NOI 2015 滞后赛解题报告

Posted 2020-09-27 lytwajue

报同步赛的时候出了些意外。于是仅仅能做一做“滞后赛”了2333
DAY1
T1离线+离散化搞，对于相等的部分直接并查集，不等部分查看是否在同一并查集中就可以，code：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
int t,n;
int father[200001];
struct hp{
    int kind,x,y;
    bool operator < (const hp &a) const
      {return kind>a.kind;}
}qst[200001];
int b[400001];
int find(int x)
{
    if (x!=father[x])
      father[x]=find(father[x]);
    return father[x];
}
int main()
{
    int i,st,r1,r2,size;
    bool f;
    freopen("prog.in","r",stdin);
    freopen("prog.out","w",stdout); 
    scanf("%d",&t);
    while (t--)
      {
        scanf("%d",&n);
        for (i=1;i<=n;++i)
          {
            scanf("%d%d%d",&qst[i].x,&qst[i].y,&qst[i].kind);
            b[i*2-1]=qst[i].x; b[i*2]=qst[i].y; 
          }
        sort(b+1,b+2*n+1);
        size=unique(b+1,b+2*n+1)-b-1;
        for (i=1;i<=n;++i)
          {
            qst[i].x=upper_bound(b+1,b+size+1,qst[i].x)-b-1;
            qst[i].y=upper_bound(b+1,b+size+1,qst[i].y)-b-1;
          }
        sort(qst+1,qst+n+1);
        for (i=1;i<=size;++i)
          father[i]=i;
        st=n+1; 
        for (i=1;i<=n;++i)
          {
            if (qst[i].kind==0) {st=i; break;}
            r1=find(qst[i].x);
            r2=find(qst[i].y);
            if (r1!=r2) father[r1]=r2;  
          }
        f=false;
        for (i=st;i<=n;++i)
          {
            r1=find(qst[i].x);
            r2=find(qst[i].y);
            if (r1==r2) {f=true; break;}
          }
        if (f) printf("NO\n");
        else printf("YES\n"); 
      }
}

T2裸链剖啊。同【HAOI 2015】T2，对于子树问题能够在建树的时候记录一下子树的左右边界。（据说这题现场200+人AC，大天朝的数据结构啊） code：

#include<iostream>
#include<cstdio>
#include<cstring>
#define mid (l+r)/2
#define lch i<<1,l,mid
#define rch i<<1|1,mid+1,r
using namespace std;
struct hp{
    int size,top,wson,dep,fat; 
}tree[100001];
int plc[100001],ls[100001],rs[100001]; 
int seg[400001],delta[400001]; 
int totw,e,n,m,ans;
struct hq{
    int u,v;
}a[200001];
int point[100001],next[200001];
void add(int u,int v)
{
    e++; a[e].u=u; a[e].v=v; next[e]=point[u]; point[u]=e;
}
void build_tree(int now,int last,int depth)
{
    int i;
    tree[now].dep=depth;
    tree[now].size=1;
    tree[now].wson=0;
    tree[now].fat=last;
    for (i=point[now];i;i=next[i])
      if (a[i].v!=last)
        {
          build_tree(a[i].v,now,depth+1);
          tree[now].size+=tree[a[i].v].size;
          if (tree[tree[now].wson].size<tree[a[i].v].size)
            tree[now].wson=a[i].v;
        }
}
void build_seg(int now,int tp)
{
    int i;
    tree[now].top=tp; plc[now]=++totw;
    ls[now]=totw;
    if (tree[now].wson!=0)
      build_seg(tree[now].wson,tp);
    for (i=point[now];i;i=next[i])
      if (a[i].v!=tree[now].fat&&a[i].v!=tree[now].wson)
        build_seg(a[i].v,a[i].v);
    rs[now]=totw;
}
void updata(int i)
{
    seg[i]=seg[i<<1]+seg[i<<1|1];
}
void paint(int i,int l,int r,int a)
{
    seg[i]=a*(r-l+1);
    delta[i]=a;
}
void pushdown(int i,int l,int r)
{
    paint(lch,delta[i]);
    paint(rch,delta[i]);
    delta[i]=-1;
}
void insert(int i,int l,int r,int x,int y,int a)
{
    if (x<=l&&y>=r)
      {
        paint(i,l,r,a);
        return;
      }
    if (delta[i]!=-1)
      pushdown(i,l,r);
    if (x<=mid) insert(lch,x,y,a);
    if (y>mid) insert(rch,x,y,a);
    updata(i);
}
void query(int i,int l,int r,int x,int y,int a)
{
    if (x<=l&&y>=r)
      {
        if (a==1) ans=ans+seg[i];
        if (a==0) ans=ans+r-l+1-seg[i];
        return;
      }
    if (delta[i]!=-1)
      pushdown(i,l,r);
    if (x<=mid) query(lch,x,y,a);
    if (y>mid) query(rch,x,y,a);
}
void work(int x,int y)
{
    int f1=tree[x].top,f2=tree[y].top;
    ans=0;
    while (f1!=f2)
      {
        if (tree[f1].dep<tree[f2].dep) {swap(x,y); swap(f1,f2);}
        query(1,1,n,plc[f1],plc[x],0);
        insert(1,1,n,plc[f1],plc[x],1);
        x=tree[f1].fat; f1=tree[x].top;
      }
    if (tree[x].dep>tree[y].dep) swap(x,y); 
    query(1,1,n,plc[x],plc[y],0);
    insert(1,1,n,plc[x],plc[y],1);
    printf("%d\n",ans);
}
void build(int i,int l,int r)
{
    delta[i]=-1;
    if (l==r)
      {
        seg[i]=0;
        return;
      }
    build(lch); build(rch);
    updata(i);
}
int main()
{
    int i,x;
    char s[20];
    scanf("%d",&n);
    for (i=1;i<=n-1;++i)
      {
        scanf("%d",&x);
        x++; 
        add(x,i+1);  
      }
    build_tree(1,0,0);
    build_seg(1,1);
    build(1,1,n);
    scanf("%d",&m);
    for (i=1;i<=m;++i)
      {
        scanf("%s",&s);
        while (s[0]!=‘u‘&&s[0]!=‘i‘) 
          scanf("%s",&s);
        if (s[0]==‘i‘)
          {
            scanf("%d",&x);
            x++;
            work(1,x);
          } 
        if (s[0]==‘u‘)
          {
            scanf("%d",&x);
            x++; ans=0;
            query(1,1,n,ls[x],rs[x],1);
            insert(1,1,n,ls[x],rs[x],0);
            printf("%d\n",ans);
          }
      }
} 

T3。考试没想出来，交的30分还打的表，最后10分钟跑出来n=30的数据，吓尿了= =；
UPD:这题事实上就是一个状压DP，考虑每一个数加到一个集合中，相当于增加一个其质因数，而考虑每一个数$n$至多一个大于其$\sqrt n$的因子。于是我们能够仅仅考虑小质因子的情况，最后再减去同样大质因子。详见PoPoQQQ神犇的题解 code：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
int n;
long long f[301][301],p[3][301][301];
int pri[9]={0,2,3,5,7,11,13,17,19};
struct hp{
    int prime;
    int set;
    bool operator < (const hp &a) const
      {return ((prime<a.prime)||(prime==a.prime&&set<a.set));}
}num[501];
long long P;
int main()
{
    int i,j,k,t;
    long long ans;
    freopen("dinner.in","r",stdin);
    freopen("dinner.out","w",stdout);
    scanf("%d%I64d",&n,&P);
    for (i=1;i<=n;++i)
      {
        num[i].set=0; t=i;
        for (j=1;j<=8;++j)
          if (t%pri[j]==0)
            {
              num[i].set=num[i].set+(1<<(j-1)); 
              while (t%pri[j]==0)
                t/=pri[j];
            }
        num[i].prime=t;
      }
    sort(num+2,num+n+1);
    f[0][0]=1;
    for (i=2;i<=n;++i)
      {
        if (i==2||num[i].prime==1||num[i].prime!=num[i-1].prime)
          {
            memcpy(p[1],f,sizeof(f));
            memcpy(p[2],f,sizeof(f));
          }
        for (j=255;j>=0;--j)
          for (k=255;k>=0;--k)
            if ((j&k)==0)
              {
                if ((k&num[i].set)==0) p[1][j|num[i].set][k]=(p[1][j|num[i].set][k]+p[1][j][k])%P;
                if ((j&num[i].set)==0) p[2][j][k|num[i].set]=(p[2][j][k|num[i].set]+p[2][j][k])%P;
              }
        if (i==n||num[i].prime==1||num[i].prime!=num[i+1].prime)
          {
            for (j=0;j<=255;++j)
              for (k=0;k<=255;++k)
                if ((j&k)==0)
                  {
                    f[j][k]=((p[1][j][k]+p[2][j][k]-f[j][k])%P+P)%P;
                  }
          }
      }
    ans=0;
    for (i=0;i<=255;++i)
      for (j=0;j<=255;++j)
        if ((i&j)==0)
          ans=(ans+f[i][j])%P;
    printf("%I64d\n",ans);
}

DAY2：
T1，考虑两个数的编码不存在一个是还有一个前缀的，在树上就相当于不存在两个点之间有父子关系。

而长度最短就相当于让权值与到根路径的乘积和最短。

这不就是k叉哈夫曼树么，对于不够k的能够先补零，然后同合并果子code：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>
using namespace std;
struct hp{
    long long num;
    int dep;
    bool operator < (const hp &a) const
      {return (num>a.num)||(num==a.num&&dep>a.dep);}
};
int n,k;
priority_queue<hp> q;
int main()
{
    long long a,x,ans=0;
    int i,depth;
    freopen("epic.in","r",stdin);
    freopen("epic.out","w",stdout);
    hp minn;
    scanf("%d%d",&n,&k);
    for (i=1;i<=n;++i)
      {
        scanf("%lld",&a);
        q.push((hp){a,0});
      }
    if (k!=2)
      {
        while (n%(k-1)!=1)
          {
            n++;
            q.push((hp){0,0});
          }
      }
    while (n!=1)
      {
        x=0; depth=0;
        for (i=1;i<=k;++i)
          {
            minn=q.top();
            x=x+minn.num;
            depth=max(depth,minn.dep);
            q.pop();
          }
        n=n-k+1;
        ans+=x;
        q.push((hp){x,depth+1});
      }
    minn=q.top();
    printf("%lld\n%d\n",ans,minn.dep);
}

T2：学后缀数组在BZOJ上就做了两道题，【AHOI 2013】差异和【JSOI 2007】字符加密。后者是裸题，而前者差点儿与此题一模一样，都能够分治的做。做法似乎被卡了常数，按点时限可能会T一到两个点。总时限的话没有问题。

code：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#define mid (l+r)/2
#define lch i<<1,l,mid
#define rch i<<1|1,mid+1,r
#define inf 2100000000LL
using namespace std;
char s[300001];
int n,maxm;
int sa[300001],rank[300001],height[300001];
int c[300001],x[300001],y[300001];
long long val[300001];
int t,ti;
long long txl,txr,tnl,tnr;
struct hq{
    long long tot,maxn;
}ans[300001];
struct hp{
    int sim,mini;
    long long minn,maxn;
}seg[1200001];
bool cmp(int *y,int i,int j,int k) 
{ 
    int a,b,c,d; 
    a=y[i]; b=y[j]; 
    c=i+k>=n?-1:y[i+k]; d=j+k>=n?
-1:y[j+k]; 
    return a==b&&c==d; 
} 
void build_sa() 
{ 
    int i,j,k; 
    for (i=0;i<maxm;++i) c[i]=0;
    for (i=0;i<n;++i) c[x[i]=s[i]]++;
    for (i=1;i<maxm;++i) c[i]+=c[i-1];
    for (i=n-1;i>=0;--i) sa[--c[x[i]]]=i;
    for (k=1;k<=n;k<<=1)
      {
        int p=0;
        for (i=n-k;i<n;++i) y[p++]=i;
        for (i=0;i<n;++i) if (sa[i]>=k) y[p++]=sa[i]-k;
        for (i=0;i<maxm;++i) c[i]=0;
        for (i=0;i<n;++i) c[x[y[i]]]++;
        for (i=1;i<maxm;++i) c[i]+=c[i-1];
        for (i=n-1;i>=0;--i) sa[--c[x[y[i]]]]=y[i];
        swap(x,y);
        p=1; x[sa[0]]=0;
        for (i=1;i<n;++i)
          x[sa[i]]=cmp(y,sa[i],sa[i-1],k)?p-1:p++;
        if (p>=n) break;
        maxm=p;
      }
    for (i=0;i<n;++i) 
      rank[sa[i]]=i; 
    k=0; 
    for (i=0;i<n;++i) 
      { 
        if (!rank[i]) continue;
        if (k) k--; 
        j=sa[rank[i]-1];
        while (s[i+k]==s[j+k]) k++; 
        height[rank[i]]=k;   
      } 
} 
void updata(int i)
{
    seg[i].sim=min(seg[i<<1].sim,seg[i<<1|1].sim);
    if (seg[i<<1].sim<=seg[i<<1|1].sim) seg[i].mini=seg[i<<1].mini;
    else seg[i].mini=seg[i<<1|1].mini;
    seg[i].minn=min(seg[i<<1].minn,seg[i<<1|1].minn);
    seg[i].maxn=max(seg[i<<1].maxn,seg[i<<1|1].maxn);
}
void build(int i,int l,int r)
{
    if (l==r)
      {
        seg[i].sim=height[l];
        seg[i].mini=l;
        seg[i].minn=seg[i].maxn=val[l];
        return;
      }
    build(lch); build(rch);
    updata(i);
}
void query(int i,int l,int r,int x,int y,int kind)
{
    if (x<=l&&y>=r)
      {
        if (kind==0)
            if (seg[i].sim<t)
              {t=seg[i].sim; ti=seg[i].mini;}
        if (kind==1)
          {txl=max(txl,seg[i].maxn); tnl=min(tnl,seg[i].minn);}
        if (kind==2)
          {txr=max(txr,seg[i].maxn); tnr=min(tnr,seg[i].minn);}
        return;
      }
    if (x<=mid) query(lch,x,y,kind);
    if (y>mid) query(rch,x,y,kind);
}
void work(int l,int r)
{
    int tti;
    if (l==r) return;
    t=n+1; ti=-1;
    query(1,0,n-1,l+1,r,0);
    tti=ti;
    ans[t+1].tot=ans[t+1].tot+(long long)((long long)(r-ti+1)*(long long)(ti-l));
    txl=-inf; tnl=inf; query(1,0,n-1,l,ti-1,1);
    txr=-inf; tnr=inf; query(1,0,n-1,ti,r,2);
    ans[t+1].maxn=max(ans[t+1].maxn,max(txl*txr,tnl*tnr));
    work(l,tti-1); work(tti,r);
}
int main()
{
    int i;
    int a;
    freopen("savour.in","r",stdin);
    freopen("savour.out","w",stdout);
    scanf("%d",&n);
    scanf("%s",&s);
    while (s[0]<‘a‘||s[0]>‘z‘)
      scanf("%s",&s);
    for (i=0;i<n;++i)
      maxm=max(maxm,(int)(s[i]));
    maxm++;
    build_sa();
    for (i=0;i<n;++i)
      {
        scanf("%d",&a);
        val[rank[i]]=(long long)a;
        ans[i+1].tot=0;
        ans[i+1].maxn=-1000000000000000000LL;
      }
    build(1,0,n-1);
    work(0,n-1);
    for (i=n-1;i>=1;--i)
     {ans[i].tot=ans[i].tot+ans[i+1].tot; ans[i].maxn=max(ans[i].maxn,ans[i+1].maxn);}
    for (i=1;i<=n;++i)
      {
        if (ans[i].tot==0) ans[i].maxn=0;
        printf("%lld %lld\n",ans[i].tot,ans[i].maxn);
      }
}

T3并没有看懂题解，（我会说我连题目都看得迷迷糊糊的么= =）
以后再研究吧= =