传送门
文章目录
- 题意:
- 思路:
题意:
思路:
考虑最终的序列是什么鸭子的,首先序列肯定单调不降,也就是a1≤a2≤a3≤...≤ana_1\le a_2\le a_3\le ...\le a_na1≤a2≤a3≤...≤an,显然不可能都是≤\le≤号,因为如果插入的话是有可能产生<<<号的。假设我们现在有xxx个<<< 号,那么应该有(n+n−1−xn)\binom{n+n-1-x}{n}(nn+n−1−x)个序列,因为我们将ai≤ai+1a_i\le a_{i+1}ai≤ai+1转换成ai<ai+1+1a_i< a_i+1+1ai<ai+1+1,也就是扩展了一下值域,最多扩展了n−1−xn-1-xn−1−x个值域,也就是目前值域是[1,n∗2−1−x][1,n*2-1-x][1,n∗2−1−x],从中选nnn个数构成一个严格递增的序列方案就是(n∗2−1−xn)\binom{n*2-1-x}{n}(nn∗2−1−x),那么现在问题就是如何知道xxx。
首先插入一组最多能产生一个<<<号,因为有可能若干数插到了同一个数后面,比如这个例子n=3,m=2,(2,1),(3,2)n=3,m=2,(2,1),(3,2)n=3,m=2,(2,1),(3,2),这样在a1a_1a1前面插入了两个数a2,a3a_2,a_3a2,a3,能确定a1>a2,a1>a3a_1>a_2,a_1>a_3a1>a2,a1>a3,但是不能确定a2a_2a2和a3a_3a3的关系,所以这样的话我们就认为a2,a3a_2,a_3a2,a3之间是一个≤\le≤ 号。我们考虑倒着来处理这mmm组,首先最终序列长度是nnn,假设现在有(xi,yi)(x_i,y_i)(xi,yi),那么先找到第yiy_iyi个位置ppp和第yi+1y_i+1yi+1的位置qqq,此时将ppp删除,将qqq标记一下代表有个数在他前面有<<<即可。至于为什么是yi,yi+1y_i,y_{i+1}yi,yi+1而不是yi−1,yiy_{i-1},y_{i}yi−1,yi,是因为如果yi=1y_i=1yi=1的时候,yi−1=0y_i-1=0yi−1=0,这个时候在线段树上二分就会出错。
在线段树上二分即可。
如果正着来,可以建一个基于权值的平衡树,让后模拟插入即可。
由于题目的问题,线段树不能每次建树。。。
// Problem: F. Top-Notch Insertions
// Contest: Codeforces - Codeforces Round #740 (Div. 2, based on VK Cup 2021 - Final (Engine))
// URL: https://codeforces.com/contest/1561/problem/F
// Memory Limit: 512 MB
// Time Limit: 3000 ms
//
// Powered by CP Editor (https://cpeditor.org)//#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")
//#pragma GCC optimize(2)
#include<cstdio>
#include<iostream>
#include<string>
#include<cstring>
#include<map>
#include<cmath>
#include<cctype>
#include<vector>
#include<set>
#include<queue>
#include<algorithm>
#include<sstream>
#include<ctime>
#include<cstdlib>
#include<random>
#include<cassert>
#define X first
#define Y second
#define L (u<<1)
#define R (u<<1|1)
#define pb push_back
#define mk make_pair
#define Mid ((tr[u].l+tr[u].r)>>1)
#define Len(u) (tr[u].r-tr[u].l+1)
#define random(a,b) ((a)+rand()%((b)-(a)+1))
#define db puts("---")
using namespace std;//void rd_cre() { freopen("d://dp//data.txt","w",stdout); srand(time(NULL)); }
//void rd_ac() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//AC.txt","w",stdout); }
//void rd_wa() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//WA.txt","w",stdout); }typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int,int> PII;const int N=1000010,mod=998244353,INF=0x3f3f3f3f;
const double eps=1e-6;int n,m;
PII p[N];
struct Node {int l,r;int cnt;
}tr[N<<2];void pushup(int u) {tr[u].cnt=tr[L].cnt+tr[R].cnt;
}void build(int u,int l,int r) {tr[u]={l,r};if(l==r) {tr[u].cnt=1;return;}build(L,l,Mid); build(R,Mid+1,r);pushup(u);
}void change(int u,int pos,int x) {if(tr[u].l==tr[u].r) {tr[u].cnt=x;return;}if(pos<=Mid) change(L,pos,x);else change(R,pos,x);pushup(u);
}int query(int u,int cnt) {if(tr[u].l==tr[u].r) return tr[u].l;if(cnt<=tr[L].cnt) return query(L,cnt);else return query(R,cnt-tr[L].cnt);
}LL fun[N],inv[N];LL qmi(LL a,LL b) {LL ans=1;while(b) {if(b&1) ans=ans*a%mod;a=a*a%mod;b>>=1;}return ans%mod;
}LL C(int a,int b) {if(a<0||b<0||a<b) return 0;return fun[a]*inv[b]%mod*inv[a-b]%mod;
}int main()
{
// ios::sync_with_stdio(false);
// cin.tie(0);fun[0]=1;for(int i=1;i<N;i++) fun[i]=fun[i-1]*i%mod;inv[N-1]=qmi(fun[N-1],mod-2);for(int i=N-2;i>=0;i--) inv[i]=inv[i+1]*(i+1)%mod;build(1,1,N-1);int _; scanf("%d",&_);while(_--) {scanf("%d%d",&n,&m);for(int i=1;i<=m;i++) scanf("%d%d",&p[i].X,&p[i].Y);set<int>s; vector<int>v;for(int i=m;i>=1;i--) {int a=query(1,p[i].Y),b=query(1,p[i].Y+1);change(1,a,0); s.insert(b); v.pb(a);}for(auto x:v) change(1,x,1);int cnt=s.size();printf("%lld\n",C(n*2-1-cnt,n));}return 0;
}
/**/
// Problem: F. Top-Notch Insertions
// Contest: Codeforces - Codeforces Round #740 (Div. 2, based on VK Cup 2021 - Final (Engine))
// URL: https://codeforces.com/contest/1561/problem/F
// Memory Limit: 512 MB
// Time Limit: 3000 ms
//
// Powered by CP Editor (https://cpeditor.org)//#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")
//#pragma GCC optimize(2)
#include<cstdio>
#include<iostream>
#include<string>
#include<cstring>
#include<map>
#include<cmath>
#include<cctype>
#include<vector>
#include<set>
#include<queue>
#include<algorithm>
#include<sstream>
#include<ctime>
#include<cstdlib>
#include<random>
#include<cassert>
#define X first
#define Y second
#define L (u<<1)
#define R (u<<1|1)
#define pb push_back
#define mk make_pair
#define Mid ((tr[u].l+tr[u].r)>>1)
#define Len(u) (tr[u].r-tr[u].l+1)
#define random(a,b) ((a)+rand()%((b)-(a)+1))
#define db puts("---")
using namespace std;//void rd_cre() { freopen("d://dp//data.txt","w",stdout); srand(time(NULL)); }
//void rd_ac() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//AC.txt","w",stdout); }
//void rd_wa() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//WA.txt","w",stdout); }typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int,int> PII;const int N=1000010,mod=998244353,INF=0x3f3f3f3f;
const double eps=1e-6;int n,m;
int root,tot;
int x,y;
struct Node {int l,r;int size,fa,val,rank,tag;
}tr[N];int newnode(int v) {tr[++tot].val=v,tr[tot].size=1,tr[tot].rank=rand();tr[tot].l=tr[tot].r=tr[tot].tag=0;return tot;
}void pushup(int u) {if(!u) return;tr[u].size=tr[tr[u].l].size+tr[tr[u].r].size+1;// tr[u].val=max(tr[tr[u].l].val,tr[tr[u].r].val);
}void pushdown(int u) {if(!u) return;int tag=tr[u].tag; tr[u].tag=0;if(tr[u].l) tr[tr[u].l].tag+=tag,tr[tr[u].l].val+=tag;if(tr[u].r) tr[tr[u].r].tag+=tag,tr[tr[u].r].val+=tag;
}void split(int u,int k,int &x,int &y) {if(!u) { x=y=0; return; }pushdown(u);if(tr[u].val<=k) x=u,split(tr[u].r,k,tr[u].r,y);else y=u,split(tr[u].l,k,x,tr[u].l);pushup(u);
}int merge(int u,int v) {if(!u||!v) return u+v;if(tr[u].rank<tr[v].rank) {pushdown(u);tr[u].r=merge(tr[u].r,v);pushup(u);return u;}else {pushdown(v);tr[v].l=merge(u,tr[v].l);pushup(v);return v;}
}int find(int u,int x) {if(!u) return 0;if(tr[u].val==x) return 1;pushdown(u);if(x<tr[u].val) return find(tr[u].l,x);else return find(tr[u].r,x);
}LL fun[N],inv[N];LL qmi(LL a,LL b) {LL ans=1;while(b) {if(b&1) ans=ans*a%mod;a=a*a%mod;b>>=1;}return ans%mod;
}LL C(int a,int b) {if(a<0||b<0||a<b) return 0;return fun[a]*inv[b]%mod*inv[a-b]%mod;
}int main()
{
// ios::sync_with_stdio(false);
// cin.tie(0);fun[0]=1;for(int i=1;i<N;i++) fun[i]=fun[i-1]*i%mod;inv[N-1]=qmi(fun[N-1],mod-2);for(int i=N-2;i>=0;i--) inv[i]=inv[i+1]*(i+1)%mod;int _; scanf("%d",&_);while(_--) {root=tot=0;scanf("%d%d",&n,&m);for(int i=1;i<=m;i++) {int a,b; scanf("%d%d",&a,&b);int flag=find(root,b);split(root,b-1,x,y);if(y) tr[y].val++,tr[y].tag++;root=merge(x,flag? y:merge(newnode(b+1),y));}printf("%lld\n",C(n*2-1-tot,n));}return 0;
}
/**/