1. 程式人生 > >POJ 2429 GCD & LCM Inverse(大數分解)

POJ 2429 GCD & LCM Inverse(大數分解)

用大數分解的模板把lcm/gcd分解了,然後就會得到十幾個質因子,然後我們dfs一次找到和最小的a和b就成了。

#pragma warning(disable:4996)
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<ctime>
#include<iostream>
#include<algorithm>
using namespace std;


//****************************************************************
// Miller_Rabin 演算法進行素數測試
//速度快,而且可以判斷 <2^63的數
//****************************************************************
const int S = 20;//隨機演算法判定次數,S越大,判錯概率越小


				 //計算 (a*b)%c.   a,b都是long long的數,直接相乘可能溢位的
				 //  a,b,c <2^63
long long mult_mod(long long a, long long b, long long c)
{
	a %= c;
	b %= c;
	long long ret = 0;
	while (b)
	{
		if (b & 1) { ret += a; ret %= c; }
		a <<= 1;
		if (a >= c)a %= c;
		b >>= 1;
	}
	return ret;
}



//計算  x^n %c
long long pow_mod(long long x, long long n, long long mod)//x^n%c
{
	if (n == 1)return x%mod;
	x %= mod;
	long long tmp = x;
	long long ret = 1;
	while (n)
	{
		if (n & 1) ret = mult_mod(ret, tmp, mod);
		tmp = mult_mod(tmp, tmp, mod);
		n >>= 1;
	}
	return ret;
}





//以a為基,n-1=x*2^t      a^(n-1)=1(mod n)  驗證n是不是合數
//一定是合數返回true,不一定返回false
bool check(long long a, long long n, long long x, long long t)
{
	long long ret = pow_mod(a, x, n);
	long long last = ret;
	for (int i = 1; i <= t; i++)
	{
		ret = mult_mod(ret, ret, n);
		if (ret == 1 && last != 1 && last != n - 1) return true;//合數
		last = ret;
	}
	if (ret != 1) return true;
	return false;
}

// Miller_Rabin()演算法素數判定
//是素數返回true.(可能是偽素數,但概率極小)
//合數返回false;

bool Miller_Rabin(long long n)
{
	if (n<2)return false;
	if (n == 2)return true;
	if ((n & 1) == 0) return false;//偶數
	long long x = n - 1;
	long long t = 0;
	while ((x & 1) == 0) { x >>= 1; t++; }
	for (int i = 0; i<S; i++)
	{
		long long a = rand() % (n - 1) + 1;//rand()需要stdlib.h標頭檔案
		if (check(a, n, x, t))
			return false;//合數
	}
	return true;
}


//************************************************
//pollard_rho 演算法進行質因數分解
//************************************************
long long factor[100];//質因數分解結果(剛返回時是無序的)
int tol;//質因數的個數。陣列小標從0開始

long long gcd(long long a, long long b)
{
	if (a == 0)return 1;//???????
	if (a<0) return gcd(-a, b);
	while (b)
	{
		long long t = a%b;
		a = b;
		b = t;
	}
	return a;
}

long long Pollard_rho(long long x, long long c)
{
	long long i = 1, k = 2;
	long long x0 = rand() % x;
	long long y = x0;
	while (1)
	{
		i++;
		x0 = (mult_mod(x0, x0, x) + c) % x;
		long long d = gcd(y - x0, x);
		if (d != 1 && d != x) return d;
		if (y == x0) return x;
		if (i == k) { y = x0; k += k; }
	}
}
//對n進行素因子分解
void findfac(long long n)
{
	if (Miller_Rabin(n))//素數
	{
		factor[tol++] = n;
		return;
	}
	long long p = n;
	while (p >= n)p = Pollard_rho(p, rand() % (n - 1) + 1);
	findfac(p);
	findfac(n / p);
}

long long fac[20], a, b, sum;
int num;

void dfs(long long x,long long y, int i) {
	if (num == i) {
		if (x + y < sum) {
			a = x, b = y;
			sum = a + b;
		}
		return;
	}
	dfs(x*fac[i], y, i + 1);
	dfs(x, y*fac[i], i + 1);
}

int main()
{
	long long g, l, n;
	while (~scanf("%I64d%I64d", &g, &l)) {
		if (g == l) {
			printf("%I64d %I64d\n", g, g);
			continue;
		}
		n = l / g;
		tol = 0;
		findfac(n);
		sort(factor, factor + tol);

		num = 0;
		int i = 0, j;
		while (i < tol) {
			fac[num] = factor[i];
			j = i + 1;
			while (j < tol&&factor[j] == factor[i])fac[num] *= factor[i], j++;
			i = j;
			num++;
		}

		a = fac[0];
		b = n / a;
		sum = a + b;
		dfs(1, 1, 0);

		printf("%I64d %I64d\n", g*min(a, b), g*max(a, b));

	}
	return 0;
}