[ABC270Ex] add 1
Problem Statement
You are given a tuple of $N$ non-negative integers $A=(A_1,A_2,\ldots,A_N)$ such that $A_1=0$ and $A_N>0$.
Takahashi has $N$ counters. Initially, the values of all counters are $0$.
He will repeat the following operation until, for every $1\leq i\leq N$, the value of the $i$-th counter is at least $A_i$.
Choose one of the $N$ counters uniformly at random and set its value to $0$. (Each choice is independent of others.)
Increase the values of the other counters by $1$.
Print the expected value of the number of times Takahashi repeats the operation, modulo $998244353$ (see Notes).
Notes
It can be proved that the sought expected value is always finite and rational. Additionally, under the Constraints of this problem, when that value is represented as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, one can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.
Constraints
- $2\leq N\leq 2\times 10^5$
- $0=A_1\leq A_2\leq \cdots \leq A_N\leq 10^{18}$
- $A_N>0$
- All values in the input are integers.
Input
The input is given from Standard Input in the following format:
$N$
$A_1$ $A_2$ $\ldots$ $A_N$
Output
Print the expected value of the number of times Takahashi repeats the operation, modulo $998244353$.
Sample Input 1
2
0 2
Sample Output 1
6
Let $C_i$ denote the value of the $i$-th counter.
Here is one possible progression of the process.
- Set the value of the $1$-st counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(0,1)$.
- Set the value of the $2$-nd counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(1,0)$.
- Set the value of the $1$-st counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(0,1)$.
- Set the value of the $1$-st counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(0,2)$.
In this case, the operation is performed four times.
The probabilities that the process ends after exactly $1,2,3,4,5,\ldots$ operation(s) are $0,\frac{1}{4}, \frac{1}{8}, \frac{1}{8}, \frac{3}{32},\ldots$, respectively, so the sought expected value is $2\times\frac{1}{4}+3\times\frac{1}{8}+4\times\frac{1}{8}+5\times\frac{3}{32}+\dots=6$.
Thus, $6$ should be printed.
Sample Input 2
5
0 1 3 10 1000000000000000000
Sample Output 2
874839568
概率生成函数做法,下面以 \('\) 表示求导
放几个求导公式先。
\]
\]
设最终的概率生成函数为 \(H(x)\),\(H(x)=\frac{F(x)}{G(x)}\),F表示 \(0\) 时刻 \(A\) 全为 0,在 \(i\) 时刻符合要求的概率,\(G\) 表示 \(0\) 时刻满足要求,在 \(i\) 时刻满足要求的概率。这是因为 \(G*H=F\),很容易理解。根据生成函数的经典结论,\(H'(1)\) 为最终答案。
那么 \(F\) 和 \(G\) 是可以求出来的。先看 F,一种操作序列满足要求的前提是操作序列长度至少为 \(a_n\) 且最后 \(a_i\) 次操作没有弄 \(i\)。
我们需要知道如果从某个时刻开始形成 \(A_i\),那么生成从出来的概率是多少。那么有 \(a_{i+1}-a_i\) 个时刻,生成出来的概率是 \(\frac in\)。那么定义 \(s_i=\prod_{j<i}(\frac jn)^{a_{j+1}-a_j}\),可以用一个前缀积求出。 F 就是从任意大于 \(a_n\) 的时刻都有 \(s_i\) 的概率成功,\(F(x)=s_n(x_{a_n}+x_{a_n+1}+\cdots)=\frac{s_nx^n}{1-x}\)
那么 G 的生成函数呢? G 和 F 的区别是他不一定需要 \(a_n\) 时刻才能满足,它可以只用 \(a_i\) 时刻,重构前面 \(i\) 个数。这里设时刻 \(a_i]\le j<a_{i+1}\),那么 \(j\) 时刻开始的概率就是 \(s_i(\frac in)^{j-a_i}\),用一个等比数列求和,这里省略一下过程,得到 \(g(x)=\frac{s_nx^{a_n}}{1-x}+\sum\limits_{i=1}^{n-1}n\frac{s_ix^{a_i}-s_{i+1}x^{a_{i+1}}}{n-ix}\)
然后求导,发现两个函数中 1 处的取值都是没有意义的,但是把 \(F\) 和 \(G\) 同乘 \((1-x)\)即可。后面的 F 和 G 都是乘了 \(1-x\) 之后的。\(F(1)=s_n,F'(1)=s_na_n,G(1)=s_n\)(后面的乘上 (1-x)) 后,带入 \(x=1\) 全部消掉。那么现在重点来推 \(G'(1)\)
现在要求 \(\frac{n(s_ix^{a_i}-s_{i+1}x^{a_{i+1}})(1-x)}{n-ix}\) 的导数,下面平方后出来 \((n-ix)^2=(n-i)^2\),方便起见,设 \((1-x)=A(x),s_{i}x^{a_i}-s_{i+1}x^{a_{i+1}}=B(x)\),则 \((nA*B)'(1)=n(A'*B(1)+A*B'(1))\),发现 \(A(1)=0\),那么最终式子简化问 \(nA'*B(1)=-ns_{i}+ns_{i+1}\)。\(n-ix\) 求导为 \(-i\),那么最终式子就是 \(\frac{n(s_{i+1}-s_i)(n-i)}{(n-i)^2}=\frac{n(s_{i+1}-s_i)}{n-i}\)
最后套除法求导公式就行了
#include<bits/stdc++.h>
using namespace std;
const int N=2e5+5,P=998244353;
typedef long long LL;
int n,g,f,fp,gp,fd,s[N];
LL a[N];
int pown(int x,LL y)
{
if(!y)
return 1;
int t=pown(x,y>>1);
if(y&1)
return 1LL*t*t%P*x%P;
return 1LL*t*t%P;
}
int main()
{
scanf("%d",&n);
s[0]=s[1]=1;
for(int i=1;i<=n;i++)
{
scanf("%lld",a+i);
if(i^1)
s[i]=1LL*s[i-1]*pown((i-1LL)*pown(n,P-2)%P,a[i]-a[i-1])%P;
}
fp=s[n]*1LL*(a[n]%P)%P;
f=s[n];
g=s[n],gp=s[n]*1LL*(a[n]%P)%P;
for(int i=1;i<n;i++)
(gp+=pown(n-i,P-2)*1LL*n%P*(s[i+1]-s[i]+P)%P)%=P;
printf("%lld",(fp*1LL*g%P-gp*1LL*f%P+P)*pown(g*1LL*g%P,P-2)%P);
}
[ABC270Ex] add 1的更多相关文章
- AutoMapper:Unmapped members were found. Review the types and members below. Add a custom mapping expression, ignore, add a custom resolver, or modify the source/destination type
异常处理汇总-后端系列 http://www.cnblogs.com/dunitian/p/4523006.html 应用场景:ViewModel==>Mode映射的时候出错 AutoMappe ...
- EntityFramework Core 1.1 Add、Attach、Update、Remove方法如何高效使用详解
前言 我比较喜欢安静,大概和我喜欢研究和琢磨技术原因相关吧,刚好到了元旦节,这几天可以好好学习下EF Core,同时在项目当中用到EF Core,借此机会给予比较深入的理解,这里我们只讲解和EF 6. ...
- ASP.NET Core: You must add a reference to assembly mscorlib, version=4.0.0.0
ASP.NET Core 引用外部程序包的时候,有时会出现下面的错误: The type 'Object' is defined in an assembly that is not referenc ...
- [转]NopCommerce How to add a menu item into the administration area from a plugin
本文转自:http://docs.nopcommerce.com/display/nc/How+to+code+my+own+shipping+rate+computation+method Go t ...
- [deviceone开发]-动态添加组件add方法的示例
一.简介 这个示例详细介绍ALayout的add方法的使用(原理也适用于Linearlayout),以及add上去的新ui和已有的ui如何数据交换,初学者推荐.二.效果图 三.相关下载 https:/ ...
- [LeetCode] Add Two Numbers II 两个数字相加之二
You are given two linked lists representing two non-negative numbers. The most significant digit com ...
- [LeetCode] Add Strings 字符串相加
Given two non-negative numbers num1 and num2 represented as string, return the sum of num1 and num2. ...
- [LeetCode] Expression Add Operators 表达式增加操作符
Given a string that contains only digits 0-9 and a target value, return all possibilities to add ope ...
- [LeetCode] Add Digits 加数字
Given a non-negative integer num, repeatedly add all its digits until the result has only one digit. ...
- [LeetCode] Different Ways to Add Parentheses 添加括号的不同方式
Given a string of numbers and operators, return all possible results from computing all the differen ...
随机推荐
- 《Linux基础》02. 目录结构 · vi、vim · 关机 · 重启
@ 目录 1:目录结构 2:vi.vim快速入门 2.1:vi 和 vim 的三种模式 2.1.1:一般模式 2.1.2:编辑模式 2.1.3:命令模式 2.2:常用快捷键 2.2.1:一般模式 2. ...
- Jenkins 忘记密码|密码重置
I. 当前环境 OS Version : AlmaLinux release 8.8 Jenkins Version : 2.414.1 II. 操作步骤 2.1 修改配置文件 1. SSH 登录服务 ...
- SQL Server用户的设置与授权
SQL Server用户的设置与授权 SSMS 登陆方式有两种,一是直接使用Windows身份验证,二是SQL Server身份验证.使用SQL Server用户设置与授权不仅可以将不同的数据库开放给 ...
- destoon根据标题删除重复数据
因为采集数据比较庞大,难免出现重复数据,所以写了一个根据标题进行删除重复数据的mysql命令,需要的朋友可以使用. 1 2 3 4 DELETE from destoon_article_36 whe ...
- 《机器人SLAM导航核心技术与实战》第1季:第6章_机器人底盘
<机器人SLAM导航核心技术与实战>第1季:第6章_机器人底盘 视频讲解 [第1季]6.第6章_机器人底盘-视频讲解 [第1季]6.1.第6章_机器人底盘_底盘运动学模型-视频讲解 [第1 ...
- 带宽优化新思路:RoCE网卡聚合实现X2增长
本文分享自华为云社区<2个RoCE网卡Bond聚合,实现带宽X2>,作者: tsjsdbd . 我们知道操作系统里面,可以将2个实际的物理网卡,合体形成一个"逻辑网卡" ...
- 征集 meme
当你每次兴致勃勃地和好友分享自己喜欢的歌但 Ta 不屑一顾 / 不喜欢时:
- Macos下用Clion调试chromium源码
1:下载CLion 2021.1.3(网上有破解版) 2:选择File->Open 导入chromium源码 3:在Src同级目录新建一个CMakeLists.txt 4:点击clion编译按钮 ...
- 简单实现.NET Hook与事件模拟
最近玩<星露谷物语>上瘾,本来是看着个休闲游戏,现在玩成修仙游戏了,上百个小时浑身是肝,中午午休习惯都强行给改了. 虽然挺有意思,但是太肝了,入坑前请谨慎.补充一下,这个游戏应该是基于 X ...
- git报错fatal: unable to access 'https://github.com/hxx.git/': LibreSSL SSL_connect: SSL_ERROR_SYSCALL in connection to github.com:443
今天拉git代码报错 弄好了,下面是解决方法: 在网上查了很多办法都没有解决,有的方法是https连接模式改成ssh模式,或者是修改代理,比如: git config --global http.pr ...