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<M<N<1e6, 0<K<1000.
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<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
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<<calc(N,M,K)<<endl;
}
int main(){
prepare();
int T;scanf("%d",&T);
while(T--) work();
return 0;
}
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<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
#include<vector>
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<int> &A){
LL ret=0;
int n=A.size();
int r=2,l=0;
for(int i=0;i<n;i++){//different from A
for(int d=0;d<A[i];d++){
int r1=r,l1=l;
if(add_sta(r1,l1,d)){
ret+=F[r1][l1][n-i-1];
}
}
if(!add_sta(r,l,A[i])) return ret;
}
ret+=F[r][l][0];
return ret;
}
vector<int> getdec(LL x){
vector<int> 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。
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<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
#include<vector>
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<N/2) printf(" ");
}
}
if(cnt) printf("\n");
}
}
void floyd(void){
for(int k=1;k<=N;k++){
for(int i=1;i<=N;i++){
for(int j=1;j<=N;j++){
dis[i][j]=min(dis[i][j],dis[i][k]+dis[k][j]);
}
}
}
for(int i=1;i<=N;i++){
for(int j=1;j<=N;j++){
dis[i][j]=(dis[i][j]<=K);
}
}
}
void read(void){
scanf("%d%d%d",&N,&M,&K);
for(int i=1;i<=N;i++){for(int j=1;j<=N;j++){dis[i][j]=INF;}dis[i][i]=0;}
int a,b,w;
for(int i=1;i<=M;i++){
scanf("%d%d%d",&a,&b,&w);
dis[a][b]=dis[b][a]=w;
}
}
int main(){
//freopen("input.in","r",stdin);
int T;scanf("%d",&T);
while(T--){
read();
floyd();
work();
}
return 0;
}
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<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
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<N;i++){
if(ncp(key[i],key[i+1])){
F[i][i+1][0]=(LL)val[i]+val[i+1];
}
else{
F[i][i+1][1]=0;
}
}
for(int t=3;t<=N;t++){
for(int i=1;i+t-1<=N;i++){
int j=i+t-1;
if(ncp(key[i],key[j])){
update(F[i][j][0],F[i+1][j-1][0]+val[i]+val[j]);
}
for(int k=i;k<j;k++){
update(F[i][j][0],F[i][k][0]+F[k+1][j][0]);
update(F[i][j][1],F[i][k][0]+F[k+1][j][1]);
update(F[i][j][1],F[i][k][1]+F[k+1][j][0]);
update(F[i][j][1],F[i][k][1]+F[k+1][j][1]);
}
}
}
LL ans=max(F[1][N][0],F[1][N][1]);
cout<<ans<<endl;
}
void read(void){
scanf("%d",&N);
for(int i=1;i<=N;i++) scanf("%d",&key[i]);
for(int i=1;i<=N;i++) scanf("%d",&val[i]);
}
int main(){
int T;scanf("%d",&T);
while(T--){
read();
work();
}
return 0;
}
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<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
#include<vector>
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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const int TN=8334;
void prepare(void){
for(int i=1;i<TN;i++){
S[i]+=S[i-1];
}
}
const int SIZE=1000010;
const LL step=1e6*12;
int flag[SIZE];
int tot=0;
vector<int> 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<primes.size();i++){
int p=primes[i];
LL start=max((LL)p*p,LL((L+p-1)/p)*p);
int *it=flag+start-L;
for(;start<=R;start+=p,it+=p){
if((*it)!=tot){
(*it)=tot;
ans--;
}
}
}
return ans;
}
LL calc(LL n){
LL k=n/step;
LL start=n/step*step;
LL ret=S[k];
while(start<=n){
LL R=min((LL)(start+1e6-1),n);
ret+=count_sieve(start,R);
start=R+1;
}
return ret;
}
int main(){
sieve(1e6);
prepare();
LL N;
while(scanf("%lld",&N)!=EOF){
printf("%lld\n",calc(N));
}
return 0;
}
