each test case, output one line containing ‘‘Case #x: y z′′``Case #x: y z''‘‘Case #x: y z′′, where x is the test case number (starting from 1), y is the maximum number of visitors, and z is the maximum possible profits.

示例1

输入

2
3
2 -1 3
3 2 1
4
3 -2 3 -1
4 2 1 2

输出

Case #1: 3 8
Case #2: 4 13

说明

For test case 1, the maximum number of visitors is 3, one of a possible solution is:
The first visitor received food 1, the profit is 2.
The second visitor received food 1, the profit is 2.
The third visitor received food 1 + food 2 + food 3, the profit is 2 + (-1) + 3.

题目大意：

      给出每种菜品的利润以及碟数，要求我们给每个客人至少一碟菜，要求从1号菜品开始给，给的菜品的号码是连续的，每个客人同号码的菜都只能给一碟。求能招待客人的最大数量以及在客人最多的情况下能获得的最大利润。

思路：

1.首先，因为要求从1号菜品开始给，所以能招待的客人的最大数量就是1号菜品的碟数。

2.用结构体记录1-x号菜品各一碟可以获得的利润以及x的位置，后续按照利润从大到小的顺序对结构体进行排序。

3.给客人的菜一定是1-x各一盘这种情况，可以用一个time数组对能够上几次1-x的菜进行计数，time[x]即1-x中b[x]的最小值。、

4.最关键的一步。核心代码就是 ans=ans+e[j].sum*(time[e[j].loc]-nowt);

为啥是time[e[j].loc]-nowt呢.举个例子，8 6 7 4 5 3 假设time[e[j].loc]=3那么则要把改系列的菜都上完，然后寻找下一次的，得到time[e[j].loc]=4，这个时候就需要time[e[j].loc]-nowt（上一次上菜的数量）用当前菜的数量的最小最减去上一次上菜的数量的就是这次的上菜数量了。因为保证每次选择到的那个系列的菜都是上完的，所以不需要上一次菜把所有的time[]数组进行处理。例如：假设数量是4的已经上完啦，那么我前面的菜肯定也花费了4个，我们直接用6-4就可以得到这次能上多少菜了。

#include <iostream>
#include <cstdio>
#include <algorithm>
using namespace std;
struct noid
{
    int loc;//取得当前利润的位置
    long long sum;//1-x菜各一碟利润总和
} e[100005];
bool cmp(struct noid a,struct noid b)
{
    return a.sum>b.sum;
}
//输入
inline __int128 read()
{
    __int128 x=0,f=1;
    char ch=getchar();
    while(ch<'0'||ch>'9')
    {
        if(ch=='-')
            f=-1;
        ch=getchar();
    }
    while(ch>='0'&&ch<='9')
    {
        x=x*10+ch-'0';
        ch=getchar();
    }
    return x*f;
}
//输出
inline void print(__int128 x)
{
    if(x<0)
    {
        putchar('-');
        x=-x;
    }
    if(x>9)
        print(x/10);
    putchar(x%10+'0');
}
int main()
{
    int i,j,t,n,k,now,nowt;
    __int128 a[100005],b[100005],time[100005],ans;
    scanf("%d",&t);
    for(i=1; i<=t; i++)
    {
        ans=0;
        scanf("%d",&n);
        e[0].sum=0;
        for(j=1; j<=n; j++)
        {
            a[j]=read();
            e[j].sum=e[j-1].sum+a[j];
            e[j].loc=j;
        }
        for(j=1; j<=n; j++)
        {
            b[j]=read();
            if(j==1)
                time[1]=b[1];
            if(j>1)
                time[j]=min(time[j-1],b[j]);//求1-j菜品数量的最小值
        }
        sort(e+1,e+1+n,cmp);//对利润进行排序
        nowt=0;
        now=100005;
        for(j=1; j<=n; j++)
        {
            if(e[j].loc>=now)//因为now前面的菜品一定会出现部分空缺，位置在now后面上不了菜
                continue;
            ans=ans+e[j].sum*(time[e[j].loc]-nowt);
            int cnt=e[j].loc;
            if(cnt<=now)
                now=e[j].loc;//记录目前的位置
            nowt=time[e[j].loc];//记录当前1号菜品被消耗的次数
        }

        printf("Case #%d: %lld ",i,b[1]);
        print(ans);
        printf("
");
    }
    return 0;
}

 链接：https://ac.nowcoder.com/acm/contest/5673/I
来源：牛客网

Apollo is playing an interesting computer game. There are N rounds in the game.
At each round, the computer will give Apollo two integers (aia_iai​ and bib_ibi​), and Apollo can do exactly one of the following three actions.

•  Apollo can do nothing.
•  If integer aia_iai​ has not been selected in all previous rounds, Apollo can select integer aia_iai​.
•  If integer bib_ibi​ has not been selected in all previous rounds, Apollo can select integer bib_ibi​.

Apollo has cracked the game, and he has known all the candidate numbers of each round before the game starts. Now he wants to know the maximum number of integers he can select with the optimal strategy.
I believe it would be very simple question for you, please help Apollo solve this question.

输入描述:

The first line is an integer Tmathbf{T}T (1≤T≤101 leq mathbf{T} leq 101≤T≤10), which is the number of test cases.

Each test case begins with a line containing a positive integer N (1≤N≤1051 le N le 10^51≤N≤105), indicating the number of rounds in this game.

Then N lines follow. The i-th line contains two integers aia_iai​ and bib_ibi​ (1≤ai≤1091 le a_i le 10^91≤ai​≤109, 1≤bi≤1091 le b_i le 10^91≤bi​≤109), indicating that two integers of the i-th round.

输出描述:

For each test case, output one line containing ‘‘Case #x: y′′``Case #x: y''‘‘Case #x: y′′, where x is the test case number and y is the answer.

示例1

输入

2
6
1 2
2 3
3 4
1 4
1 3
2 4
5
1 2
1 2
1 3
2 3
5 6

输出

    Case #1: 4

Case #2: 4
题意：给n个数的a，b，对于每个i可取a[i]或b[i]或不取，输出最大取出的数的集合大小。
思路： 把a[i],b[i]看成一条边，然后把所有的边进行连接，构成一个个的堆，如果某个堆里面有环那么这个环里面所有的点都可以选，否则只能选这个环里面所有的点数-1.
有一个需要优化的地方，就是在并查集找祖先的时候，如果相等则说明有环，然后把这个根节点记录下来，如果不相等，也要把根节点记录下来。比如
1 3
2 3
1 2
3 4
可以看出一直到1 2根节点一直都是3而且已经成环了，但是到3 4的时候发现根节点发生了变化，根节点变成了4。
我们发现4没有被记录，所以这里就要改动一下，如果俩个点a,b的根节点不同，
但是其中的一个已经是某个环的根节点了(假设是a)，而且
（a<-b）,但是b又是a的根节点，这个时候就需要把b标记一下，说明在以b为根节点的堆中，存在一个环。

f1=fin(le[i]);
            f2=fin(ri[i]);
            if(f1==f2)
            {
                fla[f1]=1;
            }
            else
            {
                f[f1]=f2;
                if(fla[f1]==1||fla[f2]==1)
                {
                    fla[f2]=1;
                }
            }

最后一点数据比较大，需要离散化一下。

#include<bits/stdc++.h>
#include<queue>
using namespace std;
typedef long long ll;
const int N=2e5+7,inf=0x3f3f3f3f,mod=1e9+7;
int num[200007],fla[200007],f[200007];
int fin(int p)
{
    if(f[p]==p)
        return p;
    else
    {
        return f[p]=fin(f[p]);
    }
}
int main()
{
    int t,Case=0;
    scanf("%d",&t);
    while(t--)
    {
        int cnt=0,ans;
        int n,le[100007],ri[100007];
        cin>>n;
        for(int i=1; i<=n; i++)
        {
            scanf("%d%d",&le[i],&ri[i]);
            num[++cnt]=le[i];
            num[++cnt]=ri[i];
        }
        sort(num+1,num+1+cnt);
        ans=cnt=unique(num+1,num+1+cnt)-num-1;//去重
        for(int i=1; i<=n; i++)
        {
            le[i]=lower_bound(num+1,num+cnt+1,le[i])-num;  //建立左端点新的映射关系
            ri[i]=lower_bound(num+1,num+cnt+1,ri[i])-num;  //建立右端点新的映射关系
        }
        for(int i=1; i<=N; i++)
        {
            f[i]=i;
            fla[i]=0;
        }
        int f1,f2;
        for(int i=1; i<=n; i++)
        {
            f1=fin(le[i]);
            f2=fin(ri[i]);
            if(f1==f2)
            {
                fla[f1]=1;
            }
            else
            {
                f[f1]=f2;
                if(fla[f1]==1||fla[f2]==1)
                {
                    fla[f2]=1;
                }
            }
        }
        for(int i=1; i<=cnt; i++)
        {
            if(f[i]==i&&fla[i]==0)   //如果遇到一棵树
            {
                ans--;
            }
        }
        cout<<"Case #"<<++Case<<": "<<ans<<endl;
    }

}