[HNOI 2011]数学作业
Description
小 C 数学成绩优异,于是老师给小 C 留了一道非常难的数学作业题:
给定正整数 N 和 M,要求计算 Concatenate (1 .. N) Mod M 的值,其中 Concatenate (1 ..N)是将所有正整数 1, 2, …, N 顺序连接起来得到的数。例如,N = 13, Concatenate (1 .. N)=12345678910111213.小C 想了大半天终于意识到这是一道不可能手算出来的题目,于是他只好向你求助,希望你能编写一个程序帮他解决这个问题。
Input
从文件input.txt中读入数据,输入文件只有一行且为用空格隔开的两个正整数N和M,其中30%的数据满足1≤N≤1000000;100%的数据满足1≤N≤1018且1≤M≤109.
Output
输出文件 output.txt 仅包含一个非负整数,表示 Concatenate (1 .. N) Mod M 的值。
Sample Input
13 13
Sample Output
4
题解
首先写出递推关系式:令 $s_i$ 为前 i 个数连接得到的数,$c(i)$ 表示 $i$ 的位数,有
$s_i=10^{c(i)}×s_{i−1}+i$
$c(i)$ 随 $i$ 变化,无法在转移矩阵 $T$ 中表示。如果 $c(i)$ 固定,就可以矩阵优化了!
考虑位数随着 $i$ 的变化最多只会变化 $18$ 次,因此可以按位数分段进行矩阵快速幂。
枚举位数,那么有
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" alt="" />
这里在状态矩阵中加了一个永远为 $1$ 的值,用于辅助 $i$ 每次加一。
#include<map>
#include<cmath>
#include<ctime>
#include<queue>
#include<stack>
#include<cstdio>
#include<string>
#include<vector>
#include<cstring>
#include<cstdlib>
#include<iostream>
#include<algorithm>
#define LL long long
#define RE register
#define IL inline
using namespace std; LL n,m;
struct mat
{
LL a[][];
mat(){for (int i=;i<;i++)for (int j=;j<;j++) a[i][j]=;}
mat operator * (const mat&b)
{
mat ans;
for (int i=;i<;i++)
for (int j=;j<;j++)
for (int k=;k<;k++)
ans.a[i][j]=(ans.a[i][j]+(a[i][k]*b.a[k][j])%m)%m;
return ans;
}
}S; int main()
{
S.a[][]=S.a[][]=;
scanf("%lld%lld",&n,&m);
LL now=;
for (LL s=;now<n;s*=)
{
mat T;
T.a[][]=T.a[][]=T.a[][]=T.a[][]=;
T.a[][]=s%m;
LL nd=min(s-,n)-now;
while (nd)
{
if (nd&) S=S*T;
T=T*T;
nd>>=;
}
now=min(s-,n);
}
printf("%lld\n",S.a[][]);
return ;
}
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