[SinGuLaRiTy] 2017 百度之星程序设计大赛

【SinGuLaRiTy-1034】 Copyright (c) SinGuLaRiTy 2017. All Rights Reserved.

度度熊保护村庄

 Time Limit: 2000/1000 MS (Java/Others)   Memory Limit: 32768/32768 K (Java/Others)

Problem Description

哗啦啦村袭击了喵哈哈村！
度度熊为了拯救喵哈哈村，带着自己的伙伴去救援喵哈哈村去了！度度熊与伙伴们很快的就过来占据了喵哈哈村的各个军事要地，牢牢的守住了喵哈哈村。
但是度度熊发现，这是一场旷日持久的战斗，所以度度熊决定要以逸待劳，保存尽量多的体力，去迎战哗啦啦村的战士。
于是度度熊决定派尽量多的人去休息，但是同时也不能松懈对喵哈哈村的保护。
换句话而言，度度熊希望尽量多的人休息，而且存在一个包围圈由剩下的人组成，且能够恰好的包围住喵哈哈村的所有住房（包括边界）。
请问最多能让多少个人休息呢？

Input

本题包含若干组测试数据。
第一行一个整数n，表示喵哈哈村的住房数量。
接下来n行，每行两个整数(x1[i],y1[i])，表示喵哈哈村的住房坐标。
第n+1行一个整数m，表示度度熊的士兵数量。
接下来m行，每行两个整数(x2[i],y2[i])，表示度度熊伙伴的坐标。
满足:
1<=n,m<=500
-10000<=x1[i],x2[i],y1[i],y2[i]<=10000

Output

请输出最多的人员休息的数目。
如果无法保护整个村庄的话，输出"ToT"

Sample Input

2
1 1
2 2
4
0 0
0 4
4 2
4 0
1
1 1
2
0 0
0 1

Sample Output

1
ToT

解析

首先我们要知道，由于是"包围"村庄，那么用来包围这个村庄的守卫连接的边都不能穿过村庄。如下图所示：在所有满足条件的边中，绿色边的包围方案最佳。

到这时，我们的问题就可以从找最少点变成找最少边——而这一点有非常好办，只要处理出上图这样的样子，给每一个边都赋值为1，求一个最小环就行了。那么问题来了，怎样才能选出满足条件的边呢？显然的是，我们首先需要n^2的枚举两个包围的端点，从而确定一条边，接着才是重点：判断相交。显然，如果对村庄的房子也采用先前的方法m^2枚举，总的时间复杂度是O(n^2*m^2)，肯定超时。这时，我们就要稍稍转换一下：如果一条边没有穿过村庄，那么所有的房子一定在它的一侧呀！于是，我们在判定每一条边时，只用枚举m次，看看这m个点是否都在同一侧就行了，如果不在同一侧，就去掉这条边。经过这个O(n^2*m)的判定过程，我们就可以处理出仅包含可行边的图，接下来跑一次最短路就可以啦。其它的问题嘛：就要看看代码实现了。

好吧，还是提一下关于"判定线段是否相交"的问题，有两个试验：快速排斥实验和跨立实验，通过这两个试验可以在不求交点的情况下快速判断线段是否相交。那么怎么实现呢？网上的博客有很多，自己去查吧！

度度熊的王国战略

 Time Limit: 40000/20000 MS (Java/Others)   Memory Limit: 32768/132768 K (Java/Others)

Problem Description

度度熊国王率领着喵哈哈族的勇士，准备进攻哗啦啦族。
哗啦啦族是一个强悍的民族，里面有充满智慧的谋士，拥有无穷力量的战士。
所以这一场战争，将会十分艰难。
为了更好的进攻哗啦啦族，度度熊决定首先应该从内部瓦解哗啦啦族。
第一步就是应该使得哗啦啦族内部不能同心齐力，需要内部有间隙。
哗啦啦族一共有n个将领，他们一共有m个强关系，摧毁每一个强关系都需要一定的代价。
现在度度熊命令你需要摧毁一些强关系，使得内部的将领，不能通过这些强关系，连成一个完整的连通块，以保证战争的顺利进行。
请问最少应该付出多少的代价。

Input

本题包含若干组测试数据。
第一行两个整数n，m，表示有n个将领，m个关系。
接下来m行，每行三个整数u,v,w。表示u将领和v将领之间存在一个强关系，摧毁这个强关系需要代价w
数据范围：
2<=n<=3000
1<=m<=100000
1<=u,v<=n
1<=w<=1000

Output

对于每组测试数据，输出最小需要的代价。

Sample Input

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

Sample Output

1
3

解析

由题意得，对于本来就不是强连通图的数据，我们直接输出"0"就行了。对于另外的大部分情况，我们很容易发现:只要将边权和最小的点从图中删去就行了。在实现过程中，可以考虑使用并查集(谁说图论题就一定要建一个图)。

Code

#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<iostream>
#include<cstdlib>

using namespace std;

int sum[3010],ufs[3010];

int Find(int x)
{
if(ufs[x]==0)
return x;
return ufs[x]=Find(ufs[x]);
}

int main(void)
{
int n,m,i,x,y,z,cnt,t1,t2;
while(scanf("%d%d",&n,&m)!=EOF)
    {
        cnt=n-1;
        memset(ufs,0,sizeof(ufs));
        memset(sum,0,sizeof(sum));
for(i=1;i<=m;i++)
        {
            cin>>x>>y>>z;
if(x==y)
continue;
            sum[x]+=z;
            sum[y]+=z;
            t1=Find(x);
            t2=Find(y);
if(t1!=t2)
            {
                ufs[t1]=t2;
                cnt--;
            }
        }
if(cnt==0)
        {
            sort(sum+1,sum+n+1);
            printf("%d\n",sum[1]);
        }
else
            printf("0\n");
    }
return 0;
}

度度熊与邪恶大魔王

 Time Limit: 2000/1000 MS (Java/Others)   Memory Limit: 32768/32768 K (Java/Others)

Problem Description

度度熊为了拯救可爱的公主，于是与邪恶大魔王战斗起来。
邪恶大魔王的麾下有n个怪兽，每个怪兽有a[i]的生命值，以及b[i]的防御力。
度度熊一共拥有m种攻击方式，第i种攻击方式，需要消耗k[i]的晶石，造成p[i]点伤害。
当然，如果度度熊使用第i个技能打在第j个怪兽上面的话，会使得第j个怪兽的生命值减少p[i]-b[j]，当然如果伤害小于防御，那么攻击就不会奏效。
如果怪兽的生命值降为0或以下，那么怪兽就会被消灭。
当然每个技能都可以使用无限次。
请问度度熊最少携带多少晶石，就可以消灭所有的怪兽。

Input

本题包含若干组测试数据。
第一行两个整数n，m,表示有n个怪兽，m种技能。
接下来n行，每行两个整数，a[i],b[i]，分别表示怪兽的生命值和防御力。
再接下来m行，每行两个整数k[i]和p[i]，分别表示技能的消耗晶石数目和技能的伤害值。
数据范围:
1<=n<=100000
1<=m<=1000
1<=a[i]<=1000
0<=b[i]<=10
0<=k[i]<=100000
0<=p[i]<=1000

Output

对于每组测试数据，输出最小的晶石消耗数量，如果不能击败所有的怪兽，输出-1

Sample Input

1 2
3 5
7 10
6 8
1 2
3 5
10 7
8 6

Sample Output

6
18

解析

注意到防御值最大才为10，所以肯定是用防御力来遍历。设dp[j][i]为防御力为 i ,打出 j点伤害以上时所需的最少晶石。
对于第u个技能来说，如果p[u]<= i，说明根本打不出伤害，不用管。
反之，伤害则为 dmg=p[u]-i, 这时候 又有两种情况：
如果dmg>=j，说明靠这一个技能就够打出足够伤害了，那么肯定是用消耗晶石最少的那个技能，dp[j][i]=min{k[u]}；
反之，光靠这个技能不足以打出足够的伤害，那么就需要借助前面的dp值来计算，dp[j-dmg][i]代表同在i防御力，打出j-dmg的伤害的最少晶石，因为dp[j-dmg][i]数量的晶石已经可以打出 j-dmg 的伤害了，此时再加上这第u个技能的伤害，就可以打到 j 以上，晶石数则为dp[j-dmg][i]+k[u],与上面一样，因为不知道哪个技能消耗的晶石最少，所以这里也取一个最小值。

Code

#include <iostream>
#include <cstring>
#include <algorithm>
#include <cstdio>

#define mem(a,b) memset(a,b,sizeof(a))
using namespace std;
typedef long long LL;
const int maxn=100000+5;
const int maxm=1000+5;

LL a[maxn],b[maxn];
LL k[maxm],p[maxm];
LL dp[maxm][15];

LL max(LL a,LL b)
{
return a>b?a:b;
}
LL min(LL a,LL b)
{
return a<b?a:b;
}


int main()
{
int n,m;
while(scanf("%d%d",&n,&m)!=EOF)
    {
        LL up1=0,up2=0,hp=0;
for(int i=0; i<n; i++)
        {
            scanf("%I64d%I64d",&a[i],&b[i]);
            up1=max(up1,b[i]);
            hp=max(hp,a[i]);
        }

for(int i=0; i<m; i++)
        {
            scanf("%I64d%I64d",&k[i],&p[i]);
            up2=max(up2,p[i]);
        }
if(up1>=up2)
        {
            printf("-1\n");
continue;
        }
        mem(dp,0);
for(int i=0; i<=10; i++)
        {
for(int j=1;j<=hp;j++)
            {
                dp[j][i]=1e18;
for(int u=0;u<m;u++)
                {
                    LL dmg=p[u]-i;
if(dmg<=0)
continue;
if(dmg>=j)
                    {
                        dp[j][i]=min(dp[j][i],k[u]);
                    }
else
                    {
                        dp[j][i]=min(dp[j][i],dp[j-dmg][i]+k[u]);
                    }
                }
            }
        }
        LL ans=0;
for(int i=0;i<n;i++)
        {
            ans+=dp[a[i]][b[i]];
        }
        printf("%I64d\n",ans);

    }
return 0;
}

 

度度熊的午饭时光

 Time Limit: 2000/1000 MS (Java/Others)   Memory Limit: 32768/32768 K (Java/Others)

Problem Description

度度熊最期待每天的午饭时光，因为早饭菜品清淡，晚饭减肥不敢吃太多（胖纸的忧伤T.T）。
百度食堂的午餐超级丰富，祖国各大菜系应有尽有，度度熊在每个窗口都有爱吃的菜品，而且他还为喜爱的菜品打了分，吃货的情怀呀(>.<)。
但是，好吃的饭菜总是很贵，每天的午饭预算有限，请帮度度熊算一算，怎样打饭才能买到的最好吃的饭菜?(不超过预算、不重样、午餐等分最高的情况下，选择菜品序号加和最小，加和相等时字典序最小的组合)

Input

第一行一个整数T，表示T组数据。 每组测试数据将以如下格式从标准输入读入：
B
N
score_1 cost_1
score_2 cost_2
:
score_N cost_N
第一行，正整数B（0 <= B <= 1000），代表午餐的预算。
第二行，正整数N (0 <= N <= 100)，代表午餐可选的菜品数量
从第三行到第 (N + 2) 行，每行两个正整数，以空格分隔，score_i表示菜品的得分，cost_i表示菜品的价格(0 <= score_i, cost_i <= 100)。

Output

对于每组数据，输出两行： 第一行输出："Case #i:"。i代表第i组测试数据。 第二行输出菜品的总得分和总花费，以空格分隔。 第三行输出所选菜品的序号，菜品序号从1开始，以空格分隔。

Sample Input

2
29
6
9 10
3 4
6 5
7 20
10 9
15 11
0
2
2 23
10 12

Sample Output

Case #1:
34 29
2 3 5 6
Case #2:
0 0

解析

读完题，应该都可以看出这是一道01背包。DP的方程式都是模板，不解释了。这题的所谓的"不同之处"就在于要求求出DP的路径，其实也和简单，对于dp[i][j]的每一个状态，都对应一个vis[i][j]来记录到达这一状态的路径就可以了，在DP过程中边递推，边维护，最后再顺着一个一个printf出来就可以了。

Code

#include<cstring>
#include<cmath>
#include<algorithm>
#include<cstdio>
#include<cstdlib>
#include<iostream>

#define INF 0x3f3f3f3f

using namespace std;

typedef long long ll;

const ll mod=1e9+7;

struct meal
{
int score,cost;
}a[110];

bool ans[110];
int dp[1010];
bool vis[110][1010];

int main()
{
int T,Case=1;
    cin>>T;
while(T--)
    {
int b,n;
        cin>>b>>n;
for(int i=1;i<=n;i++)
            cin>>a[i].score>>a[i].cost;
        memset(dp,0,sizeof(dp));
        memset(vis,false,sizeof(vis));
        memset(ans,0,sizeof(ans));
for(int i=1;i<=n;i++)
        {
for(int j=b;j>=0;j--)
            {
if(j>=a[i].cost)
                {
if(dp[j]<dp[j-a[i].cost]+a[i].score)
                    {
                        dp[j]=dp[j-a[i].cost]+a[i].score;
                        vis[i][j]=true;
                    }
else
                        vis[i][j]=false;
                }
            }
        }
int p=b,cnt=0;
for(int i=n;i>=1;i--)
        {
if(vis[i][p])
            {
                ans[i]=true;
                p-=a[i].cost;
                cnt++;
            }
        }
int ANS=0;
for(int i=1;i<=n;i++)
        {
if(ans[i])
                ANS+=a[i].cost;
        }
        printf("Case #%d:\n",Case++);
        printf("%d %d\n",dp[b],ANS);;
for(int i=1;i<=n;i++)
        {
if(ans[i])
            {
                cout<<i;
if(--cnt)
                    cout<<" ";
else
                    cout<<endl;
            }
        }
    }
return 0;
}

寻找母串

 Time Limit: 2000/1000 MS (Java/Others)   Memory Limit: 32768/32768 K (Java/Others)

Problem Description

对于一个串S，当它同时满足如下条件时，它就是一个01偏串： 1、只由0和1两种符组成； 2、在S的每一个前缀中，0的个数不超过1的个数； 3、S中0的个数和1的个数相等。
现在给定01偏串S，请计算一下S在所有长度为n的01偏串中作为子串出现的次数的总和。 由于结果比较大，结果对1e9+7取余后输出。
样例解释： 在第二个样例中，长度为4的偏串共两个1010，1100。10在1010中出现了两次，在1100中出现了1次。所以答案是3。

Input

第一行给出一个整数T(1<=T<=40),表示测试数据的数目。 每一组测试包含一个整数n和字符串S，中间用空格分开。(1<=|S|<=100000，1<=n<=1000000000)
输入保证S是一个01偏串。

Output

对于每一组数据，输出一个整数占一行，表示答案。

Sample Input

2
2 10
4 10

Sample Output

1
3

解析

首先，写一个暴力，观察观察——这一看不得了，似乎有规律！我们发现，固定好n，只要字符串的长度固定，无论字符串的内容是什么，答案都是一样的。接着再继续深入地研究一下，我们竟然发现答案就是。

接下来，我们将它化简，可以得到：原式=F((n-|S|)/2+1)*((n-|S|)/2+1+1)/2。其中的F(n)指的就是卡特兰数列的第n项。我们又会欣喜地发现：卡特兰数列有O(n)的递推公式！

 

不过我们高兴的太早了：这里的n可是很大的，这么大的n，一个数组也开不下呀。于是我们就要用到分块的思想：n不是有很大吗，我们把1~(n-|S|)/2+1)的大区间划分成许多个小区间，后台暴力打表搞出第100000,200000,300000......500000000个卡特兰数，接着判断输入的 (n-|S|)/2+1)在哪一个区间里，再直接在这个区间里面递推，这样单次的递推的时间复杂度就变为了O(100000)。注意，由于在计算答案时数据可能会比较大，需要用到乘法逆元，不过这很好办，稍稍观察就会发现，所求的逆元的值恒为500000004，直接代入就行了。

另外大家还要注意两点: 1>对于n<|S|这一类情况，要提前特判输"0"；2>对于n%2!=0的情况，也要特判输"0"，因为01串要满足0和1的个数相同，那么长度一定为偶数。表示本人为此WA掉了20多次......qwq。

Code

#include<cstring>
#include<cstdio>
#include<cmath>
#include<algorithm>
#include<iostream>
#include<cstdlib>

#define ll long long
#define mod 1000000007
#define inverse_element 500000004

using namespace std;

int c[5010]={1,945729344,874271714,728937729,488957151,213941567,621948608,165410919,130152966,477849061,70646122,104331417,804334677,112228257,406385851,373610584,959189339,129232103,596011761,653302265,502350404,254254274,23933464,200023312,437339905,838455202,353529507,181826790,268707842,918554769,361346624,475114772,551764204,264122255,348587014,339775734,190610810,147039338,576212642,958452888,907080272,897468755,485757524,926001334,253159604,720502565,795701677,485379561,512490195,62869610,44647373,170729326,181183678,223602264,14266485,546070015,669748641,266401585,722432477,844276843,806809752,215407850,514089950,48258832,388402233,287386547,184275733,39087220,221157750,796058916,274451201,551481903,468840632,654694122,349643823,123450237,466554578,613371117,104229634,534520804,340376885,53768135,665006362,362834988,113383960,458629513,877935697,990187981,24010351,477599350,491321663,619368665,730486899,302800222,248364522,617385859,136706557,988949479,681384128,140672609,39443745,327846413,353172956,108275948,378351259,482275913,432213793,904945283,871063203,757590126,541675614,649654378,339103071,153511190,307061975,535998427,953278988,315396128,629843192,166102135,623766767,9922024,962639966,258645380,484156059,75913669,837794293,574075634,404396328,768092500,903633183,512519672,44285819,538902069,621789723,71341556,81438997,28847533,40513591,849551327,716939958,942552133,395636757,409774346,383255288,612104482,95851638,696370843,884779483,524922348,220269701,390881347,114227221,395767320,614338019,380172060,180822541,691300228,891829295,967678016,701386253,162940737,994175698,908221589,19098271,438672588,631720803,573850469,339056525,473356393,859071294,949727021,98529638,168890563,563975847,7483795,276476730,39489314,514728211,87529508,963753157,380031470,856696972,158703490,515908206,227044626,959511111,993078622,80360936,888779384,323285665,543535546,550763954,601069950,458756853,717452438,141044757,197501322,634124805,238583278,200414323,553793080,66504653,42069842,722598090,913608032,740683059,742050754,961341417,612741304,91785690,526087340,214299335,290988315,920349511,650032143,826861662,655676777,832872230,243503280,428075391,780176613,762279490,958553222,298583588,683539087,245173409,561149407,172900450,994863083,373712992,577710034,676813275,65061172,354158163,474587452,779048700,252731521,513247255,63716789,589280106,300449755,843219077,437936157,617131692,778975010,166446598,341293627,939851882,377019792,974135048,18159888,617899054,824539871,922256819,904951680,298436189,877366713,763013330,69488053,73161945,968841940,502298660,815060335,585478837,52225572,617841877,964952304,864682249,537138023,438032312,507344261,611203109,211965075,510592131,253042462,662211654,306668985,993054101,886017682,550897294,748379779,600811784,662510825,923914254,398819263,459843924,563578785,800110936,890507874,833696238,67588077,662523469,355339772,897895240,326276953,66185485,707119691,543399352,359252814,343020869,435649016,326479513,313711009,180736295,933688638,341979700,832484396,852821336,938966456,327938691,897898594,514258765,67631305,832209634,147167201,933864871,764372915,290610995,894446445,552066969,955550128,899538317,98587581,791520663,371485442,250166899,386277652,850700239,24883422,365069578,542759096,393648297,338311897,998392022,963038213,184331347,216025562,616142588,243671535,922469047,642896156,173840546,617065897,502916270,399859525,626708892,203438585,794335044,466847771,485435744,176762177,370414892,387922877,421040690,506803021,831791096,536761402,659624740,205742555,354512880,992080290,655638247,441557017,45234385,602453687,642559130,907907516,150309994,959349261,667245533,336984599,7514180,293021893,527065808,716061908,183191397,64976751,577967909,833760679,814950379,238991246,714290102,902231891,437427307,149073027,103173270,685925477,694198173,725256565,71168984,261760780,55757530,112359206,436686978,516723667,537954237,916255072,34043384,456908325,881291767,146092388,189805926,498850664,471576564,977024064,502710283,230937450,166248501,100683033,725464484,750575586,525293805,107750423,541103694,100780834,184498203,767490572,317419646,113029816,182757677,138614676,694365799,832641460,450632411,301670090,972578606,809851941,356139480,741575534,319109663,491688831,946837771,820876500,944731056,441691410,439779726,878100434,110259934,781810231,822737100,122949140,185157201,147977057,78103215,98159640,570747872,913214189,526507036,614910594,606219102,234699129,882201646,184318349,623751699,220799331,756153289,390599249,416019375,679472812,608212465,946818526,676784388,848462543,701625645,631586665,279738981,507735946,983863013,797150469,667058957,302361878,11123699,265956565,759232458,630434107,603234456,651468455,184824418,604726700,101186449,24954474,873710205,172883575,299713548,965606121,589841373,782619807,545740664,94087806,846145323,99057958,275906334,98324641,712417499,506415720,153372884,988856645,995466415,137196894,296743168,461577383,447394859,350928543,800328411,359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ll Pow(ll a,ll b)
{
    ll sum=1;
while(b)
    {
if(b%2)
            sum=sum*a%mod;
        a=a*a%mod;
        b/=2;
    }
return sum;
}

int n;
char S[100010];

int main()
{
int T;
    cin>>T;
for(int t=1;t<=T;t++)
    {
        memset(S,0,sizeof(S));
        cin>>n;
        cin>>S;
int len=strlen(S);
if(n<len||n%2!=0)
        {
            cout<<"0"<<endl;
continue;
        }
        ll tool=(n-len)/2+1;
int i;
for(i=1;i<=5010;i++)
        {
if(i*100000>tool)
            {
break;
            }
        }
        ll now=c[i-1];
for(ll j=(i-1)*100000+1;j<=tool;j++)
        {
            now=2*(2*j-1)%mod*now%mod*Pow(j+1,mod-2)%mod;
        }
        ll ans=now%mod*(((tool+1)%mod*inverse_element)%mod)%mod;
        cout<<ans<<endl;
    }
return 0;
}

 

Time: 2017-08-06