ACM/ICPC2016沈阳网络赛（不完全）解题报告

比赛地址：


1003.hannnnah_j’s Biological Test

题目大意：

圆周上有N个不同的椅子，要让M个相同的人坐在上面，使得两人之间至少隔K把空椅子，求方案数（模1e9+7）。

0

题解：

考虑每两个人之间隔了几把椅子。可以发现，一共有M个数，和为N-M，且每个数都>=K.将每个数都减去K-1，即得到：M个正数之和为N-K*M，方案数为C(N-K*M-1,M-1).需要乘以圆排列的N，同时每个方案被算了M次，再除以M。
答案为：C(N-K*M-1,M-1)*N/M。需要用预处理阶乘的方法计算组合数。注意特判M=1的情况。

代码：

#include
#include
#include
#include
using namespace std;
typedef long long LL;
const LL MOD=1000000000+7;
LL quickpow(LL a,LL n){
    LL ans=1;
    while(n){
        if(n&1) ans=(ans*a)%MOD;
        a=(a*a)%MOD;
        n>>=1;
    }
    return ans;
}
LL inv(LL a){
    return quickpow(a,MOD-2);
}
LL fact[2000010];
void prepare(void){
    fact[0]=1;
    for(LL i=1;i<=2000000;i++){
        fact[i]=fact[i-1]*i%MOD;
    }
}
LL C(LL n,LL m){
    if(n<0||n-m<0||m<0) return 0;
    LL ret=fact[n];
    ret=ret*inv(fact[n-m])%MOD;
    ret=ret*inv(fact[m])%MOD;
    return ret;
}
LL calcadd(LL n,LL k){//k non-neg nums add up to N
    if(n<0) return 0;
    return C(n+k-1,k-1);
}
LL calc(LL N,LL M,LL K){
    if(M==1) return N;
    LL ret=C(N-M*K-1,M-1);
    ret=ret*N%MOD;
    ret=ret*inv(M)%MOD;
    return ret;
}
void work(void){
    LL N,M,K;
    cin>>N>>M>>K;
    cout<



1007.odd-even number

题目大意：

对于一个十进制整数，如果它的数位表示中，连续的奇数总有偶数个，连续的偶数总有奇数个，就称它是“好”的。给定1 <= L,R <= 9e18，求[L,R]内“好数”的数量。

题解：

首先容斥一下，只需要求“<=x的好数”这样一个问题。


数位DP，枚举“从哪一位开始和x不同”。例如，x为47397456，我们就需要求一些例如“有多少个形如47395???的好数”这样的问题。特别注意位数比x少的那些数，即类似于“有多少个形如????????的好数”这种问题。对此，我们可以先DP出来一个数组F：F[i][j][n]，其中i=0/1/2代表开头一段是偶数/奇数/未开始，j=0/1代表开头一段有偶数/奇数个（如果i=2就规定j=0），n代表后面有n个未定位，F[i][j][n]就是这种情况下的“好数”个数。例如，形如“47395???”的好数就有F[1][0][3]个，形如“????”的好数就有F[2][0][4]个。

代码：

#include
#include
#include
#include
#include
using namespace std;
typedef long long LL;
LL F[3][2][30]={0};
void DP(void){
    F[0][1][0]=F[1][0][0]=1;
    for(int i=1;i<=20;i++){
        F[0][0][i]=F[0][1][i-1]*5;
        F[0][1][i]=F[0][0][i-1]*5+F[1][1][i-1]*5;
        F[1][0][i]=F[1][1][i-1]*5+F[0][1][i-1]*5;
        F[1][1][i]=F[1][0][i-1]*5;
        F[2][0][i]=F[0][1][i-1]*4+F[1][1][i-1]*5+F[2][0][i-1];
    }
}
bool add_sta(int &r,int &l,int x){
    if(r==2){
        if(x!=0){
            r=(x&1);
            l=1;
        }
        return true;
    }
    if(!(x&1)){
        if(r==1&&l==1) return false;
        if(r==0) l^=1;
        else{
            r=0;
            l=1;
        }
        return true;
    }
    else if(x&1){
        if(r==0&&l==0) return false;
        if(r==1) l^=1;
        else{
            r=1;
            l=1;
        }
        return true;
    }
}
LL calc(const vector &A){
    LL ret=0;
    int n=A.size();
    int r=2,l=0;
    for(int i=0;i getdec(LL x){
    vector ret;
    while(x){
        ret.push_back(x%10);
        x/=10;
    }
    reverse(ret.begin(),ret.end());
    return ret;
}
LL calc(LL x){
    return calc(getdec(x));
}
LL calc(LL L,LL R){
    return calc(R)-calc(L-1);
}
int main(){
    DP();
    int T,kase=0;scanf("%d",&T);
    while(T--){
        LL l,r;cin>>l>>r;
        printf("Case #%d: %lld\n",++kase,calc(l,r));
    }
    return 0;
}

1008.oasis in desert

题目大意：

给定一张N<=500的边带权无向图和一个整数K。定义：

1. “最大危险集”：是顶点集的一个子集，该集合中任意两顶点的最短路大于K。最大危险集是所有这种集合中最大的。

2. “最小安全集”：是顶点集的一个子集，如果全部顶点中，任意二者的最短路小于等于K，该集合中必须至少包含其中的一个。最小安全集是所有这种集合中最小的。

要求找出一个集合，它既是最大危险集，又是最小安全集。若有多组解，输出字典序最小者。无解输出“Impossible”。

题解：

2016/11/23更新：

感谢@cmershen 的评论，他指出了这个做法的错误：

“楼主你好，你这个1008题的题解是错的，而且代码交上去也是wa。
我认为错在这里：“注意，这张图有可能不连通，但这并不影响结论，因为每一个连通块都应符合上述条件。”

如果两个子图不连通，其中每个子图都有符合题意的集合，那么按LZ代码的思路，答案应为两个子图的并集。如果我从这两个子图里分别拿出一个点，他俩根本就是不可达的，题中的“minimum distance longer than K”就没有意义。


还有，二分图就算是一边有N/2个点，也不一定存在完美匹配的。。。。。。我在LZ的代码上补充了这两种情况，提交到hdu 5899上，确实AC了。”

详见评论区。就是说，下面的题解和代码都少考虑了这些情况，请大家不要被误导（我也懒得改代码了(逃）。

首先建一张新图，其中两点之间连一条单位边当且仅当它们在原图中的最短路<=K。

我们不妨假设一个点在集合中，看看会发生什么：

1. 它本身在集合中。
2. 和它相邻的点不在集合中。
3. 和2中点相邻的点在集合中（这是最小安全集的规定）。
4. 和3中点相邻的点不在集合中。
5. 和4中点相邻的点在集合中。
……

可以发现，这实质上是对图进行二分染色。换言之，这张图必须是二分图。

在此基础上，显然“最大危险集”就是二分图的最大独立集，“最小安全集”就是二分图的最小支配集。这二者应当相等。我们注意到，最小支配集的大小等于匹配数，而最大独立集的大小等于N-匹配数，换言之，应当有匹配数=N/2。也就是说，图的左右两边应当一样大，并且有完备匹配。

因此我们总结出三条（其实是两条）判据：

1. 图是二分图。
2. 二分图的两边均有N/2个点。
3. 二分图的匹配数为N/2.

注意，这张图有可能不连通，但这并不影响结论，因为每一个连通块都应符合上述条件。

如果判断图是合法的，那么取字典序较小的那一边输出即可。注意输出格式：行末无空格，而且如果输出0个点，就不应换行。我们队当时PE了三次。

代码：

#include
#include
#include
#include
#include
using namespace std;
const int INF=0x7fffffff/2;
const int SIZEN=510;
int N,M,K;
int dis[SIZEN][SIZEN];
bool vis[SIZEN];
int match[SIZEN];
int col[SIZEN];
int find(int x){
    for(int i=1;i<=N;i++){
        if(dis[x][i]&&col[i]==1&&!vis[i]){
            vis[i]=true;
            if(match[i]==-1||find(match[i])){
                match[i]=x;
                return true;
            }
        }
    }
    return false;
}
bool DFS(int x){
    for(int i=1;i<=N;i++){
        if(i!=x&&dis[x][i]){
            if(col[i]==-1){
                col[i]=(col[x]^1);
                if(!DFS(i)) return false;
            }
            else{
                if(col[i]==col[x]){
                    return false;
                }
            }
        }
    }
    return true;
}
bool check(void){
    memset(col,-1,sizeof(col));
    for(int i=1;i<=N;i++){
        if(col[i]==-1){
            col[i]=0;
            if(!DFS(i)) return false;
        }
    }
    int cnt=0;
    for(int i=1;i<=N;i++) cnt+=col[i];
    if(cnt*2!=N) return false;
    memset(match,-1,sizeof(match));
    memset(vis,0,sizeof(vis));
    int ans=0;
    for(int i=1;i<=N;i++){
        if(!col[i]){
            memset(vis,0,sizeof(vis));
            if(find(i)) ans++;
        }
    }
    return ans==cnt;
}
void work(void){
    if(!check()){
        printf("Impossible\n");
    }
    else{
        printf("%d\n",N/2);
        int cnt=0;
        for(int i=1;i<=N;i++){
            if(col[i]==0){
                printf("%d",i);
                cnt++;
                if(cnt






1009.QSC and Master


题目大意：
在黑板上写一排N<=300个数，每个数都有一个价值val。若相邻两个数不互质，就可以把它们擦去，并获得它们的价值之和（注意，操作后原先不相邻的数可能变得相邻）。问最多能获得多少价值。

题解：

DP：F[i][j][0]代表把区间i~j全部消掉，在这一区间能获得的最大价值。F[i][j][1]代表不把区间i~j全部消掉，在这一区间能获得的最大价值。

转移：

1. F[i][j][0] <- F[i+1][j-1][0]+val[i]+val[j]，代表中间全部消掉以后i和j“碰头”，然后再把它们消掉。要求第i个数和第j个数不互质。
2. F[i][j][0] <- F[i][k][0]+F[k+1][j][0]，代表分别把前半截和后半截消掉。
3. F[i][j][1] <- F[i][k][0]+F[k+1][j][1]，代表前半截消掉，后半截不消掉。
4. F[i][j][1] <- F[i][k][1]+F[k+1][j][0]，前半截不消掉，后半截消掉。
5. F[i][j][1] <- F[i][k][1]+F[k+1][j][1]，前后都不消掉。

最后答案就是max(F[1][N][0], F[1][N][1]).

代码：

#include
#include
#include
#include
using namespace std;
typedef long long LL;
const LL INF=1e16;
const int SIZEN=310;
int gcd(int a,int b){
    return !b?a:gcd(b,a%b);
}
int ncp(int a,int b){
    return gcd(a,b)>1;
}
int N;
int key[SIZEN],val[SIZEN];
LL F[SIZEN][SIZEN][2];
void update(LL &a,LL b){
    a=max(a,b);
}
void work(void){
    //memset(F,0,sizeof(F));
    for(int i=1;i<=N;i++){for(int j=1;j<=N;j++){F[i][j][0]=F[i][j][1]=-INF;}}
    for(int i=1;i<=N;i++) F[i][i][1]=0;
    for(int i=1;i



1010. Count Primes 

题目大意：

计算[1..n]内的素数个数，n<=1e11.

题解：

一开始没敢写……其实就是打表。首先观察到一个事实：只需要先筛出1e6以内的素数，就可以判断一个1e11以内的数是不是素数。

每隔1.2e7打表，这样表大概有60KB，正好不超过限制。注意，表里存“这一段有多少个素数”，而不是“1..i内有多少个素数”，因为后者数比较大，占空间多。然后求个前缀和。查询时，先查表，然后暴力筛剩下的部分（至多1.2e7）即可。筛法的的效率大概是每秒筛1e8个数，因此不会超时。我筛法的数组只开了1e6，但是用记录“时间戳”的方法，每次不用memset整个数组，因此筛的速度和数组大小无关。

#include
#include
#include
#include
#include
using namespace std;
typedef long long LL;
LL S[]={0,788060,719062,697140,683882,673971,666543,660481,655062,650762,646685,643541,640215,637221,634839,632921,629642,628146,626220,624144,623007,621138,619537,618148,616476,615482,614305,612803,611568,610715,609572,608349,608169,606481,605663,604529,604072,603273,602289,601600,600711,599777,598971,597752,598222,597235,596579,595844,595003,594603,594293,593511,592636,592338,591685,591297,590568,589729,590262,588956,588363,588082,587759,586931,586251,586405,586488,585210,585400,584383,584453,583642,583108,583072,582279,581704,582455,581569,581006,580544,580527,580160,579508,579416,579125,578796,578462,578185,577082,577739,576846,577250,576740,576079,575653,575057,574904,575333,574815,574040,574199,573700,573703,572935,573044,572879,572918,572201,571881,571814,571502,570775,571778,571174,570549,569849,570233,569584,569305,569503,568970,569094,569036,568221,568743,568432,567552,567777,567512,567177,567147,566103,567107,566319,566109,565464,566067,566007,564584,565332,565020,565135,564233,564742,564414,563772,563890,563758,563303,563658,563712,562755,562538,562726,562341,562122,561820,561881,561773,561861,561184,561193,560959,561334,560506,561025,560486,560728,560636,559698,559883,559844,559217,559372,559113,559109,559691,558591,557843,559005,558287,558347,557918,558131,557388,557608,557572,557197,557542,557668,557337,556963,556756,556794,556078,556104,555851,555863,556001,555959,555427,555164,555445,555133,555581,554615,554676,555534,554063,554457,554642,554445,554432,554260,554249,554058,553930,553247,553163,553478,552871,552700,552889,553142,552799,552772,552709,552065,551872,551871,551979,552188,551831,551633,551898,551619,551573,550979,551376,551483,550922,550619,550946,550494,550759,550438,550607,549866,549968,550392,549258,549694,549706,549464,549895,549419,549693,549406,548832,548619,549081,549050,548468,549082,548719,548321,548484,548293,548207,547848,547963,547991,547855,547160,547800,547242,547582,547893,547388,546738,546689,547252,546533,547244,546497,546740,546880,546402,546765,546612,546053,546207,545973,546678,545989,545703,545858,545598,545552,545527,544954,545267,545395,545350,545454,544866,545186,544918,544624,544356,544845,544901,544802,543941,544460,544094,543950,544279,543944,543767,543947,543217,543568,543583,543708,543067,543671,542384,543106,543339,543264,542946,542827,542353,543034,542164,542552,542433,542534,542687,541622,542193,541983,542502,542400,541892,541833,541302,541474,541359,541624,541997,541693,541237,541130,540990,541152,541663,540781,540774,541015,541014,541134,540888,540923,540831,540287,540064,540766,539976,540106,539788,540159,539736,539663,540453,539901,539696,539790,539751,539481,539859,539111,539506,539336,539076,539627,538967,538462,539281,539200,539553,538591,538589,539012,538560,539043,538755,537824,538116,538333,538461,538048,537955,538128,537927,537850,538314,538081,537802,537138,537607,538135,537568,537316,537494,537086,536813,537978,537610,536890,536342,537333,537042,536995,536337,537217,536565,536640,536881,536904,536355,536357,536825,536122,536443,536243,536209,536029,536480,535633,536465,536222,535225,535815,535510,535877,535865,535430,535402,535358,535141,535676,534718,535867,535672,535126,534746,535337,534672,534867,534905,534926,534727,534855,535106,534903,534680,535129,534513,534449,534356,534433,534771,534918,533736,534448,533753,534289,534184,533674,533759,533693,533438,534088,533564,533595,533426,533397,533791,534001,532858,532953,533322,533448,533286,532914,533256,533216,534136,532585,533130,532242,532782,532219,532928,533067,532665,532762,532479,532420,533662,532668,531913,532416,532581,532208,532664,532044,532106,531966,531029,532556,531914,532557,532075,531637,531634,531527,531165,531236,532049,531574,531252,531128,530849,531177,531071,530891,531361,531130,531137,531371,531521,531458,531374,530423,530613,531305,530765,531542,530399,530921,530529,530381,530737,530310,530881,530450,530291,529436,531040,529534,530634,530155,530728,530111,529267,529973,530443,529743,529494,529263,530144,529110,530136,530119,529186,529362,529684,529000,529747,529344,529745,529285,529147,529620,529161,529330,529596,529255,529029,529032,528558,529279,529091,528147,528953,528690,528416,528403,528700,528564,528433,528629,528336,529020,528136,528152,528168,528429,528913,528307,528053,527961,527411,528371,528520,528546,528050,527920,528152,527663,527313,527959,528006,528037,527291,527669,527208,527738,527854,526469,526962,527138,527492,527778,527082,527271,527022,527317,527084,526812,526895,527176,527076,526923,527246,527124,526931,526310,526700,527537,527162,526801,526257,526941,526023,526492,527294,526043,526402,526474,526315,526503,526963,526463,526389,526721,526147,525764,526129,526076,526273,526569,525811,525699,525607,525444,526059,525680,526076,525255,526159,526069,525994,525437,526083,525295,525686,525967,525354,525658,524915,525026,525557,525414,525373,525528,524694,525036,525219,525281,525348,524449,524425,525035,525037,524963,524704,524952,524477,525075,525446,52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75001,474758,475494,474144,474499,474813,475085,475114,474885,474612,475130,474948,474747,474457,474499,474973,474265,474885,475243,474746,474900,474802,474702,474739,474425,474794,474920,474698,475170,475211,474897,475025,474405,475250,474688,474622,474523,474807,474198,474618,475067,474878,474379,474841,475095,474781,474399,474521,475287,474715,474608,474489,474616,474309,475152,475212,475452,474438,474719,475198,474559,474606,474573,474994,475184,474648,475037,475143,474514,474579,475020,474541,474025,474262,474627,475544,474696,474352,474794,474352,474779,473902,474876,474710,474309,474033,474265,474604,474981,474744,474952,474210,474409,474873,475507,474438,474306,475523,474049,474899,474630,474834,474484,474891,474726,474841,474675,474637,474665,474755,473985,474309,475093,474159,475125,474896,474935,474519,473932,474135,474748,475122,474895,474037,475136,474110,475047,474524,474205,474491,474463,475077,474949,474494,474716,474794,474365,474327,474600,474620,474534,474823,475258,474260,474305,474662,474627,474109,474292,474836,474481,474656,474669,474540,474997,475002,474271,474355,474144,474046,474393,474836,473777,474216,474818,474136,474653,474404,474873,474363,474644,473967,474067,474566,474269,474206,474524,474448,474898,474140,474302,474477,474388,474457,474634,475284,474104,474470,474383,474354,474432,474168,473650,474734,474300,474302,474932,473918,473298,474629,474263,474381,474672,474139,475444,473466,474367,474561,473922,474341,474330,474665,474715,473864,474911,474104,473778,475021,474107,474132,474664,475156,474252,474890,474668,474145,474084,474070,474659,473882,474232,474610,474186,474625,475236,474062,474634,474235,474072,474028,474490,474048,474330,474676,474339,474119,474571,474337,474104,473483,474533,475039,474204,473754,474424,474708,474307,473843,474581,474179,473924,473896,474176,474492,474750,473802,474013,474470,474226,473874,474687,474614,473769,474483,474982,474015,474341,474179,473830,474255,474477,474281,474303,473903,474047,474096,474627,474725,473848,473813,474434,474356,474643,474268,473898,473955,473944,474585,474704,474530,474067,473752,474522,474620,473895,473969,474788,474273,474459,474609,473667,474146,474687,473619,474568,474555,474640,473911,474441,473941,474646,473773,474078,474699,473913,473765,474224,474412,474170,474755,474146,474122,473587,473720,473747,474398,473670,473933,473926,473322,474337,473943,474407,473869,474645,474031,473710,473551,474099,474113,474333,474540,474136,474368,474728,473778,473429,474342,474167,473800,474425,473606,473994,474594,473489,474309,473468,473753,473985,473731,473975,474443,474102,473590,474357,473980,473615,474491,473856,474036,473445,474305,473665,474113,474058,473226,474286,474624,473974,473412,474492,474412,473968,473719,474019,473471,473113,473844,473923,474393,473752,473126,473434,474219,474189,473611,474017,474145,474374,473985,473762,473776,474436,473394,474089,473980,473820,474562,474196,474167,474287,473519,474004,473629,473679,474378,473925,473638,473808,473539,473482,474729,473411,473926,474457,473913,473637,474112,474390,473746,473884,473384,473148,473855,473531,473783,473672,473181,474511,473558,473641,473869,473841,473992,473874,473317,474197,473612,473609,473837,474263,473799,474062,473182,473887,473539,473789,474560,473428,473745,474630,473706,473808,473569,473713,474170,474010};
const int TN=8334;
void prepare(void){
    for(int i=1;i primes;
void sieve(int n){
    tot++;
    for(int i=2;i<=n;i++){
        if(flag[i]!=tot){
            primes.push_back(i);
            for(LL j=(LL)i*i;j<=n;j+=i){
                flag[j]=tot;
            }
        }
    }
}
int count_sieve(LL L,LL R){
    tot++;
    L=max(L,2LL);
    int ans=int(R-L)+1;
    for(int i=0;i

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