POJ 3101 Astronomy 解题报告（大数乘法+分数最小公倍数）

    题目大意：给出n的行星的周期，问n个行星在一条直线上的周期。

    解题报告：懒……就直接用Java大数。当然，时间垫底

import java.math.BigInteger;
import java.util.Scanner;

public class Main {
	static int[] a = new int[1010];
	static Scanner cin = new Scanner(System.in);
	
	public static void main(String[] args) {
		
		int Max=0;
		int pos=-1;
		int pos2=-2;
		int n=cin.nextInt();
		for(int i=0;iMax)
			{
				Max=a[i];
				pos2=pos;
				pos=i;
			}
		}
		
		BigInteger p = new BigInteger(Max+"");
		BigInteger q;
		if(pos2!=-1)
			q = new BigInteger(Max-a[pos2]+"");
		else
			q = new BigInteger(Max-a[1]+"");
		
		for(int i=0;i

    当然，常规解法应该是这样。列出公式。假设行星1的周期是t1，行星2的周期是t2，在时间T时两行星在一条直线上，必然有：

    T*(L/t1-L/t2)=0.5*L*m,m是整数。如果要求两行星在一条直线上的最短时间，必然满足m=1。对于其他任意行星都要满足该式。

    当然，以行星1为基准，如果行星3与行星1在一条直线上，那么行星3和行星1也在一条直线上。我们可以求得T，使得

    T*(1/t1-1/ti)=0.5 对于所有行星的行星都满足。2T即为所有的（1/t1-1/ti）的最小公倍数。

    分数的最小公倍数，大家可以自己百度。结论是：分子是所有分母的最小公倍数，分母是所有分子的最大公约数。

    因为有大数据，所以用到大数乘法。经过多次优化，下面的代码在POJ上是32MS，排名第3哈哈。不过事先是把素数打表的……

#include 
#include 
using namespace std;
#include 

int gcd(int a,int b)
{
    return b==0?a:gcd(b,a%b);
}

int fab(int a)
{
    return a<0?-a:a;
}

const int maxn=10001;
int num[]={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,8389,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,8677,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,9319,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9433,9437,9439,9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547,9551,9587,9601,9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,9697,9719,9721,9733,9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,9941,9949,9967,9973};
int c[maxn];
int r[maxn];
int index=1229;

bool work()
{
    int n;
    if(scanf("%d",&n)==-1)
        return false;

    int a;
    scanf("%d",&a);

    int allgcd=0;
    for(int i=1;i1;j++)
        {
            int k=0;
            while(mul%num[j]==0)
            {
                mul/=num[j];
                k++;
            }
            c[j]=max(c[j],k);
        }
    }

    if(c[0])
        c[0]--;
    else
        allgcd*=2;

    r[0]=1;
    int len=0;
    for(int i=0;i=0;len--)
        printf("%04d",r[len]);
    printf(" %d\n",allgcd);
    return true;
}

int main()
{
    while(work());
}

    鉴于以前的代码不太美观，重写了一份，速度虽然不然以前的（100MS左右），但是更具有可读性。代码如下：

#include 
#include 
#include 
using namespace std;

typedef long long LL;
const int K = 10000;    // 数组里每位代表1W
const int M = 500;       // 一共10位
const char show[] = "%04lld";

struct Bignum
{
    LL a[M*2];          // 大数数组
    int len;            // 长度
    bool negative;      // 正负

    Bignum()
    {
        clear();
    }

    void clear()
    {
        len=0;
        negative=false;
        memset(a, 0, sizeof(a));
    }

    Bignum(LL num)
    {
        *this=num;
    }

    Bignum operator=(LL num)
    {
        clear();
        if(num<0) negative=true, num=-num;
        while(num)
            a[len++]=num%K,num/=K;
        return *this;
    }

    Bignum(const Bignum& cmp)
    {
        memcpy(this, &cmp, sizeof(Bignum));
    }

    Bignum operator=(const Bignum& cmp)
    {
        memcpy(this, &cmp, sizeof(Bignum));
        return *this;
    }

    int absCmp(const Bignum& cmp)
    {
        if(len!=cmp.len)
            return len>cmp.len?1:-1;

        for(int i=len-1;i>=0;i--)
            if(a[i]!=cmp.a[i])
                return a[i]>cmp.a[i]?1:-1;

        return 0;
    }

    int absCmp(LL num)
    {
        Bignum cmp(num);
        return absCmp(cmp);
    }

    bool operator<(const Bignum& cmp)
    {
        if(negative^cmp.negative)
            return negative?true:false;

        if(negative)
            return absCmp(cmp)>0;
        else
            return absCmp(cmp)<0;
    }

    bool operator<(LL num)
    {
        Bignum cmp(num);
        return *this=K) a[i]-=K, a[i+1]++;
        }
        if(a[len]) len++;
    }

    void absSub(const Bignum& one, const Bignum& two)
    {
        len=one.len;
        for(int i=0;i0 && a[len-1]==0) len--;
    }

    void absMul(const Bignum& one, const Bignum& two)
    {
        len=one.len+two.len;
        for(int i=0;i=K)
            a[i+1]+=a[i]/K,a[i]%=K;
        while(len>0 && a[len-1]==0) len--;
    }

    Bignum operator+(const Bignum& cmp)
    {
        Bignum c;
        if(negative^cmp.negative)
        {
            bool res = absCmp(cmp)>0;
            c.negative = !(negative^res);
            if(res)
                c.absSub(*this, cmp);
            else
                c.absSub(cmp, *this);
        }
        else if(negative)
        {
            c.negative=true;
            c.absAdd(*this, cmp);
        }
        else
        {
            c.absAdd(*this, cmp);
        }
        return c;
    }

    Bignum operator-(const Bignum& cmp)
    {
        Bignum cpy;
        if(cpy==cmp)
            return *this;
        else
            cpy=cmp, cpy.negative^=true;

        return *this+cpy;
    }

    Bignum operator*(const Bignum& cmp)
    {
        Bignum c;
        if(c==cmp || c==*this)
            return c;

        c.negative = negative^cmp.negative;
        c.absMul(*this, cmp);
        return c;
    }

    void output()
    {
        if(len==0)
        {
            puts("0");
            return;
        }

        if(negative)
            printf("-");

        printf("%lld", a[len-1]);
        for(int i=len-2;i>=0;i--)
            printf(show, a[i]);
//        puts("");
    }
};

const int maxn = 10001;
bool h[maxn];
int prime[maxn];
int primeNum;

int e[maxn];

void calPrime()
{
    for(int i=2;i>=1;
    }
    return res;
}

int gcd(int a, int b)
{
    return b==0?a:gcd(b, a%b);
}

int array[1111];
void work(int n)
{
    for(int i=0;i0) e[i]--, numerator/=prime[i];

    Bignum ans=1;
    for(int i=0;i