poj 3233 Matrix Power Series (矩阵快速幂+分治)

题目链接

题意:

Given a n × n matrix A and a positive integer k, find the sum S = A + _A_2 + _A_3 + … + Ak.

思路: 对k进行二分。

比如,当k=6时,有: A + A^2 + A^3 + A^4 + A^5 + A^6 =(A + A^2 + A^3) + A^3*(A + A^2 + A^3) 应用这个式子后,规模k减小了一半。我们二分求出A^3后再递归地计算A + A^2 + A^3,即可得到原问题的答案。

以及错误的递归方式:

screenshot-from-2016-10-19-16-12-57

无形中增加了多少次。。。。。。怎么像小学生呢。。。

screenshot-from-2016-10-19-16-12-57

正确的写法。。。

screenshot-from-2016-10-19-16-14-08

  1/* ***********************************************
  2Author :111qqz
  3Created Time :Tue 18 Oct 2016 05:10:53 PM CST
  4File Name :code/poj/3233.cpp
  5************************************************ */
  6#include <cstdio>
  7#include <cstring>
  8#include <iostream>
  9#include <algorithm>
 10#include <vector>
 11#include <queue>
 12#include <set>
 13#include <map>
 14#include <string>
 15#include <cmath>
 16#include <cstdlib>
 17#include <ctime>
 18#define fst first
 19#define sec second
 20#define lson l,m,rt<<1
 21#define rson m+1,r,rt<<1|1
 22#define ms(a,x) memset(a,x,sizeof(a))
 23typedef long long LL;
 24#define pi pair < int ,int >
 25#define MP make_pair
 26using namespace std;
 27const double eps = 1E-8;
 28const int dx4[4]={1,0,0,-1};
 29const int dy4[4]={0,-1,1,0};
 30const int inf = 0x3f3f3f3f;
 31const int N=35;
 32int n,k,mod;
 33struct Mat
 34{
 35    int mat[N][N];
 36    void clear()
 37    {
 38	ms(mat,0);
 39    }
 40    void out()
 41    {
 42	for ( int i = 0 ; i < n ; i++ )
 43	{
 44	    for ( int j = 0 ; j <n-1 ; j++) printf("%d ",mat[i][j]);
 45	    printf("%d\n",mat[i][n-1]);
 46	}
 47    }
 48}M;
 49Mat operator * ( Mat a ,Mat b )
 50{
 51    Mat c;
 52    c.clear();
 53    for ( int i = 0 ; i < n ; i++)
 54	for ( int j = 0 ; j < n; j ++)
 55	    for ( int k = 0 ; k < n;  k++)
 56		c.mat[i][j] = (c.mat[i][j] + a.mat[i][k]*b.mat[k][j])%mod;
 57    return c;
 58}
 59Mat operator + (Mat a,Mat b)
 60{
 61    Mat c;
 62    c.clear();
 63    for ( int i = 0 ; i < n ;i ++)
 64	for ( int j = 0 ; j < n ; j++)
 65	    c.mat[i][j] = (a.mat[i][j] + b.mat[i][j])%mod;
 66    return c;
 67}
 68Mat operator ^ (Mat a,int b)
 69{
 70    Mat c;
 71    for ( int i = 0 ; i < n ; i++)
 72	for ( int j = 0 ;j  <n ; j++)
 73	    c.mat[i][j] = (i==j);
 74    while (b>0)
 75    {
 76	if (b&1) c = c * a;
 77	b = b >> 1;
 78	a = a* a;
 79    }
 80    return c;
 81}
 82Mat solve(  int k)
 83{
 84    if (k==1) return M;
 85    Mat res;
 86    res.clear();
 87    for ( int i = 0 ; i < n ;  i++) res.mat[i][i] = 1;
 88    res = res + (M^(k>>1));
 89    res = res * solve(k>>1);
 90    if (k%2==1) res = res + (M^k);
 91    return res;
 92}
 93int main()
 94{
 95#ifndef  ONLINE_JUDGE 
 96    freopen("code/in.txt","r",stdin);
 97#endif
 98        scanf("%d%d%d",&n,&k,&mod);
 99	for ( int i = 0 ; i < n ; i++)
100	    for ( int j = 0 ; j  < n ; j++)
101		scanf("%d",&M.mat[i][j]);
102	Mat ans;
103	ans.clear();
104	ans = solve(k);
105	ans.out();
106#ifndef ONLINE_JUDGE  
107	fclose(stdin);
108#endif
109	return 0;
110}