【（阶乘的质因数分解）算组合数】【TOJ4111】【Binomial efficient】

n<=10^6

m<=10^6

p=2^32

用unsigned int 可以避免取模

我写的SB超时 阶乘分解代码

#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <cstring>
#include <ctime>
#include <algorithm>
#include <iostream>
#include <sstream>
#include <string>
#define oo 0x13131313
using namespace std;
const unsigned int N=1000000+5;
unsigned int tag[N],p[N],z[N],mm[N];
unsigned int cnt = 0;
unsigned int n,m;
unsigned int quickpow(unsigned int m,unsigned int n)
{
    unsigned int b = 1;
    while (n > 0)
    {
          if (n & 1)
             b = (b*m);
          n = n >> 1 ;
          m = (m*m);
    }
    return b;
}

void get_prime()
{
    tag[1]=1;
    tag[0]=1;
    for (unsigned int i = 2; i < N; i++)
    {
        if (!tag[i])    p[cnt++] = i;
        for (unsigned int j = 0; j < cnt && p[j] * i < N; j++)
        {
            tag[i*p[j]] = 1;
            if (i % p[j] == 0)
                break;
        }
    }
}
unsigned int FIND(unsigned int x)
{
    unsigned int l=0,r=cnt-1;
    while(l<=r)
    {
        unsigned int m=(l+r)/2;
        if(p[m]==x) return m;
        else if(p[m]<x) l=m+1;
        else r=m-1;
    }
}
void fenjie(unsigned int *a,unsigned int d)
{
    for(unsigned int i=0;i<cnt&&p[i]<=sqrt(d);i++)
    {
        while(d%p[i]==0)
        {
            d=d/p[i];
            a[i]++;
        }
    }
    if(tag[d]==0)
    {
        unsigned int t=FIND(d);
        a[t]++;
    }
}
int main()
{
   // freopen("a.in","r",stdin);
    get_prime();
    int T;
    cin>>T;
    while(T--)
    {
        cin>>n>>m;
        memset(z,0,sizeof(z));
        memset(mm,0,sizeof(mm));
        for(unsigned int i=n;i>=n-m+1;i--)
            fenjie(z,i);
        for(unsigned int i=1;i<=m;i++)
            fenjie(mm,i);
        unsigned int ans=1;
        for(unsigned int i=0;i<cnt;i++)
            {
                z[i]=z[i]-mm[i];
                ans=ans*quickpow(p[i],z[i]);
            }
        printf("%u
",ans);
    }
}

利用阶乘的质因数分解！

比如２５０！

１＊２＊３＊４＊５＊６＊７＊８＊９＊１０＊１１＊１２＊１３＊１４．．．．．２５０

中３的质因子个数　除了３后变成（不是倍数的不管）计算３＾１次方的为２５０／３个

又变成　１　２　３　．．．．．２５０除３　　

重复上面　知道　３＾２　为２５０／（３＾２）　

所以阶乘的质因数分解是另外的简单算法

void getcn(int n)
{
	int ans = 0;
	int i;
	for (i = 1; i <= prime[0] && prime[i] <= n; i++)
	{
		int tmp = n;
		while (tmp)
		{
			num[i] += tmp / prime[i];
			tmp /= prime[i];
		}
	}
	num[0] = i;
}

所以最后的代码是

#include <string.h>
#include <iostream>
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string>
#include <algorithm>
#include <vector>
#include <string.h>
#include <time.h>
#include <queue>
#include <stack>
#include <map>
#include <set>
#include <sstream>
#define INF 0x3f3f3f3f
#define MAXN 1000005
#define Precision 100005
#define MAX_INT 2147483647
#define Pi acos(-1.0)
#define lowbit(x) ((x)&(-x))
#define Lson root<<1,left,mid
#define Rson root<<1|1,mid+1,right
#define LL long long
#define ULL unsigned long long
#define fresh(x) memset(x,0,sizeof(x))
using namespace std;


int prime[MAXN];
int num[MAXN];


void print()
{
<span style="white-space:pre">	</span>memset(prime, 0, sizeof(prime));
<span style="white-space:pre">	</span>for (int i = 2; i <= 1000000; i++)
<span style="white-space:pre">	</span>{
<span style="white-space:pre">		</span>if (!prime[i]) prime[++prime[0]] = i;
<span style="white-space:pre">		</span>for (int j = 1; j <= prime[0] && prime[j] <= 1000000 / i; j++)
<span style="white-space:pre">		</span>{
<span style="white-space:pre">			</span>prime[prime[j]*i] = 1;
<span style="white-space:pre">			</span>if (i % prime[j] == 0) break;
<span style="white-space:pre">		</span>}
<span style="white-space:pre">	</span>}
}


unsigned qpow(unsigned a, unsigned b)
{
<span style="white-space:pre">	</span>unsigned ans = 1;
<span style="white-space:pre">	</span>while (b)
<span style="white-space:pre">	</span>{
<span style="white-space:pre">		</span>if (b & 1)
<span style="white-space:pre">			</span>ans *= a;
<span style="white-space:pre">		</span>b >>= 1;
<span style="white-space:pre">		</span>a *= a;
<span style="white-space:pre">	</span>}
<span style="white-space:pre">	</span>return ans;
}


void getcn(int n)
{
<span style="white-space:pre">	</span>int ans = 0;
<span style="white-space:pre">	</span>int i;
<span style="white-space:pre">	</span>for (i = 1; i <= prime[0] && prime[i] <= n; i++)
<span style="white-space:pre">	</span>{
<span style="white-space:pre">		</span>int tmp = n;
<span style="white-space:pre">		</span>while (tmp)
<span style="white-space:pre">		</span>{
<span style="white-space:pre">			</span>num[i] += tmp / prime[i];
<span style="white-space:pre">			</span>tmp /= prime[i];
<span style="white-space:pre">		</span>}
<span style="white-space:pre">	</span>}
<span style="white-space:pre">	</span>num[0] = i;
}


void getcm(int m)
{
<span style="white-space:pre">	</span>int ans = 0;
<span style="white-space:pre">	</span>int i;
<span style="white-space:pre">	</span>for (i = 1; i <= prime[0] && prime[i] <= m; i++)
<span style="white-space:pre">	</span>{
<span style="white-space:pre">		</span>int tmp = m;
<span style="white-space:pre">		</span>while (tmp)
<span style="white-space:pre">		</span>{
<span style="white-space:pre">			</span>num[i] -= tmp / prime[i];
<span style="white-space:pre">			</span>tmp /= prime[i];
<span style="white-space:pre">		</span>}
<span style="white-space:pre">	</span>}
}


int main()
{
<span style="white-space:pre">	</span>int n, m, T;
<span style="white-space:pre">	</span>print();
<span style="white-space:pre">	</span>//printf("%d
", prime[prime[0]]);
<span style="white-space:pre">	</span>//cout << prime[0];
<span style="white-space:pre">	</span>scanf("%d", &T);
<span style="white-space:pre">	</span>while (T--)
<span style="white-space:pre">	</span>{
<span style="white-space:pre">		</span>memset(num, 0, sizeof(num));
<span style="white-space:pre">		</span>scanf("%d%d", &n, &m);
<span style="white-space:pre">		</span>getcn(n);
<span style="white-space:pre">		</span>getcm(m);
<span style="white-space:pre">		</span>getcm(n - m);
<span style="white-space:pre">		</span>unsigned ans = 1;
<span style="white-space:pre">		</span>for (int i = 1; i <= num[0]; i++)
<span style="white-space:pre">		</span>{
<span style="white-space:pre">			</span>ans *= qpow(prime[i], num[i]);
<span style="white-space:pre">		</span>}
<span style="white-space:pre">		</span>printf("%u
", ans);
<span style="white-space:pre">	</span>}
<span style="white-space:pre">	</span>return 0;
}