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;i<n;i++)
{
a[i]=cin.nextInt();
if(a[i]>Max)
{
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<n;i++) if(a[i]!=Max)
{
p=lcm(p,new BigInteger(a[i]+""));
q=gcd(q,new BigInteger(Max-a[i]+""));
}
q=q.multiply(new BigInteger("2"));
BigInteger c=gcd(p,q);
q=q.divide(c);
p=p.divide(c);
System.out.println(p+" "+q);
}
static BigInteger gcd(BigInteger a,BigInteger b)
{
if(b.equals(BigInteger.ZERO))
return a;
else
return gcd(b,a.mod(b));
}
static BigInteger lcm(BigInteger a,BigInteger b)
{
return a.divide(gcd(a,b)).multiply(b);
}
}
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 <cstdio>
#include <cstring>
using namespace std;
#include <algorithm>
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;i<n;i++)
{
int b;
scanf("%d",&b);
if(a==b) continue;
int mul=a*b;
int sub=fab(a-b);
int g=gcd(mul,sub);
mul/=g;
sub/=g;
allgcd=gcd(sub,allgcd);
for(int j=0;j<index && mul>1;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<index;i++)
{
for(int j=0;j<c[i];j++)
{
int temp=0;
for(int k=0;k<=len;k++)
{
r[k]=r[k]*num[i]+temp;
temp=r[k]/10000;
r[k]%=10000;
if(temp && k==len)
len++;
}
}
}
printf("%d",r[len--]);
for(;len>=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 <cstdio>
#include <cstring>
#include <algorithm>
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<cmp;
}
bool operator==(const Bignum& cmp)
{
if(negative^cmp.negative)
return false;
return absCmp(cmp)==0;
}
bool operator==(LL num)
{
Bignum cmp(num);
return *this==cmp;
}
void absAdd(const Bignum& one, const Bignum& two)
{
len=max(one.len, two.len);
for(int i=0;i<len;i++)
{
a[i]+=one.a[i]+two.a[i];
if(a[i]>=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;i<len;i++)
{
a[i]+=one.a[i]-two.a[i];
if(a[i]<0) a[i+1]--,a[i]+=K;
}
while(len>0 && a[len-1]==0) len--;
}
void absMul(const Bignum& one, const Bignum& two)
{
len=one.len+two.len;
for(int i=0;i<one.len;i++) for(int j=0;j<two.len;j++)
a[i+j]+=one.a[i]*two.a[j];
for(int i=0;i<len;i++) if(a[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<maxn;i++) if(!h[i])
{
prime[primeNum++] = i;
for(int j=i+i;j<maxn;j+=i) h[j]=true;
}
}
Bignum powBig(Bignum a, int b)
{
Bignum res=1;
while(b)
{
if(b&1)
res=res*a;
a=a*a;
b>>=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;i<n;i++)
scanf("%d", array+i);
sort(array, array+n);
n = unique(array, array+n)-array;
int numerator = array[1] - array[0];
memset(e, 0, sizeof(e));
for(int i=1;i<n;i++)
{
int num=array[i]*array[0];
int sub=array[i]-array[0];
int g=gcd(sub, num);
sub/=g;
num/=g;
numerator=gcd(sub, numerator);
for(int j=0;j<primeNum && prime[j]<=num;j++)
{
int t=0;
while(num%prime[j]==0) t++, num/=prime[j];
e[j]=max(e[j], t);
}
}
if(e[0])
e[0]--;
else
numerator*=2;
for(int i=0;i<primeNum && prime[i]<=numerator;i++)
while(numerator%prime[i]==0 && e[i]>0) e[i]--, numerator/=prime[i];
Bignum ans=1;
for(int i=0;i<primeNum;i++) if(e[i])
ans = ans*powBig(prime[i], e[i]);
ans.output();
printf(" %d\n", numerator);
}
int main()
{
calPrime();
int n;
while(~scanf("%d", &n))
work(n);
}