Cmd2001的毒瘤水题题解

怕不是我再不写题解这题就该成没人做也没人会的千古谜题了......

T1:

仔细分析题面，发现相同就是广义SAM上节点相同，相似就是广义SAM上为从根到某个点路径的前缀。、
直接SAM上跑从根开始，每个点下界为1的最小流即可。
代码:

 #include<iostream>
 #include<cstdio>
 #include<cstring>
 #include<algorithm>
 #include<map>
 #include<queue>
 #define debug cout
 using namespace std;
 const int maxn=5e3+1e2;
 const int inf=0x3f3f3f3f;
 
 int in[maxn],n,ans,sum;
 
 namespace NetworkFlow {
     int s[maxn<<],t[maxn<<],nxt[maxn<<],f[maxn<<],dep[maxn<<],deg[maxn<<],cnt=;
     int bak[maxn<<],bcnt;
     int st,ed,_s,_t;
     inline void coredge(int from,int to,int flow) {
         t[++cnt] = to , f[cnt] = flow ,
         nxt[cnt] = s[from] , s[from] = cnt;
     }
     inline void singledge(int from,int to,int flow) {
         coredge(from,to,flow) , coredge(to,from,);
     }
     inline bool bfs() {
         memset(dep,-,sizeof(dep)) , dep[st] = ;
         queue<int> q; q.push(st);
         while( q.size() ) {
             const int pos = q.front(); q.pop();
             for(int at=s[pos];at;at=nxt[at])
                 if( f[at] && !~dep[t[at]] ) {
                     dep[t[at]] = dep[pos] +  , q.push(t[at]);
                 }
         }
         return ~dep[ed];
     }
     inline int dfs(int pos,int flow) {
         if( pos == ed ) return flow;
         int ret =  , now = ;
         for(int at=s[pos];at;at=nxt[at])
             if( f[at] && dep[t[at]] > dep[pos] ) {
                 now = dfs(t[at],min(flow,f[at])) ,
                 ret += now , flow -= now ,
                 f[at] -= now , f[at^] += now;
                 if( !flow ) return ret;
             }
         if( !ret ) dep[pos] = -;
         return ret;
     }
     inline int dinic() {
         int ret =  , now = ;
         while( bfs() ) {
             while( ( now = dfs(st,inf) ) )
                 ret += now;
         }
         return ret;
     }
     inline int findflow() {
         for(int at=s[_t];at;at=nxt[at])
             if( t[at] == _s ) return f[at^];
     }
     inline void backup() {
         memcpy(bak,s,sizeof(s)) , bcnt = cnt;
     }
     inline void restore() {
         memcpy(s,bak,sizeof(bak)) , cnt = bcnt;
     }
 }
 
 namespace SAM {
     map<int,int> ch[maxn<<];
     int fa[maxn<<],len[maxn<<],root,last,cnt;
 
     inline int NewNode(int ll) {
         len[++cnt] = ll;
         return cnt;
     }
     inline void extend(int x) {
         int p = last;
         int np = NewNode(len[p]+);
         while( p && ch[p].find(x) == ch[p].end() ) ch[p][x] = np , p = fa[p];
         if( !p ) fa[np] = root;
         else {
             int q = ch[p][x];
             if( len[q] == len[p] +  ) fa[np] = q;
             else {
                 int nq = NewNode(len[p]+);
                 ch[nq] = ch[q] , fa[nq] = fa[q];
                 fa[np] = fa[q] = nq;
                 while( p && ch[p][x] == q ) ch[p][x] = nq , p = fa[p];
             }
         }
         last = np;
     }
     inline void Ex_extend(int* sou,int li) {
         last = root;
         for(int i=;i<=li;i++) {
             if( ch[last].find(sou[i]) != ch[last].end() ) last = ch[last][sou[i]];
             else extend(sou[i]);
         }
     }
 }
 
 inline void build() {
     using SAM::ch;using SAM::cnt;
     using namespace NetworkFlow;
     _s = cnt *  +  , _t = _s +  , st = _t +  , ed = st + ;
     #define cov(x) (x+cnt)
     for(int i=;i<=cnt;i++) {
         if( i !=  ) ++deg[i] , --deg[cov(i)];
         for(map<int,int>::iterator it=ch[i].begin();it!=ch[i].end();it++) {
             const int tar = it->second;
             if( i ==  ) singledge(_s,tar,);
             else singledge(cov(i),tar,);
         }
         if( i !=  ) singledge(cov(i),_t,);
     }
     backup();
     for(int i=;i<=_t;i++) {
         if( !deg[i] ) continue;
         if( deg[i] >  ) singledge(i,ed,deg[i]) , sum += deg[i];
         else singledge(st,i,-deg[i]);
     }
     singledge(_t,_s,inf);
 }
 inline int getans() {
     using namespace NetworkFlow;
     int d = dinic();
     if( d != sum ) return -; // No solution .
     int ret = findflow();
     restore();
     st = _t , ed = _s;
     int dd = dinic();
     return ret - dd;
 }
 
 int main() {
     static int m;
     SAM::root = SAM::NewNode();
     scanf("%d",&m);
     while(m--) {
         scanf("%d",&n);
         for(int i=;i<=n;i++) scanf("%d",in+i);
         SAM::Ex_extend(in,n);
     }
     build();
     ans = getans();
     printf("%d\n",ans);
     return ;
 }

T2:

观察操作数量特点，发现可持久化块状数组可过。
代码:

 #include<iostream>
 #include<cstdio>
 #include<cstring>
 #include<algorithm>
 using namespace std;
 const int maxn=1e6+1e2,maxb=1e3,maxu=1e4+1e2,maxr=1e5+1e2;
 
 inline int* NewArray() {
     static const int maxu = 1e4 + 1e3 + ;
     static int dat[maxu][maxb],cnt;
     return dat[cnt++];
 }
 
 struct PersistentBlockedArray {
     int* p[maxb];
     inline void insert(int pos,int x) { // insert into this array .
         if( !p[pos/maxb] ) p[pos/maxb] = NewArray();
         p[pos/maxb][pos%maxb] = x;
     }
     inline void modify(int pos,int x) {
         int* t = NewArray();
         memcpy(t,p[pos/maxb],sizeof(int)*maxb);
         t[pos%maxb] = x , p[pos/maxb] = t;
     }
     inline int query(int pos) {
         return p[pos/maxb][pos%maxb];
     }
 }dat[maxu];
 
 int ptr[maxu+maxr],now,cnt;
 
 inline void roll(int tar) {
     ptr[++now] = ptr[tar];
 }
 inline void modify(int pos,int x) {
     dat[++cnt] = dat[ptr[now]] , ptr[now] = cnt;
     dat[ptr[now]].modify(pos,x);
 }
 inline int query(int pos) {
     return dat[ptr[now]].query(pos);
 }
 
 namespace IO {
     const int BS =  << ;
     char ibuf[BS],obuf[BS],*ist,*ied,*oed=obuf;
     inline char nextchar() {
         if( ist == ied ) ied = ibuf + fread(ist=ibuf,,BS,stdin);
         return ist == ied ? - : *ist++;
     }
     inline int getint() {
         int ret =  , ch;
         while( !isdigit(ch=nextchar()) );
         do ret=ret*+ch-''; while( isdigit(ch=nextchar()) );
         return ret;
     }
     inline void getstr(char* s) {
         char ch;
         while( !isalpha(ch=nextchar()) ) ;
         do *s++=ch; while( isalpha(ch=nextchar()) );
     }
     inline void flush() {
         //cerr<<"in flush delta = "<<oed-obuf<<endl;
         fwrite(obuf,,oed-obuf,stdout) , oed = obuf;
     }
     inline void printchar(const char &x) {
         *oed++ = x;
         //cerr<<"delta = "<<oed-obuf<<endl;
         if( oed == obuf + BS ) flush();
     }
     inline void printint(int x) {
         //cerr<<"x = "<<x<<endl;
         static int stk[],top;
         if( !x ) printchar('');
         else {
             top = ;
             while(x) stk[++top] = x %  , x /= ;
             while(top) printchar(''+stk[top--]);
         }
         printchar('\n');
     }
 }
 using IO::getint;
 using IO::printint;
 using IO::getstr;
 using IO::flush;
 
 int main() {
     static int n,m,lastans,ope;
     static char o[];
     n = getint() , m = getint();
     for(int i=,x;i<n;i++) x = getint() , dat[].insert(i,x);
     for(int i=,p,x,t;i<=m;i++) {
         getstr(o);
         if( *o == 'Q' ) {
             p = ( getint() ^ lastans ) % n;
             printint(lastans=query(p));
         } else if( *o == 'M' ) {
             ++ope , p = ( getint() ^ lastans ) % n , x = getint();
             modify(p,x);
         } else if( *o == 'R' ) {
             ++ope , t = ( getint() ^ lastans ) % ope;
             roll(t);
         }
     }
     flush();
     return ;
 }

T3:

大力反演出phi，后面的sigma(i^k)显然是k+1次多项式，拉格朗日插值即可。
代码:

 #include<iostream>
 #include<cstdio>
 #include<cstring>
 #include<algorithm>
 #define debug cout
 #define bool unsigned char
 typedef long long int lli;
 using namespace std;
 const int maxn=1e6+1e2,lim=1e6,maxk=1e3+1e1;
 const int mod=1e9+;
 
 int n,k;
 
 namespace Sieve {
     lli sum[maxn],mem[maxn];
     bool vis[maxn];
 
     inline void pre() {
         static int prime[maxn/],cnt;
         static bool vis[maxn];
         sum[] = ;
         for(int i=;i<=lim;i++) {
             if( !vis[i] ) prime[++cnt] = i , sum[i] = i - ;
             for(int j=;j<=cnt&&(lli)i*prime[j]<=lim;j++) {
                 const int tar = i * prime[j];
                 vis[tar] = ;
                 if( i % prime[j] ) sum[tar] = sum[i] * ( prime[j] -  );
                 else {
                     sum[tar] = sum[i] * prime[j];
                     break;
                 }
             }
         }
         for(int i=;i<=lim;i++) sum[i] = ( sum[i] + sum[i-] ) % mod;
     }
     inline lli getphi(lli x) {
         if( x <= lim ) return sum[x];
         const lli t = n / x;
         if( vis[t] ) return mem[t];
         lli& ret = mem[t]; ret = x * ( x +  ) >>   , vis[t] = ;
         for(lli i=,j;i<=x;i=j+) {
             j = x / ( x / i );
             ret -= ( j - i +  ) * getphi(x/i) % mod , ret %= mod;
         }
         return ret = ( ret % mod + mod ) % mod;
     }
 }
 
 namespace Inter {
     lli in[maxk],fac[maxk],facrev[maxk],pprv[maxk],ssuf[maxk],*prv=pprv+,*suf=ssuf+;
     inline lli fastpow(lli base,int tim) {
         lli ret = ;
         while(tim) {
             if( tim &  ) ret = ret * base % mod;
             if( tim >>=  ) base = base * base % mod;
         }
         return ret;
     }
     inline void init() {
         for(int i=;i<k;i++) in[i] = fastpow(i,k-);
         for(int i=;i<k;i++) in[i] = ( in[i] + in[i-] ) % mod;
     }
     inline lli getmul(int p) {
         return p ? fac[p] * facrev[k-p-] % mod : facrev[k-];
     }
     inline lli getval(lli x) {
         lli ret = ;
         prv[-] = ;
         for(int i=;i<k;i++) prv[i] = prv[i-] * (x-i+mod) % mod;
         suf[k] = ;
         for(int i=k-;~i;i--) suf[i] = suf[i+] * (x-i+mod) % mod;
         for(int i=;i<k;i++) {
             lli now = prv[i-] * suf[i+] % mod;
             ret = ret + now * in[i] % mod * getmul(i) % mod , ret %= mod;
         }
         return ret;
     }
     inline void getinv() {
         static lli inv[maxn];
         *fac = ;
         for(int i=;i<=k;i++) fac[i] = fac[i-] * i % mod;
         inv[k] = fastpow(fac[k],mod-);
         for(int i=k;i;i--) inv[i-] = inv[i] * i % mod;
         for(int i=;i<=k;i++) inv[i] = inv[i] * fac[i-] % mod;
         for(int i=;i<=k;i++) fac[i] = fac[i-] * inv[i] % mod;
         facrev[] = ;
         for(int i=;i<=k;i++) facrev[i] = facrev[i-] * ( mod - inv[i] ) % mod;
     }
 }
 inline lli segphi(lli l,lli r) {
     return ( Sieve::getphi(r) - Sieve::getphi(l-) + mod ) % mod;
 }
 
 inline lli calc(lli n) {
     lli ret = ;
     for(lli i=,j;i<=n;i=j+) {
         j = n / ( n / i );
         ret += segphi(i,j) % mod * Inter::getval(n/i) % mod , ret %= mod;
     }
     return ret;
 }
 
 int main() {
     scanf("%d%d",&n,&k) , k +=  , Sieve::pre() , Inter::init() , Inter::getinv();
     printf("%lld\n",calc(n));
     return ;
 }

当然那个RYOI是什么意思？就不告诉你！